7. What is magnetic current density ?
Where does it come to picture?
8. ❖ magnetic currents and charges do not
exist
❖ have not been found so far in reality
❖ the concepts of such currents and
charges are useful because sometimes we
can introduce equivalent magnetic
currents and charges to simplify the
analysis of some electromagnetic
problems.
9. Given a time - harmonic source defined by electric current density J and magnetic
current density M , the electromagnetic field generated by this source satisfies
Maxwell ’ s equations
electric and magnetic fields are coupled in these equations and the strength of coupling
depends on the frequency
As the frequency decreases, the coupling weakens. When the frequency approaches zero
( ω → 0)
which indicate that the electric and magnetic fields are completely decoupled and therefore
can be solved for independently. Such fields are called static fields(Write the equations in
the absence of magnetic currents).????????????????
(ans:static field)
10. UNIQUENESS OF SOLUTIONS
After going to the trouble of finding a solution to
Maxwell’s equations for a particular problem,one
may wonder if it is the only possible solution.
This is guaranteed, under certain conditions, by the
uniqueness theorem.
To prove the theorem, we assume the existence of two
possible solutions, and derive the conditions
required to insure they are identical.
11. Uniqueness theorem
❑establishes necessary condition for a unique solution to
maxwells equation
❑A foundation to develop the image theory and the surface
equivalence principle which allow us to solve an EM
problem more easily
❑Or consider an EM problem with a different perspective
❑We combine the knowledge of the uniqueness theorem
,image theory ,surface equivalence principle,and duality
principle to deal with the problem of radiation and
scattering by aperture in an infinitely large conducting
plane
❑And discuss Babinet’s principle and the characteristics of
complementary structures
12. • Any uniqueness theorem says that an object
satisfying some set of given conditions or
properties is the only such object that exists; it
is uniquely determined by the specified
conditions
13. UNIQUENESS THEOREM
• Consider a volume V enclosed by a surface
S and occupied by a medium characterized
by permittivity ε , permeability μ , and
conductivity σ
• The volume contains time - harmonic
electric and magnetic sources specified by
electric current density Ji and magnetic
current density Mi .
• To check whether the
electromagnetic field
generated by the given
sources is unique and, if yes,
what conditions that make
the field uniqueElectric and magnetic sources in a volume.
14. we assume first that the sources generate two
different fields denoted by ( Ea , Ha ) and( Eb ,
Hb ), respectively. Both fields should satisfy
Maxwell ’s equations
and
15. • an application of the surface equivalence
principle, we derive the induction theorem
and physical equivalent and formulate integral
equations for scattering by conducting and
dielectric objects.
16. Subtracting the second set of equations fromthe
first set eliminates the source terms since the
sources are assumed to be the same yielding
• The proof of the field uniqueness is then to
show that this difference has to vanish.
These are
called source
free equations
18. To check the difference everywhere, we Integrate over volume V
and then apply Gauss theorem to obtain
It is easy to see that the surface integral would vanish under one of the following three
conditions:
(1) the tangential electric field ( ˆn × E) is specified on the entire surface S such that n ˆ ×δE = 0
on S ;
(2) the tangential magnetic field (n ˆ ×H) is specified on the entire surface S such that nˆ ×δH = 0
on S ;
and (3) the tangential electric field (nˆ × E) is specifi ed over a portion of surface S and the
tangential magnetic field ( ˆn ×H) is specified over the rest of the surface.
For source free equations, the difference fields must
satisfy Poynting’s theorem
19. Therefore, the difference fields must satisfy
Poynting’s theorem
• Therefore, under one of the three conditions
For a general lossy medium, μ = μ ′ − j μ ″ ( μ ″ ≥ 0) and ε = ε ′ − j
ε ″ ( ε ″ ≥ 0). Therefore, the real part of Equation becomes
20. With these two equations, it can be shown
easily that if the medium is lossy and ω >
0, then δE = 0 and δH = 0 regardless of
the type of loss.
This conclusion is valid for any kind of
medium since in the entire procedure
nothing is assumed about the permittivity,
permeability, and conductivity other than that
the medium is lossy and the frequency is
non - zero.
