The theory of the coupling quotient begins with the Kerns “transmission integral”
(To the authors’ best effort, the original publication of Kerns can not be founded in June 2009), which express the coupling quotient in terms of the transmitting and receiving plane-wave characteristics of each antenna:
( ) ( )
02 10
1 1
R i
x y
T R L
b s s e dk dk
a
∞ ∞
⋅
−∞ −∞
= ′ ⋅
− Γ Γ ∫ ∫ k k
k r (A.1) where s (k) and s’ (k) are the vector transmitting and receiving characteristics definedwith respect to plane waves traveling in the k direction but with phase referenced to the phase center of each antenna. Other definitions of symbols are identical to those in Chapter 2.3.1. Equation (A.1) is an exact solution from Maxwell’s equations, and the assumptions are neglecting multiple reflections between two antennas. However, (A.1) cannot be used to compute bR/aT unless the characteristic s10 and s’02 are transformed to commonly measured parameters of the antennas.
We first transform the receiving functions s’02 into its transmitting function s’20 by
Again, all parameters have been defined in Chapter 2.3.1. Substituting s’02 from (A.2) into (A.1) gives Note that the integral interval in (A.3) has been made to K < k by leaving only the radiating part of the spectrum. The radiating characteristics s10 and s’20 for K < k are related to the normalized complex electric far-field patterns by
( ) ( )
Substituting (A.4) into (A.3) produces the coupling quotient for two antennas as a
double integral over the inner product of the complex electric far-field patterns of the
where CR is consolidated notation for the mismatch factor
( 1
RFeed)
The coupling quotient bR/aT expressed in (A.5) and (A.6) are identical to (2.2) and (2.3).A.2 Series Expansion of Spherical Wave Functions
We first consider the scalar Helmholtz equation
(
∇ +2 k2)
ψ = in spherical 0To solve the solution ψ, we use the method of separation of variables and let
( ) ( ) ( )
For proper choosing constants m and n, we can completely separate (A.9) into spherical Bessel function of the first and second kinds, denoted by j krn
( )
and, respectively. Typically these two functions represent standing waves. On the
other hand, to express a plane-wave characteristic, it is convenient to define the spherical Hankel function of the first and second kinds as
n
( )
o expand the near-field coupling quotient, empirically we choose as the basis in our formulation.
The H equation is related to Legendre’s equation, and their solutions are called associated Legendre functions. We express them as
outward-traveling wave. T associated Legendre polynomials of the first and second kind, respectively. A study of
associated Legendre polynomials shows that all functions have singularities at θ = 0 or θ
= π except Pnm
(
cosθ)
with n as an integer. Since ψ should be finite in the range 0 to π on θ, then n must be an integral and we choose Pnm(
cosθ)
as our basis.The Φ equation is a so-called harmonics equation, and the solution of the harmonic
equation are called harmonic functions and denoted by h m
( )
φ . Commonly used harmonic functions in spherical coordinates aresin m φ , cos m φ , e
imφ, e
−imφ (A.12)ons, we can form product solutions
( ) ( )
are the desired elementary functions for the coupling quotient with m and n integrals. To
construc l solutions, we construc binations of the elementary
function as where Bnm are the unknown spherical wave coefficients. Here we sum up all possible
values of m and n, and the remaining work is to evaluate the unknown spherical wave
coefficients Bnm.
A.3 Orthogonality Relationship of Tesseral Harmonics
The orthogonality relationships state that an arbitrary function f
(
θ φ,)
definedover the surface of a sphere can be expanded in a series of tesseral harmonics. Here the
tesseral harmonics of nth degree and mth order are defined as the functions
( )
,(
cos)
coswhose coefficients are determined by
( ) ( )
To derive the orthogonality relationships of tesseral harmonics, we first assume
two solutions to the scalar Helmholtz as
( ) ( )
(
2 2)
same Helmholtz equation. Next, applying (A.18) to a sphere of radius r, we have
2 2 2 1
already known their orthogonality relationships
(A.22)
Finally, the overall orthogonality relationships of tesseral harmonics can be
combined and expressed as
p q
Using (A.15) and (A.16), we can consequently derive the spherical wave
coefficients Bnm as (2.11).
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