Design and Analysis of Coupling Matrix for Microwave Filter Applications

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VSRD
-
IJEECE, Vol. 2 (7), 2012
,
1
-
5


____________________________

1
,2
Assistant Professor,

Department of Electronics & Communication Engineering
,

Idela Institute of Technology, Ghaziabad,
Uttar Pradesh, INDIA.
*Correspondence :
vishalmishra2007@gmail.com

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E
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A
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R
R
R
C
C
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H
H
H



A
A
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Design and Analysis of Coupling Matrix

for
Microwave Filter Applications

1
Vishal Mishra
*

and

2
Ajit Kr. Singh

ABSTRACT

This paper mainly analyzes the concepts of the Coupling Matrix and its application for microwave filter.
Methods are presented for the generation of the transfer polynomials, and then the direct synthesis of the
corresponding canonical network coupling mat
rices for Chebyshev (i.e., prescribed
-
equiripple) filtering
functions of the most general kind. A simple recursion technique is described for the generation of the
polynomials for even
-

or odd
-
degree Chebyshev filtering functions with symmetrically or asym
metrically
prescribed transmission zeros and equalization zero pairs. Finally, a novel direct technique, involving
optimization, for reconfiguring the matrix into a practical form suitable for realization with microwave resonator
technology is introduced.

These universal methods will be useful for the design of efficient high performance
microwave filters in a wide variety of technologies for application in space and terrestrial communication
systems.

Keywords :
***

1.

INTRODUCTION

In the present scenario, e
lectromagnetic spectrum is limited. Also many applications are required for various
users across the globe within this stipulated band. Due to this overcrowding/congestions of numerous emerging
wireless applications, challenge to RF/microwave filters with
even more stringent requirement for higher
performance, smaller size, lighter weight, and lower cost continue to grow.

Very high close
-
to
-
band rejections are required to prevent interference to or from closely neighboring channels
at the same time. Narrow
band filters are key components in many systems, especially in the field of
telecommunication. In the base station of an antenna ,a duplexer employs two side by side narrow band filters to
isolate the transmit (Tx) and receive (Rx) signals. Optimal utiliza
tion of the frequency spectrum requires close
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proximity of the Tx and Rx channels. Such high isolation requires sharp cutoff in the frequency band between
Tx and Rx channels.

Section 2 gives an overview of basic filter theory (Butterworth filter response,
Chebyshev filter response and
Elliptical filter response) and network variables. In section 3, presents brief description, formation of coupling
matrix, discuss about transfer and reflection polynomial using in synthesis. Explain recursive technique for
ge
nerate nth order transfer and reflection polynomial. Section

4

is concern with overview of optimization and
Discussion about Genetic Algorithm optimization technique. Section

5

is concern with results and discussion.
Section

6

concludes the Paper.

2.

COUPLING

MATRIX FOR FILTER NETWORKS

Modeling the circuit in matrix form is particularly useful because matrix operations can then be applied, such as
inversion, similarity transformation, and partitioning. Such operations simplify the synthesis, reconfiguration of

the topology, and performance simulation of complex circuits. Moreover the coupling matrix is able to include
some of the real
-
world properties of the elements of the filter. Each element in the matrix can be identified
uniquely with an element in the fin
ished microwave device. This enables us to account for the attributions of
electrical characteristics of each element, such as the Q, values for each resonator cavity, different dispersion
characteristics for the various types of mainline coupling and cros
s
-
coupling within the filter. This is difficult or
impossible to achieve with a polynomial representation of the filter's characteristics.

In the figure.1

we are using N resonator which coupled together with lumped elements. Sharpness or selectivity
is di
rectly depends upon to number of resonator. We can

increase the selectivity with the increase the number of
resonator.

2.1.

Formation of the General N x N Coupling Matrix

Th
e two
-
port network of Figure.
2 in either its BPP or LPP form) operates between a voltage

source generating
e
g

volts and an internal impedance of R
s

ohms and a load impedance of R
L

ohms.


Fig. 1

:
Equivalent Circuit of N
-
Coupled Resonators

Th
e two
-
port network of Fig.2 in either its BPP or LPP form

operates between a voltage source generating

e
g

volts and an internal impedance of R
s

ohms and a load impedance of R
L

ohms. As a series resonator circuit with
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currents circulating in the loops, the overall circuit including the source and load terminations are represented
with the impedance matrix [
z’]
.


