Advanced Coupling Matrix Synthesis Techniques
for Microwave Filters
Richard J.Cameron ,Fellow,IEEE
Abstract—A general method is prented for the synthesis of the folded-configuration coupling matrix for Chebyshev or other filtering functions of the most general kind,including the fully canonical ca,
<,
+2”transversal network coupling matrix,which is able to accommodate multiple input/output couplings,as well as the direct source–load coupling needed for the fully canonical cas.Firstly,the direct method for building up the coupling matrix for the transversal network is described.A simple nonoptimization process is then outlined for the conversion of the transversal matrix to the equivalent
“
”coupling matrix,ready for the realization of a microwave filter with resonators arranged as
a folded cross-coupled array.It was mentioned in [1]that,although the polynomial synthesis procedure was capable
of
generating
finite-position
zeros could be realized by
the
coupling matrix.This excluded some uful filtering characteristics,including tho that require multiple input/output couplings,which have been finding applications recently [3].
In this paper,a method is prented for the synthesis of the fully-canonical or
“coupling matrix.
The
.(b)Equivalent circuit of the k th
“low-pass resonator”in the transversal array.
The
matrix has the following advantages,as compared with the conventional coupling matrix.•Multiple input/output couplings may be accommodated,
<,couplings may be made directly from the source and/or to the load to internal resonators,in addition to the main input/output couplings to the first and last resonator in the filter circuit.•Fully canonical filtering functions
(i.e.,coupling matrix,
not requiring the Gram–Schmidt orthonormalization stage.The
0018-9480/03$17.00©2003IEEE
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Fig.2.N +2fully canonical coupling matrix [M ]for the transversal array.The “core”N 2N matrix is indicated within the double lines.The matrix is symmetric about the principal ,M =M
CAMERON:ADV ANCED COUPLING MATRIX SYNTHESIS TECHNIQUES FOR MICROWA VE FILTERS3
The numerator and denominator polynomials for
the
and
for
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for
and
,
may be found with partial fraction ex-
pansions,and the purely real
eigenvalues
common
to
both
(6)
where the real
constant
.In this ca,the degree
of the numerator
of
first to reduce the degree of its numerator
polynomial
by one before its
residues may be found.Note that,
in the fully canonical ca,where the integer
quantity
is even,it is necessary to
multiply to ensure that the
unitary conditions for the scattering matrix are satisfied.
Being independent
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of
,and the
residues
of
individual first-degree low-pass ctions,connected in parallel
between the source and load terminations,but not to each other.
The direct source–load coupling
inverter
thlow-passctionisshowninFig.1(b).
Fully Canonical Filtering Functions
The direct source–load
inverter
,the driving point
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admittance
4IEEE TRANSACTIONS ON MICROWA VE THEORY AND TECHNIQUES,VOL.51,NO.1,JANUARY
everything
2003
Fig.3.Equivalent circuit of transversal array at s=6j1. Solving
for
At infinite
frequency
and
is slightly greater than unity for a fully canonical net-
work,choosing the negative sign will give a relatively small
value
for
(11)
and correctly
gives
.It can be shown that the positive sign will give a cond
solution
transfer
matrix for
the
for the par-
allel-connected transver array is the sum of
the
individual ctions,plus
the
for the direct source–load coupling
inverter
Transversal Matrix
200字小作文Now the two expressions
for
and the
eigenvalues
and
input couplings and occupy the first row and column of
the matrix from positions1
to
(output couplings and they occupy the last
row and column of.All other entries
are
zero.are equivalent to the terminating
impedances,respectively,in[1].
CAMERON:ADV ANCED COUPLING MATRIX SYNTHESIS TECHNIQUES FOR MICROWA VE FILTERS
5
(a)
兵马俑坑
(b)
Fig. 4.Folded canonical network coupling matrix form—fifth-degree example.(a)Folded coupling matrix form.“s ”and “xa ”couplings are zero for symmetric characteristics.(b)Coupling and routing schematic.
Reduction of
the
input and output couplings,the transversal topology
is clearly impractical to realize for most cas and must be trans-formed to a more suitable topology if it is to be of practical u.A more convenient form is the folded or “reflex”configuration [9],which may be realized directly or ud as the starting point for further transformations to other topologies more suitable for the technology it is intended to u for the construction of the filter.
To reduce the transversal matrix to the folded form,the formal procedure,as described in [1],may be applied,working on
the
coupling matrix.This proce-dure involves applying a ries of similarity transforms (“rota-tions”),which eliminate unwanted coupling matrix entries alter-nately right to left along rows and top to bottom down columns,starting with the outermost rows and columns and working in-wards toward the center of the matrix,until the only remaining couplings are tho that can be realized by filter resonators in a folded structure (Fig.4)As with
the
”couplings in the cross-di-agonals—they will automatically become zero if they are not required to realize the particular filter characteristic under con-sideration.Illustrative Example To illustrate
the
and
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and the are shown in Table I.Being fully
canonical,
have
been multiplied
by
and
,normalized to the highest degree coefficient
of
,
finding the associated
residues
is straightforward.How-ever,the degree of the numerator
of
,and the
factor has
to be extracted first to
reduce
in degree by one.This is easily accomplished by first
finding
which may be en is the highest degree coefficient
of in Table II.
Alternatively,
At this
stage,will be one degree less
than
]
,and the associated
eigenvectors
will be positive real for a
realizable network,
and
6IEEE TRANSACTIONS ON MICROWA VE THEORY AND TECHNIQUES,VOL.51,NO.1,JANUARY2003
TABLE I
4–4F ILTERING F UNCTION—C OEFFICIENTS OF E(s),F(s)AND P(s)P OLYNOMIALS
TABLE II
4–4F ILTERING F UNCTION—C OEFFICIENTS OF N UMERATOR AND D ENOMINATOR P OLYNOMIALS OF y
