Electromagnetic Theorems and Principles
Duality Theorem:
Ex:
A very thin linear magnetic current element of very small
length (l<<λ) is often used to represent the fields of a very
small loop radiator. It can be shown that the fields radiated
by a small linear magnetic current element are identical to
those radiated by a small loop whose area is perpendicular to
the length of the dipole. Assume that the magnetic dipole is
placed at the origin and is symmetric along the z axis with a
constant magnetic current of
̂𝑧 𝐼𝑚
𝑰𝑚 = 𝒂
Find the fields radiated by the dipole.
Ans:
Since the magnetic dipole is the dual of the electric dipole,
Uniqueness Theorem:
Whenever a problem is solved, it is important to know that
the solution is unique.
Under what condition or what information are needed for a
unique solution:
The important cases for this equation is satisfied:
Or
Image Theory:
(Vertical dipole and its image)
In summary,
(Electric and Magnetic Sources and Their Images)
Ex: Vertical Electric Dipole
Refer to the following geometry
Find the far zone fields. Plot the power patterns for h=2λ and
5λ
Ans:
The far zone component of the electric field of the
infinitesimal dipole of length l, constant current I0, and
observation point P1 is given according to the dominant terms
(βr>>1)
The reflected components can be accounted for by the
images
(since the reflection coeff. is unity)
The total field above the interface is sum of the incident and
reflected components.
To simplify the expression for the total field,
For h<<r
And
Or
Then the total field is the product of a field from a single
source and a factor, that is a function of h and θ. This product
is known as a pattern multiplication rule, and the factor is the
array factor.
Power patterns for h=2λ and 5λ:
Reciprocity Theorem:
The positions of the electric and magnetic sources can be
interchanged without affecting the fields.
Reaction Theorem:
The coupling effect in the previous equation is the basis for
the reaction theorem.
Volume Equivalence Theorem:
The sources Ji and Mi generates the fields E0 and H0 in free
space.
The same sources generates fields E and H in the dielectric
with ϵ, µ.
Substracting these equations from the former ones
Define scattered fields (disturbance fields) as
and
Define volume equivalent electric and magnetic sources as
which exists only in the region ϵ≠ϵ0 and µ≠µ0.
Then,
Surface Equivalence Theorem (Huygen’s Principle):
Actual sources are replaced by equivalent sources.
a) Actual, b)Equivalent problem
Equivalent sources
The equivalent problem of b) is
(Love’s equivalence principle)
Equivalence principle models, a) Love’s equivalent, b)
Electric conductor equivalent, c) Magnetic conductor
equivalent.
(Equivalent models for magnetic source radiation)
Induction Theorem:
Where E, H = total fields in the presence of the obstacle.
E1, H1 = total fields in the absence of the obstacle.
Es, Hs = Scattered fields owing to the obstacle.
On the boundary
or
The equivalent currents are
(Induction Theorem for PEC)
(Induction equivalent for Scattering by flat PEC of infinite
extend.)
Physical Equivalent and Physical Optics Equivalent:
Considering the problem
(Physical equivalence for scattering by a PEC, a) Actual
problem, b) Equivalence)
or
and
or
(Physical Equivalent of a flat PEC,
infinite extend)
Induction and Physical Optic Approximations:
(Approximate Induction Equivalence)
(Approximate Physical Equivalence)
Ex: A parallel polarized uniform plane wave on the xy
plane, in a free space medium, is obliquely incident on a
rectangular flat PEC as shown below. The dimensions of
the plate are a and b.
Find the far zone fields scattered by the plate. Solve the
problem using Induction and Physical equivalents.
a) Actual problem, b) Induction Equivalent, c) Physical
Equivalent
where
In summary
For θs=π/2, θs=θi
Physical Equivalent:
where
In summary
For θs=π/2, θs=θi