Title: Electromagnetism INEL 4151 Ch 10 Waves
1ElectromagnetismINEL 4151Ch 10 Waves
 Sandra CruzPol, Ph. D.
 ECE UPRM
 Mayagüez, PR
2(No Transcript)
3Electromagnetic Spectrum
4Maxwell 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
5Who was NikolaTesla?
 Find out what inventions he made
 His relation to Thomas Edison
 Why is he not well know?
6Special 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!
7Phasors complex s
 Working with harmonic fields is easier, but
requires knowledge of phasor, lets review  complex numbers and
 phasors
8COMPLEX NUMBERS
 Given a complex number z
 where
9Review
 Addition,
 Subtraction,
 Multiplication,
 Division,
 Square Root,
 Complex Conjugate
10For a time varying phase
 Real and imaginary parts are
11PHASORS
 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)
12To change back to time domain
 The phasor is multiplied by the time factor,
ejwt, and taken the real part.
13Advantages of phasors
 Time derivative is equivalent to multiplying its
phasor by jw  Time integral is equivalent to dividing by the
same term.
14TimeHarmonic 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.
15Maxwell 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
)
16A wave
 Start taking the curl of Faradays law
 Then apply the vectorial identity
 And youre left with
17A 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
18To change back to time domain
 From phasor
 to time domain
19Several Cases of Media
 Free space
 Lossless dielectric
 Lossy dielectric
 Good Conductor
eo8.854 x 1012 F/m mo 4p x 107 H/m
201. 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?
213. Lossy Dielectrics(General Case)
 In general, we had
 From this we obtain
 So , for a known material and frequency, we can
find gajb
22Intrinsic 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
23Note
 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
24Loss Tangent
 If we divide the conduction current by the
displacement current
http//fipsgold.physik.unikl.de/software/java/pol
arisation
25Relation between tanq and ec
262. Lossless dielectric
 Substituting in the general equations
27Review 1. Free Space
 Substituting in the general equations
284. Good Conductors
 Substituting in the general equations
Is water a good conductor???
29Summary
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
30Skin depth, d
 Is defined as the depth at which the electric
amplitude is decreased to 37
31Short 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).
32Waves
 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
33EM 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.
34Exercises 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,
35Power 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/polarisation
36Poynting Vector Derivation
 Start with E dot Ampere
 Apply vectorial identity
 And end up with
37Poynting Vector Derivation
 Substitute Faraday in 1rst term
38Poynting 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.
39Power 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 vector
associated to the electromagnetic wave.
40Time Average Power
 The Poynting vector averaged in time is
 For the general case wave
41Total 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)
42Exercises 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
43TEM 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
44PE 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 x1, 0
x
Answer Ptot 53mW
Hz
Ey
Answer Ptot 0mW!
45Polarization 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
46DualPol 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
 Lineal Typical
 Horizontal
 Vertical
ZHH ZVV ZHV ZVH
Dra. Leyda León
47Polarization
 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
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.
48Polarization
 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
49Several 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
50Linear polarization
Front View
 d 0
 _at_z0 in time domain
Back View
51Circular polarization
 Both components have same amplitude EoxEoy,
 d d yd x 90o Right circular polarized (RCP)
 d 90o LCP
52Elliptical polarization
 X and Y components have different amplitudes
Eox?Eoy, and d 90o  Or d ?90o and EoxEoy,
53Polarization example
54Example
 Determine the polarization state of a plane wave
with electric field  a.

 b.
 c.
 d.
 Elliptic
 90, RHEP
 90, LHCP
 90, RHCP
55Cell phone brain
 Computer model for Cell phone Radiation inside
the Human Brain
56Radar 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
57Microwave 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!
58Decibel 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.
59Attenuation rate, A
 Represents the rate of decrease of the magnitude
of Pave(z) as a function of propagation distance
60Submarine 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)?
61Exercise 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