Download ELEC 390 Theory and Applications of Electromagnetics Spring 2012

ELEC 390
Theory and Applications of Electromagnetics
Spring 2012
Review Topics for Exam #3
The following is a list of topics that could appear in one form or another on the exam. Not all of
these topics will be covered, and it is possible that an exam problem could cover a detail not
specifically listed here. However, this list has been made as comprehensive as possible. You
should also be familiar with the topics on the review sheets for the previous exams.
Maxwell’s equations in differential time-domain form (the “point-wise” equations):
- Gauss’s law:
  D  v
B
E  
- Faraday’s law:
t
- “Magnetic Gauss’s” law:
B  0
D
H  J 
- Ampére’s law:
t
Maxwell’s equations in differential time-harmonic (phasor) form:
~
- Gauss’s law:
  D  ~v
~
~
  E   jB
- Faraday’s law:
~
- “Magnetic Gauss’s” law:
B  0
~ ~
~
  H  J  jD
- Ampére’s law:
Constitutive relations (valid in time-domain form or time-harmonic, i.e., phasor, form):
- D = E, where is the permittivity of the medium
- B = H, where is the permeability of the medium
- J = E, where is the conductivity of the medium
Source-free vs. source-filled regions (i.e., are J and/or v zero or non-zero?)
Antenna analysis (determination of radiation fields from current distributions)
- all time-varying currents act as sources and potentially can radiate EM waves. Examples:
o antennas
o the sun
o lightning
o sparks and other arc flashes
o time-varying currents flowing in circuits (e.g., address bus in a computer)
o charged particles accelerated by earth’s magnetic field
- far fields of infinitesimally short filament of current with uniform current distribution
~
located at origin and aligned along z-axis (magnitude of I o is in peak units):
~
~
jkI o dz  jkR
~ ˆ jkI o dz  jkR
~
dE  θ
e
sin  , dH  φˆ
e
sin 
4R
4R
- can calculate electric (magnetic) field from magnetic (electric) field via (only guaranteed
to work with full near-field expressions):
1
~
~
o E
  H (source-free Ampére’s law)
j
1
~
~
o H
  E (source-free Faraday’s law)
j
1
-
position vectors:
o R (unprimed) defines observation point
o R′ (primed) defines point on antenna
ˆ R , where R̂ is a function of  and 
o spherical coordinates: R  R
o cylindrical coordinates: R  rˆr  zˆz , where r̂ is a function of 
o rectangular coordinates: R  xˆ x  yˆ y  zˆ z
- approximation of current distribution:
o start with current distribution on an open-circuited transmission line stub
o bend transmission line wires near end of stub (fold point is l/2 from end of stub)
so that folded wires are collinear (in line with each other)
o short dipole has a nearly triangular current distribution because the “ends” of the
sinusoidal distribution are nearly linear
o Hertzian dipole (uniform current distribution) can be approximated using
“capacity hats” (charge reservoirs), which obviate the need for the current to go to
zero at the ends of the dipole
Exact electric and magnetic fields radiated by Hertzian dipole of length l (peak units):
~ 2
~
I o k 2 l  jkR  j
j2 
1
j 
~ ˆ I o k l  jkR  2
ˆ
ER
e 

cos   θ
e  

sin 
2
3
2
3
4
4
 kR kR 
 kR kR kR 
~
I o k 2 l  jkR  j
1 
~
H  φˆ
e  
sin 
2
4
kR


kR


Far field criterion (kR >> 1 for Hertzian and short dipoles)
Far fields of Hertzian dipole (uniform current distrib.) of length l (peak units):
~
~
jkI o z  jkR
~ ˆ jkI o l  jkR
~
ˆ
Eθ
e
sin  , H  φ
e
sin 
4R
4R
Far fields of short dipole (triangular current distrib.) of length l (peak units):
~
~
jkI o z  jkR
~ ˆ jkI o l  jkR
~
ˆ
Eθ
e
sin  , H  φ
e
sin 
8R
8R
Common characteristics of all far-field expressions (for dipoles oriented along z-axis):
e  jkR
- the
factor, which implies spreading spherical waves
R
- propagation in R̂ direction (if antenna is centered at origin)
1
- speed of propagation is
(speed of light in the surrounding medium)

-
electric and magnetic fields are proportional to input current
electric field is -directed; magnetic field is-directed
~ ~ ~
~
E  H , E  S av , and H  S av (Sav = time-average Poynting vector)
electric and magnetic fields are in phase if  is purely real
~
E
~ 
H
2
Time-average Poynting vector
1
~ ~
- definition: S av  Re E  H * , if electric and magnetic fields are in peak units and
2
expressed as phasors
- gives the power density per unit area of an EM wave (unit is the W/m2)
- points in the direction of power flow and propagation of phase fronts (in lossless media)
Radiation pattern
- plot of normalized |E| or |H| vs. and/or  or plot of normalized |Sav| vs.and/or 
- normalized power pattern: F(,) = | Sav(,)|/Smax
- usually plotted using a dB (or dBi) scale
- interpretation of radiation pattern plot
- determination of half-power beamwidth
Directivity and gain
- calculation of radiated power (equal to input power if no losses)

