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## Electromagnetism INEL 4152

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Title: Electromagnetism INEL 4152

1
ElectromagnetismINEL 4152
• Sandra Cruz-Pol, Ph. D.
• ECE UPRM
• Mayagüez, PR

2
Electricity gt Magnetism
• In 1820 Oersted discovered that a steady current
produces a magnetic field while teaching a
physics class.

3
Would magnetism would produce electricity?
• Eleven years later, and at the same time, Mike
Faraday in London and Joe Henry in New York
discovered that a time-varying magnetic field
would produce an electric current!

4
Electromagnetics was born!
• This is the principle of motors, hydro-electric
generators and transformers operation.

This is what Oersted discovered accidentally
Mention some examples of em waves
5
(No Transcript)
6
Electromagnetic Spectrum
7
Some terms
• E electric field intensity V/m
• D electric field density
• H magnetic field intensity, A/m
• B magnetic field density, Teslas

8
Maxwell Equations in General Form
Differential form Integral Form
Gausss Law for E field.
Gausss Law for H field. Nonexistence of monopole
Faradays Law
Amperes Circuit Law
9
Maxwells Eqs.
• Also the equation of continuity
• Maxwell added the term to Amperes Law so
that it not only works for static conditions but
also for time-varying situations.
• This added term is called the displacement
current density, while J is the conduction
current.

10
Maxwell put them together
• And added Jd, the displacement current

I
L
At low frequencies JgtgtJd, but at radio
frequencies both terms are comparable in
magnitude.
11
Moving loop in static field
• When a conducting loop is moving inside a magnet
(static B field, in Teslas), the force on a
charge is

Encarta
12
Who was NikolaTesla?
• Find out what inventions he made
• His relation to Thomas Edison
• Why is he not well know?

13
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!

14
Phasors complex s
• Working with harmonic fields is easier, but
requires knowledge of phasor, lets review
• complex numbers and
• phasors

15
COMPLEX NUMBERS
• Given a complex number z
• where

16
Review
• Addition,
• Subtraction,
• Multiplication,
• Division,
• Square Root,
• Complex Conjugate

17
For a time varying phase
• Real and imaginary parts are

18
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)

19
To change back to time domain
• The phasor is multiplied by the time factor,
ejwt, and taken the real part.

20
Advantages of phasors
• Time derivative is equivalent to multiplying its
phasor by jw
• Time integral is equivalent to dividing by the
same term.

21
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.

22
Maxwell Equations for Harmonic fields
Differential form
Gausss Law for E field.
Gausss Law for H field. No monopole
Faradays Law
Amperes Circuit Law
(substituting and
)
23
A wave
• Start taking the curl of Faradays law
• Then apply the vectorial identity
• And youre left with

24
A Wave
• Lets look at a special case for simplicity
• without loosing generality
• The electric field has only an x-component
• The field travels in z direction
• Then we have

25
To change back to time domain
• From phasor
• to time domain

26
Ejemplo 9.23
• In free space,
• Find k, Jd and H using phasors and maxwells eqs.

27
Several Cases of Media
1. Free space
2. Lossless dielectric
3. Lossy dielectric
4. Good Conductor

Recall Permittivity eo8.854 x 10-12
F/m Permeability mo 4p x 10-7 H/m
28
1. Free space
• There are no losses, e.g.
• Lets 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?

29
3. General Case (Lossy Dielectrics)
• In general, we had
• From this we obtain
• So , for a known material and frequency, we can
find gajb

30
Intrinsic Impedance, h
• 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.

Not in-phase for a lossy medium
31
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

32
Loss Tangent
• If we divide the conduction current by the
displacement current

http//fipsgold.physik.uni-kl.de/software/java/pol
arisation
33
Relation between tanq and ec
34
2. Lossless dielectric
• Substituting in the general equations

35
Review 1. Free Space
• Substituting in the general equations

36
4. Good Conductors
• Substituting in the general equations

Is water a good conductor???
37
Summary
Any medium Lossless medium (s0) Low-loss medium (e/elt.01) Good conductor (e/egt100) Units
a 0 Np/m
b rad/m
h ohm
uc l w/b 2p/bup/f m/s m
In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m In free space eo 8.85 x 10-12 F/m mo4p x 10-7 H/m
38
Skin depth, d
We know that a wave attenuates in a lossy medium
until it vanishes, but how deep does it go?
• Is defined as the depth at which the electric
amplitude is decreased to 37

39
Short Cut
• You can use Maxwells 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).

40
Waves
• Static charges gt static electric field, E
• Steady current gt static magnetic field, H
• Static magnet gt static magnetic field, H
• Time-varying current gt time varying E(t) H(t)
that are interdependent gt electromagnetic wave
• Time-varying magnet gt time varying E(t) H(t)
that are interdependent gt electromagnetic wave

41
EM waves dont need a medium to propagate
• Sound waves need a medium like air or water to
propagate
• EM wave dont. 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.