However, if we consider the lossless and static
cases as the limiting cases when the loss
and frequency approach zero, but are not
exactly equal to zero, then the conclusion
remains valid even for the lossless and static
cases.
we can conclude that in a volume the field
produced by a given source is unique
when either the tangential component of
the electric field is specified over the
surface of the volume,
or the tangential component of the magnetic
field is specified over the surface of the
volume,
or the tangential component of the electric
field is specified over one portion of the
surface of the volume and the tangential
component of the magnetic field is
specified over the remaining portion of
the surface.
This statement is the uniqueness theorem for
electromagnetic fields
21. ➢The uniqueness theorem has a number of applications.
➢it guarantees the same solution to a uniquely defined electromagnetic
problem no matter what method is used to find such a solution.
➢The uniqueness theorem establishes a one - to - one correspondence
between the source and the field, which makes it possible to determine a
source from its field.
➢ But the most important application for our purpose here is that the
uniqueness theorem provides a solid foundation for developing the very
useful image theory and the surface equivalence principle.
➢Although the image theory and the surface equivalence principle can be
derived rigorously using sophisticated mathematics, the uniqueness
theorem provides a more intuitive approach to developing them without
resorting to complicated mathematical manipulations.
24. ➢To analyze the performance of an antenna near an infinite plane conductor, virtual
sources (images) will be introduced to account for the reflections.
➢these are not real sources but imaginary ones, which when combined with the real
sources, form an equivalent system.
➢the equivalent system gives the same radiated field on and above the conductor as the actual
system itself.
➢Below the conductor, the equivalent system does not give the correct field. However,
In this region the field is zero and there is no need for the equivalent.
➢To begin let us assume that a vertical electric dipole is placed a distance h above an infinite,
flat, perfect electric conductor
➢The arrow indicates the polarity of the source. Energy from the actual source is radiated in all
directions in a manner determined by its unbounded medium directional
properties.
➢For anobservation point P1, there is a direct wave.
➢ Inaddition , a wave from the actual source radiated toward point R1 of the interface
undergoes a reflection.
25.
26. • The directionis determined by the
law of reflection (θi1= θr1 ) which
assures that the energy in
homogeneous media travels in
straight lines along the shortest
paths. This wave will pass through
the observationpoin t P1.
• By extending its actual path below
the interface, it will seem to originate
from a virtual source positioned a
distance h below the boundary.
• For another observation point P2 the
point of reflection is R2, but the
virtual source is the same as before.
The same is concluded for all other
observation points above the
interface.
27. • The amount of reflection is generally
determined by the respective
constitutive parameters of the media
below and above the interface.
• For a perfect electric conductor below
the interface, the incident wave is
completely reflected and the field below
the boundary is zero.
• According to the boundary conditions,
the tangential components of the
electric field must vanish at all points
along the interface. Thus for an incident
electric field with vertical polarization
shown by the arrows, the polarization of
the reflected waves must be as indicated
in the figure to satisfy the boundary
conditions.
• To excite the polarizationof the reflected
waves, the virtual source must also be
vertical and with a polarity in the same
direction as that of the actual source
(thus a reflection coefficient of +1).
28. • Another orientation of the
source will be to have the
radiating element in a
horizontal position.
• Following a procedure similar
to that of the vertical dipole,
the virtual source (image) is
also placed a distance h below
the interface but with a 180◦
polarity difference relative to
the actual source (thus a
reflection coefficient of −1).
• The single arrow indicates an
electric element and the
double a magnetic one. The
direction of the arrow
identifies the polarity.
31. ❑Actual sources, such as an antenna
and transmitter, are replaced by
equivalent sources.
❑The fictitious sources are said to be
equivalent within a region because they
produce the same fields within that
region.
32. Huygens’ principle
➢Each point on a primary wavefront can be
considered to be a new source of a secondary
spherical wave
➢A secondary wavefront can be constructed as the
envelope of these secondary spherical waves
33. Huygens principle
Each elementary aperture spot dS (with equivalent sources Je a Jm) is a
source for further radiation.
