The above equations are called the differential and the integral forms of the principle of reciprocity, respectively.
Reciprocity leads to a relationship between two sets of sources of the same frequency and the fields they generate.
S [(Ea Hb ) (Eb Ea )] dS V [Eb Ja Ea Jb Ha Jmb Hb Jma ]dV
If the closed surface encloses all the sources a and b, then no
matter what range of the closed surface S, as long as it encloses
Since the above equation holds, we have
V [Eb Ja Ea Jb H a Jmb Hb Jma ]dV 0
Or it is rewritten as
Va [Eb J a Hb J ma ]dV Vb [Ea Jb H a J mb ]dV
all of the sources, the surface integral is equal to the volume
integr(Vaal ovVeb r)
.
Hence, the surface integral should be a constant.
In order to find this constant, we expand the surface S to the
far-zone field region. Since the far-zone field is TEM wave, with
E ZH er
er
, where Z is the intrinsic iemr pedSance and is the unit
vectorSinubtshteitduitrinecgtitohnisorfepsruoltpaingtaotitohne, equation. , two terms in the
Using ( A B) B A A B , we obtain
[(Ea Hb ) (Eb Ha )] Eb Ja Ea Jb Ha Jmb Hb Jma
[(
S
E
a
Hb
)
(Eb
Ea
)]
dS
V [Eb J a Ea Jb H a J mb Hb J ma ]dV
If we take the above integration over Va or Vb only, we have
Sa [(Ea Hb ) (Eb H a )] dS Va [Eb J a Hb J ma ]dV Sb [(Ea Hb ) (Eb H a )] dS Vb [H a J mb Ea Jb ]dV
8. Principle of Reciprocity
In a linear isotropic medium, there are two sets of sources Ja , Jma and Jb , Jmb with the same frequency in a finite region V .
which is called the Carson reciprocity relation.
The above reciprocity relations hold regardless of whether the space medium is homogeneous or not. We can prove that the Carson reciprocity relation still holds if there is a perfect electric or magnetic conductor in the region V.
If there are no any sources in the closed surface S, we have
S [(Ea Hb ) (Eb H a )] dS 0
If the closed surface S encloses all sources, then the above equation stil if a set of sources and the fields are known, then the relationship between another set of sources and the fields can be found.
S [(Ea Hb ) (Eb Ea )] dS V [Eb Ja Ea Jb Ha Jmb Hb Jma ]dV
integrand of the surface integral will cancel each other. The
surface integral is therefore zero, namely, the equation holds.
Hence, as long as the closed surface S encloses all sources, or all
sources are outside the closed surface S, then the following equation
will hold
S [(Ea Hb ) (Eb H a )] dS 0
which is called the Lorentz reciprocity relation.
S
en
Sa
Ea Ha; EbHb
Sb
Va
V
Vb
J a , J ma
Jb , J mb
These sources and the fields satisfy the following Maxwell’s equations:
H a Ja j Ea Ea Jma j H a
Hb Jb j Eb Eb Jmb j Hb