Kirchhoff's nodal law (stating that the vector sum of all the currents entering a node is equal to zero) is
applied to the currents circulating in the series resonators of the circuit shown in Figure 3.1a, leading to a series
of equations that may

be represented with the matrix equation.


[


]
=
[


]
[

]

Fig. 2 :
Two
-
Port Network With Source And Load Impedance R
S
and R
L

Where [z
t
] is the impedance matrix of the N
-
loop network plus its terminations.
Equation is expanded as
follows:



[









]
t

=
[






]
.
[


















]
t



… (1)

Where
[.]
t

denotes matrix transpose and I is the unit matrix,



is the source voltage, and



















are the
currents in each of the N loops of the network. It is evident that the impedance matrix [z] is itself

the sum of
three N x N matrices.

2.2.

Main Coupling Matrix jM

This is the N x N matrix containing the values of the mutual couplings between the nodes of the network
(provided by the transformers in Fig). If the coupling is between sequentially numbered nodes
,






, it is
referred to as a mainline coupling. The entries on the main diagonal





(=


, the FIR at each node) are the self
-
couplings, whereas all the other couplings between the non sequentially numbered nodes are known as cross
-
couplings. Because
of the reciprocity of the passive network,





=




, and generally, all the entries are
nonzero. In the RF domain, any variation of the coupling values with the frequency (dispersion
) may be
included at this stage.

Jm

=j






… (
2
)

2.3.

Frequency Variable
Matrix SI

This diagonal matrix contains the frequency variable portion (either the low pass prototype or the band pass
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prototype) of the impedance in each loop, giving rise to an N x N matrix with all entries at zero except for the
di
agonal filled with s = j

as follows :

SI =













… (3
)

2.4.

Termination Impedance Matrix R

This N x N matrix contains the values of the source and load impedances in the



and



positions;

all the
other entries are zero :

R =





… (4
)

The N x N impedance matrixes for the series resonator network are separated out into the matrix's purely
resistive and purely reactive parts.

[z] = R
+ [jM +sI] = R + [z]





… (5
)

A purely reactive network
is
operating between a voltage source with internal impedance



and a load



.

2.5.

Transfer and Reflection Polynomial Synthesis

The reflection coefficient and transfer function of any lossless two port network composed of n interconnected
resonator cavities
is expressed as the ratio of two n
th

degree polynomials.




(

)





(

)


(

)








… (6
)




(

)





(

)


(

)








… (7
)

For a chebyshev transfer function ε is a constant normalizing



to the chosen equiripple level at ω=1rad/s. For
an ideal filter, we cannot have a zero loss over the entire passband .It is possible to have a zero loss at finite
number of frequencies, such frequencies are referred to as reflection zeros. Similarly in st
opband frequencies at
which no power is transmitted, and loss is infinite are known as transmission zeros.

The synthesis procedure start with the specification of a normalized transfer function with transmission zeroes.
The aim is therefore to determin
e


(

)
,





(

)

and




(

)

in terms of transmission zeros. Due to the fact that ,
for a coupled resonator structure, it is not possible to have direct coupling between the input and output ports,
the transfer function may have a minimum of n
-
2 finite trans
mission zeros. The remaining zeros are placed at
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infinity. Additionally, in order for




(

)

and




(

)

to have real coefficient, the prescribed transmission zeros
have to be symmetrical about the imaginary axis of s
-
plane.

By applying the conservation
of energy formula for a lossless network

:










(

)




(

)








… (8
)

Therefore,






(
ω
)




ε




(
ω
)



(



ε


(
ω
)
(



ε


(
ω
)
)




… (9)

With





(
ω
)





(
ω
)


(
ω
)








… (10
)



(
ω
)

Is defined as the filtering function on degree n with the g
eneral chebys
hev characteristic :



(

)

[




(


)




]





… (11)

Where






ω


ω




ω
ω





(

)







[




(

)
]

From chebyshev function :






(

)


[


(









)




]




… (12)

Where













(





)











(

)



[


(


(





)
)



(



(





)
)
]




[

(









)







(









)





]



… (13)

By multiplying the numerator and denominator of the second by


(









)




,



(

)



[

(









)






(









)




]




… (14)

Putting
















(





)












(

)



[

(









)






(









)




]

(






)









… (15)

With





ω


ω



,



ω

(



ω


)




and



(




)



.