Total radiated power to the space is integral of all the contributions of
individual elementary sources over the whole aperture.
Penetration of a wave through an aperture according to: a) geometrical optic, b) wave optics
34. The equivalence principle is based on the uniqueness
theorem
“a field in a lossy region is uniquely specified by the sources within the region plus
the tangential components of the electric field over the boundary,
Or the tangential components of the magnetic field over the boundary,
or the former over part of the boundary and the latter over the rest of the
boundary
The field in a lossless medium is considered to be the limit, as the losses go to
zero, of the corresponding field in a lossy medium.
Thus if the tangential electric and magnetic fields are completely known over a
closed surface, the fields in the source-free region can be determined
35. ❖By the equivalence principle, the fields outside an imaginary closed surface are
obtained by placing over the closed surface suitable electric- and magnetic-current
densities which satisfy the boundary conditions.
❖The current densities are selected so that the fields inside the closed surface are
zero
❖and outside they are equal to the radiation produced by the actual sources.
❖Thus the technique can be used to obtain the fields radiated outside a closed
surface by sources enclosed within it.
❖The formulation requires integration over the closed surface.
❖The degree of accuracy depends on the knowledge of the tangential components
of the fields over the closed surface.
❖In most applications, the closed surface is selected so that most of it coincides with
the conducting parts of the physical structure
36. • The equivalence principle is
developed by considering an
actual radiating source,which
electrically is represented by
current densities J1 and M1
• The source radiates fields E1
and H1 everywhere.
• it is desired to develop a
method that will yield the
fields outside a closed surface.
37. • To accomplish this, a closed
surface S is chosen, shown
dashed in Figure ,which
encloses the current densities
J1 and M1.
• The volume within S is
denoted by V1 and outside S
by V2.
• The primary task will be to
replace the original problem,
by an equivalent one which yields
the same fields E1 and H1
outside S (within V2).
38. ▪ The formulation of the problem can
be aided eminently if the closed
surface is judiciously chosen so that
fields over most, if not the entire
surface, are known a priori.
▪ The original sources J1 and M1 are
removed, and we assume that there
exist fields E and H inside S and fields
E1 and H1 outside of S. For these
fields to exist within an d outside S,
▪ they must satisfy the boundary
conditions on the tangential electric
and magnetic field components.
▪ Thus on the imaginary surface S there
must exist the equivalent sources
▪ Js = ˆn × [H1 − H]
▪ Ms = −ˆn × [E1 − E]
39. • Slot antennas are used typically at frequencies between 300
MHz and 24 GHz. The slot antenna is popular because they
can be cut out of whatever surface they are to be mounted
on, and have radiation patterns that are roughly
omnidirectional (similar to a linear wire antenna, as we'll see).
The polarization of the slot antenna is linear. The slot size,
shape and what is behind it (the cavity) offer design variables
that can be used to tune performance.
40. Babinet's Principle
• A theorem called Babinet's Principle states that the diffraction pattern for
an aperture is the same as the pattern for an opaque object of the same
shape illuminated in the same manner. That is, except for the intensity of
the central spot, the pattern produced by a diffracting opening of arbitrary
shape is the same as a conjugate of the opening would produce.
• This principle can be very useful for making measurements of very small
objects. For example a circular hole and a droplet of the same size will
produce the same diffraction pattern.
41.
42. SLOT ANTENNA
• This principle relates the radiated fields
and impedance of an aperture or slot
antenna to that of the field of its dual
antenna. The dual of a slot antenna
would be if the conductive material and
air were interchanged - that is, the slot
antenna became a metal slab in space.
• a voltage source is applied across the
short end of the slot antenna. This induces
an E-field distribution within the slot, and
currents that travel around the slot
perimeter, both contributed to radiation.