42
Exercises Wave Propagation in Lossless materials
• A wave in a nonmagnetic material is given by

• Find
• direction of wave propagation,
• wavelength in the material
• phase velocity
• Relative permittivity of material
• Electric field phasor
• Answer y, up 2x108 m/s, 1.26m, 2.25,

43
Power in a wave
• A wave carries power and transmits it wherever it
goes

The power density per area carried by a wave is
given by the Poynting vector.
See Applet by Daniel Roth at http//fipsgold.physi
k.uni-kl.de/software/java/oszillator/index.html
44
Poynting Vector Derivation
• Start with E dot Amperes
• Apply vector identity
• And end up with

45
Poynting Vector Derivation
• Substitute Faraday in 1rst term

46
Poynting Vector Derivation
• Taking the integral wrt volume
• Applying Theorem of Divergence
• Which 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.

47
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.

Represents the instantaneous power density vector
associated to the electromagnetic wave.
48
Time Average Power
• The Poynting vector averaged in time is
• For the general case wave

49
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)

50
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 er9 is characterized by
Determine the
direction of wave travel and the average power
density carried by the wave
• Answer

51
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

52
PE 10.7
• In free space, H0.2 cos (wt-bx) z A/m. Find the
total power passing through a
• square plate of side 10cm on plane xz1
• square plate at z3

x
Answer Ptot 53mW
Hz
Ey
Answer Ptot 0mW!
53
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)
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 more 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.

54
Polarization of a wave
• IEEE 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

x
y
x
y
http//fipsgold.physik.uni-kl.de/software/java/pol
arisation/
55
Polarization
• In general, plane wave has 2 components in x y
• And y-component might be out of phase wrt to
x-component, d is the phase difference between x
and y.

Front View
56
Several Cases
• Linear polarization ddy-dx 0o or 180on
• Circular polarization dy-dx 90o EoxEoy
• Elliptical polarization dy-dx90o Eox?Eoy, or
d?0o or ?180on even if EoxEoy
• Unpolarized- natural radiation

57
Linear polarization
Front View
• d 0
• _at_z0 in time domain

Back View
58
Circular polarization
• Both components have same amplitude EoxEoy,
• d d y-d x -90o Right circular polarized (RCP)
• d 90o LCP

59
Elliptical polarization
• X and Y components have different amplitudes
Eox?Eoy, and d 90o or d ?90o and EoxEoy
• Or d ?0,180o,
• Or any other phase difference, for example d 56o

60
Polarization example
61
Example
• Determine the polarization state of a plane wave
with electric field
• a.
• b.
• c.
• d.
1. d105, Elliptic
2. d0, linear a 30o
3. 180, LP a 45o
4. -90, RHCP

62
Cell phone brain
• Computer model for Cell phone Radiation inside
the Human Brain
• SAR Specific Absorption Rate W/Kg FCC limit
1.6W/kg, .2mW/cm2 for 30mins
• http//www.ewg.org/cellphoneradiation/Get-a-Safer-
Phone/Samsung/Impression28SGH-a87729/

63
Human absorption
• The FCC limit in the US for public exposure
from cellular telephones at the ear level is a
SAR level of 1.6 watts per kilogram (1.6 W/kg) as
averaged over one gram of tissue.
•  The ICNIRP limit in Europe for public exposure
from cellular telephones at the ear level is a
SAR level of 2.0 watts per kilogram (2.0 W/kg) as
averaged over ten grams of tissue.
• 30-300 MHz is where the human body absorbs RF
energy most efficiently
• http//handheld-safety.com/SAR.aspx
• http//www.fcc.gov/Bureaus/Engineering_Technology/
Documents/bulletins/oet56/oet56e4.pdf

64
Radar bands
Band 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 GHz2.7-3.7gt Weather observations Cellular phones
C 4-8 GHz (4-8 cm) 5.25-5.925 GHz TV stations, short range Weather
X 8-12 GHz (2.54 cm) 8.5-10.68 GHz Cloud, 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 chasers
65
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 meat which at 2.45
GHz has the complex permittivity given.
• The power reaches the inside as soon as the oven
in turned on!

66
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, fields, and electric
currents, the log is multiplied by 20 instead of
10.

67
Attenuation rate, A
• Represents the rate of decrease of the magnitude
of Pave(z) as a function of propagation distance

68
Submarine antenna
• A 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 er81, s4.
• At what depth the amplitude of E has decreased to
1 its initial value at z0 (sea surface)?

69
Exercise Lossy media propagation
• For each of the following determine if the
material is low-loss dielectric, good conductor,
etc.
• Glass with mr1, er5 and s10-12 S/m at 10 GHZ
• Animal tissue with mr1, er12 and s0.3 S/m at
100 MHZ
• Wood with mr1, er3 and s10-4 S/m at 1 kHZ
• Answer
• low-loss, a 8.4x10-11 Np/m, b 468 r/m, l 1.34
cm, up1.34x108, hc168 W
• general, a 9.75, b12, l52 cm, up0.5x108 m/s,
hc39.5j31.7 W
• Good conductor, a 6.3x10-4, b 6.3x10-4, l
10km, up0.1x108, hc6.28(1j) W
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