electromagnetic theorems & principles inel 6216 dr. sandra x-pol
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Electromagnetic Electromagnetic Theorems & Theorems &
PrinciplesPrinciples
INEL 6216INEL 6216
Dr. Sandra X-PolDr. Sandra X-Pol
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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
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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
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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
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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
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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
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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.
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Electric perfect conductorElectric perfect conductor
electric
image
oo ,
h
h
magnetic
sources
Infinite losser
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MagneticMagnetic perfect conductorperfect conductor
electric
image
oo ,h
h
magnetic
Magnetic conductor
sources
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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
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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 ,
,
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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
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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."
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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
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11 HE
HE 0sJ
EEnM s 1ˆ
11 HE
HE HHnJ s 1ˆ
0sM
Perfect conductor
surface
Perfect magnetic conductor
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Love’s PrincipleLove’s Principle• If inside field are zero
11
ˆ
ˆ
EnM
HnJ
s
s
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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
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Induction TheoremInduction Theorem[[similar to Huygen’s]similar to Huygen’s]
• Used mostly for scattering from obstacles
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Homework problems- Ch.7Homework problems- Ch.7
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