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## Electromagnetism INEL 4151 Ch 10 Waves

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Title: Electromagnetism INEL 4151 Ch 10 Waves

1
ElectromagnetismINEL 4151Ch 10 Waves
• Sandra Cruz-Pol, Ph. D.
• ECE UPRM
• Mayagüez, PR

2
(No Transcript)
3
Electromagnetic Spectrum
4
Maxwell Equations in General Form
Differential form Integral Form
Gausss Law for E field.
Gausss Law for H field. Nonexistence of monopole
Amperes Circuit Law
5
Who was NikolaTesla?
• Find out what inventions he made
• His relation to Thomas Edison
• Why is he not well know?

6
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!

7
Phasors complex s
• Working with harmonic fields is easier, but
requires knowledge of phasor, lets review
• complex numbers and
• phasors

8
COMPLEX NUMBERS
• Given a complex number z
• where

9
Review
• Subtraction,
• Multiplication,
• Division,
• Square Root,
• Complex Conjugate

10
For a time varying phase
• Real and imaginary parts are

11
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)

12
To change back to time domain
• The phasor is multiplied by the time factor,
ejwt, and taken the real part.

13
• Time derivative is equivalent to multiplying its
phasor by jw
• Time integral is equivalent to dividing by the
same term.

14
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.

15
Maxwell Equations for Harmonic fields
Differential form
Gausss Law for E field.
Gausss Law for H field. No monopole
Amperes Circuit Law
(substituting and
)
16
A wave
• Start taking the curl of Faradays law
• Then apply the vectorial identity
• And youre left with

17
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

18
To change back to time domain
• From phasor
• to time domain

19
Several Cases of Media
1. Free space
2. Lossless dielectric
3. Lossy dielectric
4. Good Conductor

eo8.854 x 10-12 F/m mo 4p x 10-7 H/m
20
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?

21
3. Lossy Dielectrics(General Case)
• From this we obtain
• So , for a known material and frequency, we can
find gajb

22
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
23
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

24
Loss Tangent
• If we divide the conduction current by the
displacement current

http//fipsgold.physik.uni-kl.de/software/java/pol
arisation
25
Relation between tanq and ec
26
2. Lossless dielectric
• Substituting in the general equations

27
Review 1. Free Space
• Substituting in the general equations

28
4. Good Conductors
• Substituting in the general equations

Is water a good conductor???
29
Summary
Any medium Lossless medium (s0) Low-loss medium (e/elt.01) Good conductor (e/egt100) Units
a 0 Np/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
30
Skin depth, d
• Is defined as the depth at which the electric
amplitude is decreased to 37

31
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).

32
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

33
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.

34
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,

35
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/polarisation
36
Poynting Vector Derivation
• Apply vectorial identity
• And end up with

37
Poynting Vector Derivation
• Substitute Faraday in 1rst term

38
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.

39
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 vector
associated to the electromagnetic wave.
40
Time Average Power
• The Poynting vector averaged in time is
• For the general case wave

41
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)

42
Exercises Power
• 1. At microwave frequencies, the power density
considered safe for human exposure is 1 mW/cm2.
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?
• 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

43
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

44
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 x1, 0

x
Hz
Ey
45
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
46
• 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
47
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.

48
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
49
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

50
Linear polarization
Front View
• d 0
• _at_z0 in time domain

Back View
51
Circular polarization
• Both components have same amplitude EoxEoy,
• d d y-d x -90o Right circular polarized (RCP)
• d 90o LCP

52
Elliptical polarization
• X and Y components have different amplitudes
Eox?Eoy, and d 90o
• Or d ?90o and EoxEoy,

53
Polarization example
54
Example
• Determine the polarization state of a plane wave
with electric field
• a.
• b.
• c.
• d.
1. Elliptic
2. -90, RHEP
3. 90, LHCP
4. -90, RHCP

55
Cell phone brain
• Computer model for Cell phone Radiation inside
the Human Brain

56
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
57
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!

58
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

59
Attenuation rate, A
• Represents the rate of decrease of the magnitude
of Pave(z) as a function of propagation distance

60
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)?

61
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