2.6.

Recursive Technique

The aim of above manipulation is to implement a recursive technique with which one can determine




(

)

from the specified asymmetrical transmission zeros. Accordingly, eqn can be

rewritten in the following way :

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Num
[


(

)
]



(

)



[


(

)




(

)
]





… (16
)

Where




(
ω
)


(









)




=

[
(
ω


ω


)

ω

(



ω


)



]







(17)

A
nd





(

)


(









)








[
(
ω


ω


)

ω

(



ω


)



]







(18)

Further, in



one can group all the terms in ω and ω

as


(
ω
)

and



(
ω
)
, respectively.



(

)



(

)



(

)








(19
)

W
ith




(
ω
)










ω




ω





+


ω







(20
)

and




(
ω
)

ω

(





ω



ω





ω








ω

)




… (21
)

To construct


(
ω
)

in a systematic way, the addition o
f each product in equation (19
) will be followed by
regrouping of terms in ω and ω
1
.The recursive cycle is begun with the first prescribed transmission zero



(

)

[








]





=
(
ω


ω


)

ω

(



ω


)











(
ω
)



(
ω
)








(22)

Next, the terms corresponding to the second prescribed transmission zero are multiplied with the




and re
-
ordered.





(

)



(

)

[







]






(

)



(

)
[
(






)



(






)



]







(

)



(

)








(23)

W
here



(

)




(
ω
)



(
ω
)
ω


+


(



ω


)





(

)



(

)




(

)



(

)



+


(






)





(

)



(
ω
)

can be constructed by continuing the process for all the of the remaining transmission zeros, including
those at infinity. By then the process for



(
ω
)

, it can be written as :







(

)
=



(

)
+



(

)








(24)

d
ue to sign difference between the de
finitions of


(

)

and




(

)

in equation
.



(

)
=



(

)

and


(

)
=



(

)

substitution of this result into equation (3a) leads to an expression for

:

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(

)
=


[


(

)





(

)
]







(25)

If first transmission zero







than :



(

)










(

)



(



)

If second transmission zero







than :



(

)


















(

)



(








)

This completes the synthesis of the transfer and reflection polynomial in terms of the prescribed transmission
zeros. The next step in the creation o
f the coupling matrix, is the determine of rational polynomials for the short
-
circuit admittance parameter in terms of the transfer polynomials explained in next chapter.

3.

DIRECT SYNTHESIS OF THE COUPLING MATRIX

In this chapter, two methods for the direct
synthesis of the coupling matrix are presented, the first for the N x N
matrix and the second for the N+2 matrix. In both cases, the approach is the same, namely, to formulate the two
-
port short
-
circuit admittance parameters in two ways:

(1)

From the coeff
icients of the polynomials F(s)/


, P(s)/ε, and E(s) that make up the desired transfer and
reflection characteristics



(

)

and



(

)

.

(2)

From the elements of the coupling matrix itself. By equating the two formulations, the coupling values of the
m
atrix are related to the coefficients of the transfer and reflection polynomials.

Synthesis of the Transversal Coupling Matrix

:
To synthesize by N+ 2 transversal coupling matrixes, we
need to construct the two
-
port short
-
circuit admittance parameter mat
rix
[


]

for the overall network in two
ways. First, the matrix is constructed from the coefficients of the rational polynomials of the transfer and
reflection scattering parameters


(s) and



(s), which represent the characteristics of the filter to be
realized,
and the second from the circuit elements of the transversal array network. By equating the
[


]

matrices,
derived by these two methods, the elements of the coupling matrix, associated with the transversal array
network, are related to the coefficie
nts of the


(s) and



(s) polynomials.

4.

OPTIMIZATION

First of all question arises what optimization is, so to make it clear in general way we can say that optimization
is just a way to solve any considered problem in such a way that it give not only a f
easible design but also a
design objective. In other words by optimization we can reduce any particular size while the consider property
will remain safe.

So to consider it we use a lot of algorithm. Now a day’s optimization algorithm is becoming increasi
ngly
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popular in engineering design activity primarily because of the availability and affordability of high speed
computer. They are extensively used in those engineering design problems where the emphasis is on
maximizing or minimizing a certain goal.

4.1.

