Title: Electromagnetism INEL 4152
1ElectromagnetismINEL 4152
 Sandra CruzPol, Ph. D.
 ECE UPRM
 Mayagüez, PR
2Electricity gt Magnetism
 In 1820 Oersted discovered that a steady current
produces a magnetic field while teaching a
physics class.
3Would 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 timevarying magnetic field
would produce an electric current!
4Electromagnetics was born!
 This is the principle of motors, hydroelectric
generators and transformers operation.
This is what Oersted discovered accidentally
Mention some examples of em waves
5(No Transcript)
6Electromagnetic Spectrum
7Some terms
 E electric field intensity V/m
 D electric field density
 H magnetic field intensity, A/m
 B magnetic field density, Teslas
8Maxwell 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
9Maxwells 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 timevarying situations.  This added term is called the displacement
current density, while J is the conduction
current.
10Maxwell 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.
11Moving 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
12Who was NikolaTesla?
 Find out what inventions he made
 His relation to Thomas Edison
 Why is he not well know?
13Special 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!
14Phasors complex s
 Working with harmonic fields is easier, but
requires knowledge of phasor, lets review  complex numbers and
 phasors
15COMPLEX NUMBERS
 Given a complex number z
 where
16Review
 Addition,
 Subtraction,
 Multiplication,
 Division,
 Square Root,
 Complex Conjugate
17For a time varying phase
 Real and imaginary parts are
18PHASORS
 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)
19To change back to time domain
 The phasor is multiplied by the time factor,
ejwt, and taken the real part.
20Advantages of phasors
 Time derivative is equivalent to multiplying its
phasor by jw  Time integral is equivalent to dividing by the
same term.
21TimeHarmonic 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.
22Maxwell 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
)
23A wave
 Start taking the curl of Faradays law
 Then apply the vectorial identity
 And youre left with
24A Wave
 Lets look at a special case for simplicity
 without loosing generality
 The electric field has only an xcomponent
 The field travels in z direction
 Then we have
25To change back to time domain
 From phasor
 to time domain
26Ejemplo 9.23
 In free space,
 Find k, Jd and H using phasors and maxwells eqs.
27Several Cases of Media
 Free space
 Lossless dielectric
 Lossy dielectric
 Good Conductor
Recall Permittivity eo8.854 x 1012
F/m Permeability mo 4p x 107 H/m
281. 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?
293. General Case (Lossy Dielectrics)
 In general, we had
 From this we obtain
 So , for a known material and frequency, we can
find gajb
30Intrinsic 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 inphase for a lossy medium
31Note
 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
32Loss Tangent
 If we divide the conduction current by the
displacement current
http//fipsgold.physik.unikl.de/software/java/pol
arisation
33Relation between tanq and ec
342. Lossless dielectric
 Substituting in the general equations
35Review 1. Free Space
 Substituting in the general equations
364. Good Conductors
 Substituting in the general equations
Is water a good conductor???
37Summary
Any medium Lossless medium (s0) Lowloss 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 1012 F/m mo4p x 107 H/m In free space eo 8.85 x 1012 F/m mo4p x 107 H/m In free space eo 8.85 x 1012 F/m mo4p x 107 H/m In free space eo 8.85 x 1012 F/m mo4p x 107 H/m In free space eo 8.85 x 1012 F/m mo4p x 107 H/m In free space eo 8.85 x 1012 F/m mo4p x 107 H/m
38Skin 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
39Short 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).
40Waves
 Static charges gt static electric field, E
 Steady current gt static magnetic field, H
 Static magnet gt static magnetic field, H
 Timevarying current gt time varying E(t) H(t)
that are interdependent gt electromagnetic wave  Timevarying magnet gt time varying E(t) H(t)
that are interdependent gt electromagnetic wave
41EM 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.
42Exercises 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,
43Power 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.unikl.de/software/java/oszillator/index.html
44Poynting Vector Derivation
 Start with E dot Amperes
 Apply vector identity
 And end up with
45Poynting Vector Derivation
 Substitute Faraday in 1rst term
46Poynting 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.
47Power 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.
48Time Average Power
 The Poynting vector averaged in time is
 For the general case wave
49Total 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)
50Exercises 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
51TEM 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
52PE 10.7
 In free space, H0.2 cos (wtbx) 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!
53Polarization
 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
sidetoside, 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.
54Polarization of a wave
 IEEE Definition
 The trace of the tip of the Efield vector as a
function of time seen from behind.  Simple cases
 Vertical, Ex
 Horizontal, Ey
x
y
x
y
http//fipsgold.physik.unikl.de/software/java/pol
arisation/
55Polarization
 In general, plane wave has 2 components in x y
 And ycomponent might be out of phase wrt to
xcomponent, d is the phase difference between x
and y.
Front View
56Several Cases
 Linear polarization ddydx 0o or 180on
 Circular polarization dydx 90o EoxEoy
 Elliptical polarization dydx90o Eox?Eoy, or
d?0o or ?180on even if EoxEoy  Unpolarized natural radiation
57Linear polarization
Front View
 d 0
 _at_z0 in time domain
Back View
58Circular polarization
 Both components have same amplitude EoxEoy,
 d d yd x 90o Right circular polarized (RCP)
 d 90o LCP
59Elliptical 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
60Polarization example
61Example
 Determine the polarization state of a plane wave
with electric field  a.

 b.
 c.
 d.
 d105, Elliptic
 d0, linear a 30o
 180, LP a 45o
 90, RHCP
62Cell 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/GetaSafer
Phone/Samsung/Impression28SGHa87729/
63Human 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.
 30300 MHz is where the human body absorbs RF
energy most efficiently
 http//handheldsafety.com/SAR.aspx
 http//www.fcc.gov/Bureaus/Engineering_Technology/
Documents/bulletins/oet56/oet56e4.pdf
64Radar bands
Band Name Nominal FreqRange Specific Bands Application
HF, VHF, UHF 330 MHz0, 30300 MHz, 3001000MHz 138144 MHz216225, 420450 MHz890942 TV, Radio,
L 12 GHz (1530 cm) 1.2151.4 GHz Clear air, soil moist
S 24 GHz (815 cm) 2.32.5 GHz2.73.7gt Weather observations Cellular phones
C 48 GHz (48 cm) 5.255.925 GHz TV stations, short range Weather
X 812 GHz (2.54 cm) 8.510.68 GHz Cloud, light rain, airplane weather. Police radar.
Ku 1218 GHz 13.414.0 GHz, 15.717.7 Weather studies
K 1827 GHz 24.0524.25 GHz Water vapor content
Ka 2740 GHz 33.436.0 GHz Cloud, rain
V 4075 GHz 5964 GHz Intrabuilding comm.
W 75110 GHz 7681 GH, 92100 GHz Rain, tornadoes
millimeter 110300 GHz Tornado chasers
65Microwave 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!
66Decibel 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.
67Attenuation rate, A
 Represents the rate of decrease of the magnitude
of Pave(z) as a function of propagation distance
68Submarine 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)?
69Exercise Lossy media propagation
 For each of the following determine if the
material is lowloss dielectric, good conductor,
etc.  Glass with mr1, er5 and s1012 S/m at 10 GHZ
 Animal tissue with mr1, er12 and s0.3 S/m at
100 MHZ  Wood with mr1, er3 and s104 S/m at 1 kHZ
 Answer
 lowloss, a 8.4x1011 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 6.3x104, b 6.3x104, l
10km, up0.1x108, hc6.28(1j) W