Prad
-
-

2 
2 
0 0
0 0
ˆ R 2 sin  d d  S R 2
   S av  ,    R
max
  F  , sin  d d
concept of isotropic radiator
o hypothetical antenna that radiates with equal intensity in all directions
o radiated fields have no specific polarization (not realistic)
ˆ Pin
o Poynting vector of isotropic radiator: S iso  R
4R 2
directivity calculated from power pattern
S
S
4
D  2 
 max  4R 2 max
S iso
Prad
  F  , sin  d d
0 0
gain (G) and efficiency ()
o G = D
o Prad = Pin
Rrad
o 
, if Rrad and Rloss are in series
Rrad  Rloss
o loss resistance usually represents finite conductivity of antenna structure and/or
ground beneath it
- dBi unit (D referenced to isotropic radiator)
- directivities of short dipole, Hertzian dipole, and small loop are 1.5 (1.76 dBi) because
normalized power patterns are all sin2
- calculation of power density at a distance given gain or directivity and input power:
P G P D
S max  in 2  in 2
4R
4R
Radiation resistance
- real part of equivalent input impedance of antenna that represents radiated power
- accounts for power delivered by transmission line and radiated by antenna
2P
- definition Rrad  rad2 , if Iin represents peak (not rms) input current; however, Iin might
I in
not be the peak value of the current distribution along the antenna
-
3
-
 z 
short dipole: Rrad  20 2  
  
2
 z 
Hertzian dipole: Rrad  80 2  
  
Loss resistance due to finite conductivity of antenna
- real part of equivalent input impedance of antenna that represents power lost in antenna
structure
- accounts for power delivered by transmission line and absorbed (not radiated) by antenna
structure
2P
- definition Rloss  loss2 , with Iin in peak (not rms) units
I in
2
-
-
Hertzian dipole: Rloss 
short dipole: Rloss 
/2 dipole: Rloss 
 f c
2 a  c
l
 f c
6 a  c
l
  f c
8 a  c
arbitrary-length dipole: Rloss 
 f c
c
1
2 a I in
2

l 2
l 2
I z  dz
2
where l = length of wire; a = radius of wire; f = operating frequency; c = permeability of
wire (usually o); c = conductivity of wire
Specialized computational methods like the one used in EZNEC are required to find accurate
current distributions along real antennas.
Half-wave dipole
- expression for far field if dipole is center fed and oriented along z-axis
~ ˆ
~ e  jkR  cos0.5 cos  
E  θ j 60I o

R 
sin 
- Io = Iin only for half-wave dipoles (and any odd multiple of /2 if current distribution is
assumed to be perfectly sinusoidal)
- directivity is 1.64 (2.15 dBi)
- radiation resistance is approx. 73 
- input reactance behaves much like that of an open-circuited transmission line stub in the
neighborhood of /4 in length
- current distrib. is not exactly sinusoidal, so actual resonant length is slightly less than /2
Center-fed dipoles of arbitrary length l
- assumed current distribution:
I o sin k 0.5l  z , 0  z  0.5l
I z   
I o sin k 0.5l  z ,  0.5l  z  0
-
approximations of R  R used to find expression for far field:
o magnitude R  R  R
o phase e
 jk R  R
 e  jkR e jkz cos
4
-
-
expression for far field if dipole is center fed and oriented along z-axis:
~
~ e  jkR  cos0.5kl cos    cos0.5kl
E  θˆ j 60I o
 ,
R 
sin 
~
where k = 2/ and I o is the peak current magnitude along wire (it’s not necessarily
equal to the input current)
Iin = I(0) (but maybe not Io) for center-fed dipole
normalized power pattern if l < 1.44, in which case max. is in  = /2 direction:
cos0.5kl cos   cos0.5kl
F   
2
sin 
1  cos0.5kl
Normalization factor is more difficult to determine for longer dipoles.
power pattern is not a function of  for dipoles oriented along z-axis
multiple main lobes (directions where F() = 1) for longer dipoles, which makes them not
very desirable for most applications
Rrad and Xin are strong functions of length, frequency, and conductor thickness
2
1
-
Relevant homework, readings, and other resources:
HW:
Textbook:
Assignments #7 and #8
Chap. 3 (review of relevant vector analysis and vector calculus methods)
Sections 4-1 and 4-2 (light on 4-2)
Sections 9-1 through 9-4
Lecture Notes: “Radiation Power and Directivity of Antennas”
“Radiation Resistance, Efficiency, and Gain of Antennas”
“Loss Resistance Calculations for Arbitrary Current Distributions”
Web Links: (none)
Mathcad:
(none)
Matlab:
(none)
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