Gen
etic Algo
rithm

Genetic Algorithm mimics the principles of natural genetics and natural selection to constitute search and
optimization procedures. Genetic algorithms are computerized search and optimization algorithms based on the
mechanics of natural gene
tics and natural selection. Professor Ohn Holland of the University of Ann Arbor
envisaged the concept of these algorithms in the mid sixties and published his seminal work (Holland 1975).

In other words Genetic Algorithm (GA) optimizers are robust, stocha
stic search methods modeled on the
principles and concepts of natural selection and evolution. As an optimizer, the powerful heuristic of the GA is
effective at solving complex, combinatorial and related problems.GA optimizers are particularly effective wh
en
the goal is to find approximate global maxima in a high dimension, multi modal function domain in a near
optimal manner.

4.2.

Basic Elements of GA

Genetic Algorithm (GA) is a robust and stochastic search method based on the principles and concepts of
natural

selection and evolution. It is a direct search optimizer, which makes it effective to find an approximate
global maximum in a multi
-
variable, multi
-
model function domain compared with the conventional optimization
methods (e.g. the gradient based method).

In GA, a set of potential solutions is caused to evolve toward a global
optimal solution. Evolution toward a global optimum occurs as a result of pressure exerted by a fitness
-
weighted
selection process and exploration of the solution space is accomplishe
d by recombination (in GA, it is often
called ‘crossover’) and mutation of existing characteristics present in the current population. The flowchart of
the proposed GA in this chapter is illustrated in Fig. 5.1. To make it clear, some key GA terminologies
are
explained here.

In general a GA optimizer must be able to perform six basic tasks



Encode the solution parameters as genes.



Create a string of genes to form a chromosome.



Initialize a starting point.



Evaluate and assign fitness values to individuals in

the population.



Perform reproduction through the fitness weighted selection of individuals from the population.



Perform recombination and mutation to produce members of the next generation

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5.

RESULTS AND DISCUSSION

The major steps for synthesis the filter of

different order follows the “Flow chart” given below. As the first
example
,

four pole chebyshev filter

with one transmission zero at 1.8 added in flow chart.
Roots of polynomial
F
N
(s) gives

reflection zeros and roots of
P
N
(
s) gives transmission zeros.

As the second example, a fourth
-
order general Chebyshev filter with a pair of finite transmission zeros is
considered.

For this example, the positions of the two transmission zeros are at Ω1,2 = ±1.8 and the in
-
band return loss is
-
21dB.

ε = [10^RL/10
-
1]
-
1/2

Since it is a fourth
-
order filter, it has four transmission zeros in theory. In this case, with two transmission zeros
assigned to the finite frequencies, the other two transmission zeros are at Ω3,4 = ± ∞.





(17)


Fig.

3

:

Four
Pole With Single Pair Transmission Zero

As the next example, a fourth
-
order general Chebyshev filter with at two finite

transmission zeros with different
frequencies is considered. For this example, the positions of the two transmission zeros are at Ω
1,2 =
-
1.5,1.7
and the in
-
band return loss is
-
21dB. Since it is a fourth
-
order filter, it has four transmission zeros in theory. In
this case, with two transmission zeros assigned to the finite frequencies, the other two transmission zeros are at
Ω3,4 = ±

∞.

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Fig.

4

:
Coupling and
Routing Diagram

As the next example, a fourth
-
order general Chebyshev filter with at two finite transmission zeros with different
frequencies at same side is considered.

For this example, the positions of the two transmission
zeros are at Ω1,2 = 1.25,1.7 and the in
-
band return loss is
-
21dB.Since it is a fourth
-
order filter, it has four transmission zeros in theory. In this case, with two transmission
zeros assigned to the finite frequencies, the other two transm
ission zeros ar
e at Ω3,4 = ± ∞.

Cost func
tion of this filter is given as :







(18)

As the next example, a fifth
-
order general Chebyshev filter with at three finite transmission zeros with one pair
and one at different frequency is considered. For this example, the positions of the three transmission zeros are
at Ω1,2 =
-
1.5,
-
1.25,1.25 an
d the in
-
band return loss is
-
21dB. Since it is a fifth order
-
order filter, it has fifth
transmission zeros in theory. In this case, with three transmission zeros assigned to the finite frequencies, the
other two transmission zeros are at Ω4,5 = ± ∞.






(18)


S 1 2 5 6 L



3 4




Fig.

