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

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