• Babinet's principle relates these two
antennas. The first result states that the
impedance of the slot antenna (Zs) is
related to the impedance of its dual
antenna (Zc) by the relation:
43. SLOT ANTENNA
• The second major result of
Babinet's/Booker's principle
is that the fields of the dual
antenna are almost the
same as the slot antenna
(the fields components are
interchanged, and called
"duals"). That is, the fields
of the slot antenna are
related to the fields of it's
complement
• Hence, if we know the fields
from one antenna we know
the fields of the other
antenna. Hence, since it is
easy to visualize the fields
from a dipole antenna, the
fields and impedance from
a slot antenna can become
intuitive if Babinet's
principle is understood.
the polarization of the two antennas are reversed. That is, since the dipole
antenna is vertically polarized, the slot antenna will be horizontally polarized.
46. The horn is widely used as a feed element for large
radio astronomy, satellite tracking, and
communication dishes
it is a common element of phased arrays
serves as a universal standard for calibration and gain
measurements of other high gain antennas. I
its widespread applicability stems from its simplicity
in construction, ease of excitation, versatility, large
gain, and preferred overall performance
Horn antennas
47. Horn Antennas :
▪ flared waveguides that produce a nearly uniform
phase front larger than the waveguide itself
▪ constructed in a variety of shapes such as
sectoral E-plane, sectoral H-plane, pyramidal,
conical, etc.
49. CORRUGATED HORN
reduce spillover efficiency and cross-polarization losses and increase aperture
efficiencies of large reflectors used in radio astronomy and satellite communications.
In the 1970s, high-efficiency and rotationally symmetric antennas were needed in
microwave radiometry. Using conventional feeds, aperture efficiencies of 50–60%
were obtained. However, efficiencies of the order of 75–80% can be obtained with
improved feed systems utilizing corrugated horns.
Corrugated conical horn
Corrugated horn
50. E-plane Sectoral Horn
▪ Fields expressions OVER THE horn are similar to the fields
of a TE10 mode for a rectangular waveguide with the
aperture dimensions of a and b1.
▪ difference is in the complex exponential term, parabolic
phase error, due to plane of the aperture not
corresponding to cylindrical wave-front.
)2/(
1
1
2
cos),( ykj
y ex
a
EyxE −
)2/(
1
1
2
sin),(
ykj
z ex
aka
jEyxH −
)2/(1 1
2
cos),(
ykj
x ex
a
E
yxH −
55. 55
Why Reflectors ?
While using aperture antennas we always need to increase the aperture
Area to increase its directivity ,but as this is not practical , instead of using
Large apertures we place a reflecting surface face to face with the aperture
( or any other antenna ) , the reflecting surface collimates radiation to
The small aperture and thus we satisfied high directivity with a small
Aperture , and overcame space limitations.
A side view of
An aperture of
A large area
A side view of
An aperture of
A small area
And a reflecting
Surface used.
57. 57
Types according to geometry : 90 degree corner
To better collimate the energy in the forward direction , the geometrical shape
Of the plane reflector must be changed to prohibit radiation in the back and
Side directions .
The 90 degree – corner reflector has a unique property , the ray incident on
It reflects exactly in the same direction , so it is not used in military applications
To prevent radars from detecting airplanes positions.
61. 61
Methods of feeding parabolic reflectors
Why we use Offset reflectors ( single and dual ) ?
To avoid blockage caused by struts , we use half a dish and adjust the
Feeding element in a way that makes the antenna equivalent to a single
Reflector .
Why we use cassegrain fed reflectors ?
This increases the focal length and thus increases the directivity .
62.
63. Reflecting Antennas
• Corner reflector
– Practical size at 222 MHz and up
– Simple to construct, broadbanded, gains 10-15dBd
• Pyramidal Horn
– Practical at 902 MHz and up
– Sides of horn are fed for up to 15 dBi, 13dBd gain
• Parabolic dish
– Gain is a function of reflector diameter, surface
accuracy and illumination
65. The largest radio telescopes
• Effelsberg (Germany), 100-m paraboloidal
reflector
• The Green Bank Telescope (the National Radio
Astronomy Observatory) – paraboloid of
aperture 100 m
66. The Arecibo Observatory Antenna
System
The world’s
largest single
radio telescope
304.8-m
spherical
reflector
National
Astronomy and
Ionosphere
Center (USA),
Arecibo,
Puerto Rico