5

:
Coupling and
Routing Diagram

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Fig. 6 :
Six Pole With Two Pair Of Transmission Zeros
8

As the next example, a sixth
-
order general Chebyshev filter with four finite transmission zeros (two pair) at
different frequencies is considered. For this example, the positions of the four transmi
ssion zeros are at Ω1,4 =
-
1.25,1.25,1.5,
-
1.5 and the in
-
band return loss is
-
21dB. Since it is a sixth order
-
order filter, it has six
transmission zeros

in theory.

6.

DISCUSSION

Result of various cross
-
coupled filters frequency characteristics are discussed
here. Numerous filter
configurations are obtained by choosing different number of resonators and different number of transmission
zeros at finite frequencies. Results are presented, starting with filter configurations based on less number of
resonators to
larger number of resonators.

From the analysis of all these results, it has been observed that with the increase in number of resonators, the
filters selectivity or the sharpness of roll
-
off from pass band to stop band is increased.

It is also observed tha
t putting transmission zeros at finite frequencies around the pass band, the filters
selectivity is increased. The degree of selectivity depends upon the number of transmission zeros and their
positions. In this synthesis methodology, maximum number of tra
nsmission zeros, one can add is limited to the
value of N
-
2, where N is the order of the filter.

7.

CONCLUSION

In present scenario frequency spectrum is limited. Many

applications are required for various users across the
globe within this stipulated band. Du
e to this overcrowding/congestions of numerous emerging wireless
applications, challenge to RF/microwave filters with ever more stringent requirement higher performance,
smaller size, lighter weight, and lower cost continue to grow.

Cross
-
coupled resonator

circuits are of importance
for design of RF/microwave filters, in particular the narrow
-
band band pass filters that play a significant role in
many modern applications
. Transmission zeros and reflection zeros has great importance in cross
-
coupled
resonato
r filters design. In this thesis work, generated optimized coupling matrix was used in the cross coupled
resonator synthesis and results shows selectivity of the filter has been increased.

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From the results it can be concluded that the increase in number of

resonators, the filters selectivity or the
sharpness of roll
-
off from pass band to stop band is increased.

8.

FUTURE WORK

Still, it needs a lot of improv
ement in the field of filter design techniques
. By
Developing novel software tool for
“Coupling matrix synthesis” for higher order of filters according user requirement.

9.

REFERENCES

[1]

R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,”
IEEE
Trans. Microwave Theory Te
ch
.
, vol. 47, pp. 433

442, Apr. 1999.

[2]

S. Amari, “Direct synthesis of folded symmetric resonator filters with source

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.
, vol. 11,pp. 264

266, June 2001.

[3]

R. J. Cameron and J. D. Rhodes, “Asymmetric realiza
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-
mode bandpass filters,”
IEEE Trans.
Microwave Theory Tech
.
,vol. MTT
-
29, pp. 51

58, Jan. 1981.

[4]

Richard J. Cameron” Advanced Coupling Matrix Synthesis Techniques for Microwave Filters” IEEE
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[5]

M. E. Van Valkenburg,
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Network Synthesis
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[6]

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-
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-
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-
21, pp. 64
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655, Sept. 1974.

[7]

A. E. Atia and A. E. Williams, “Narrow
-
bandpass waveguide filters,”
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Tech
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-
20, pp. 258

265, Apr.1972.

[8]

R
. J. Cameron, “Fast generation of Chebychev filter prototypes with asymmetrically
-
prescribed
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ESA J.,
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95, 1982.

[9]

H. C. Bell, “Canonical asymmetric coupled
-
resonator filters,”
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Microwave Theory Tech.,

vol.
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-
30
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1340, Sept.1982.

[10]

Smain Amari
,
Member
, IEEE
, Uwe Rosenberg
,
Senior Member, IEEE
, and Jens Bornemann
, Fellow, IEEE

“Adaptive Synthesis and Design of Resonator Filters with Source/Load
-
Multiresonator Coupling” IEEE
TRANSACTIONS ON MICROWAVE THEORY
AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002.

[11]

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-
coupled resonator filters using an analytical gradient
-
based optimization technique,”
IEEE Trans. Microwave Theory

Tech
.
, vol. 48, pp. 1559

1564, Sept. 2000.

[12]

S. Amari and J. Bornemann, “Maxim
um number of finite transmission zeros of coupled resonator filters
with source/load
-
multi
-
resonato
r coupling and a given topology

,

in
Proc
.
Asia

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