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Electromagnetic Electromagnetic Theorems & Theorems &
PrinciplesPrinciples
INEL 6216INEL 6216
Dr. Sandra X-PolDr. Sandra X-Pol
Theoremas que ayudan a Theoremas que ayudan a resolver problemas de EMresolver problemas de EM
• Dualidad• Unicidad• Imágenes• Reciprocidad• Equivalencia de Volumen• Equivalencia de superficie:
– Principio de Huygen• Inducción
DualidadDualidad Electric source (J) Magnetic source (M)
Same mathematical formSustematic interchange of symbols
FFAA
FFAA
HjMEEjJH
EjHHjE
FEAH FA
11
FA
MJ
EH
HE
FA
FA
Uniqueness theoremUniqueness theorem• Solution is unique given certain conditions:
If you find two solutions, E1 & E2, H1 & H2, then over a surface S,
00ˆˆ
0ˆˆ
21
21
HnHHn
EnEEn
Image TheoryImage Theory
• When the ground, corner reflector or other obstacle is close to a radiating element, there will be reflections from it.
• This can be accounted for by image.
• Earth is more lossy at high frequency and moisture
• Assume ground is a perfect electric conductor, flat, and infinite in extent
Vertical (electric) DipoleVertical (electric) Dipole
ri
Real source
reflection
direct
image
oo ,
The direction of the Image is such that the E tangent to perfect conductor is zero.
h
h
Horizontal (electric) DipoleHorizontal (electric) Dipole
ri
Real source
reflection
direct
image
oo ,
The direction of the Image is such that the E tangent to perfect conductor is zero.
Electric perfect conductorElectric perfect conductor
electric
image
oo ,
h
h
magnetic
sources
Infinite losser
MagneticMagnetic perfect conductorperfect conductor
electric
image
oo ,h
h
magnetic
Magnetic conductor
sources
Reciprocity TheoremReciprocity TheoremRelated to transmitting and receiving properties of radiating
systems.
• Fields for sources J1, M1
• Fields for sources J2, M2
• By reciprocity then,
111
111
EjJH
HjME
222
222
EjJH
HjME
'' 12122121 dvMHJEdvMHJEVV
Volume Equivalence Th.Volume Equivalence Th.
Used to find the fields scattered by dielectric obstacles.
If in free-space ( )we have:
But then, the same sources find another medium ( ):
We define the difference as the scattered fields:
ooio
ooio
EjJH
HjME
EjJH
HjME
i
i
os
os
HHH
EEE
oo ,
,
Volume Equivalence (cont.)Volume Equivalence (cont.)
Using this, we can derive:
Where the volume equivalent current densities sources
soeqs
soeqs
EjJH
HjME
EjJ
HjM
oeq
oeq
Huygen's PrincipleHuygen's Principle• predicts the future position
of a wave when its earlier position is known.
“Every point on a wave front can be considered as a source of tiny wavelets that spread out in the forward direction at the speed of the wave itself. The new wave front is the envelope of all the wavelets - that is, tangent to them."
Volume Equivalence Volume Equivalence (Huygen’s Principle)(Huygen’s Principle)
• Used mostly for aperture radiation
• Here actual sources are replaced by equivalent sources (J,M) within a region to simplify solution
• An imaginary closed surface is chosen (usually so that it coincides with conducting structure) but fields outside are the same.
• Chosen current densities source on the surface create the same fields outside.
EEnM
HHnJ
s
s
1
1
ˆ
ˆ
11 HE
11 HE
1J
1M
11 HE
HE HHnJ s 1ˆ
EEnM s 1ˆ
11 HE
HE 0sJ
EEnM s 1ˆ
11 HE
HE HHnJ s 1ˆ
0sM
Perfect conductor
surface
Perfect magnetic conductor
Love’s PrincipleLove’s Principle• If inside field are zero
11
ˆ
ˆ
EnM
HnJ
s
s
Equivalent model for magnetic sourceEquivalent model for magnetic source radiating near perfect conductorradiating near perfect conductor
1ˆ EnM s
1ˆ2 EnM s
1ˆ EnM s
, , , , ,
Remove plateSolution valid only on this side
Induction TheoremInduction Theorem[[similar to Huygen’s]similar to Huygen’s]
• Used mostly for scattering from obstacles
Homework problems- Ch.7Homework problems- Ch.7
•5•11•13
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