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Synthesis and design of microwave filters and duplexers with single and dual band responses

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SYNTHESIS AND DESIGN OF MICROWAVE FILTERS AND DUPLEXERS
WITH SINGLE AND DUAL BAND RESPONSES
Iman K. Mandal
Thesis Prepared for the Degree of
MASTER OF SCIENCE
UNIVERSITY OF NORTH TEXAS
August 2013
APPROVED:
Hualiang Zhang, Major Professor
Yan Wan, Co-Major Professor
Xinrong Li, Committee Member
Shengli Fu, Chair of the Department of
Electrical Engineering
Dr. Costas Tsatsoulis, Dean of College of
Engineering
Mark Wardell, Dean of the Toulouse Graduate
School
UMI Number: 1526852
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UMI 1526852
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Mandal, Iman K. Synthesis and Design of Microwave Filters and Duplexers with Single
and Dual Band Responses. Master of Science (Electrical Engineering), August 2013, 61 pp., 5
tables, 28 figures, bibliography, 52 titles.
In this thesis the general Chebyshev filter synthesis procedure to generate transfer and
reflection polynomials and coupling matrices were described. Key concepts such as coupled
resonators, non-resonant nodes have been included. This is followed by microwave duplexer
synthesis. Next, a technique to design dual band filter has been described including ways to
achieve desired return loss and rejection levels at specific bands by manipulating the stopbands
and transmission zeros. The concept of dual band filter synthesis has been applied on the
synthesis of microwave duplexer to propose a method to synthesize dual band duplexers.
Finally a numerical procedure using Cauchy method has been described to estimate the filter
and duplexer polynomials from measured responses. The concepts in this thesis can be used to
make microwave filters and duplexers more compact, efficient and cost effective.
Copyright 2013
by
Iman K. Mandal
ii
ACKNOWLEDGEMENTS
I would like to greatly express my gratitude to my advisor, Dr. Hualiang Zhang for
countless academic and professional guidance. He has had enormous patience in guiding me
through a research area which I was completely unacquainted with. I would also like to thank
my co-advisor, Dr. Yan Wan, for her kind help and support through this process.
I want to express my thanks to Dr. Xinrong Li for his support as committee member for
my thesis. I would also like to thank all my professors and friends at my lab which made my
research experience so much fun. I am also highly obliged to Professor G. Macchiarella, Richard
Cameron and Microwave Filter community for providing assistance and advice on numerous
occasions.
Finally, none of what I have done or have been so far would have been possible without
the support of my family, especially my brother Dr. Suman Mandal for being constant source of
inspiration and support.
iii
TABLE OF CONTENTS
ACKNOWLEGEMENTS ..................................................................................................................... iii
TABLE OF CONTENTS....................................................................................................................... iv
LIST OF TABLES ................................................................................................................................ vi
LIST OF FIGURES ............................................................................................................................. vii
CHAPTER 1 SYNTHESIS OF NARROW BAND MICROWAVE FILTERS ................................................ 1
1.1
Introduction........................................................................................................... 1
1.2
Original Contributions ........................................................................................... 2
1.3
Coupled Resonator Filters ..................................................................................... 3
1.4
Filters using Non-Resonant Nodes ........................................................................ 7
1.5
Transfer and Reflection Polynomial Synthesis .................................................... 11
1.6
Evaluation of Coupling Matrix ............................................................................. 12
1.7
Coupling Matrix Reconfiguration ........................................................................ 16
1.8
Example of Synthesis........................................................................................... 17
CHAPTER 2 SYNTHESIS OF MICROWAVE DUPLEXERS ................................................................... 20
2.1
Introduction......................................................................................................... 20
2.2
General Structure ................................................................................................ 20
2.3
Polynomial Synthesis........................................................................................... 21
2.4
Example of Synthesis........................................................................................... 23
CHAPTER 3 SYNTHESIS OF DUAL BAND FILTERS ........................................................................... 27
3.1
Introduction......................................................................................................... 27
3.2
Polynomial Synthesis........................................................................................... 27
iv
Chapter 4 DESIGN OF DUAL BAND DUPLEXERS ............................................................................ 34
4.1
Introduction......................................................................................................... 34
4.2
Synthesis Procedure ............................................................................................ 34
4.3
Implementation in Cavity Type and Non-Resonating Node Type....................... 39
CHAPTER 5 MODEL ORDER REDUCTION OF MICROWAVE DUPLEXERS ....................................... 45
5.1
Introduction......................................................................................................... 45
5.2
Formulation of Cauchy Method .......................................................................... 45
5.3
Application in General Polynomial Function:...................................................... 47
5.4
Application in General Microwave Filter Function ............................................. 49
5.5
Application in Microwave Duplexers .................................................................. 51
5.6
Conclusion ........................................................................................................... 53
CHAPTER 6 CONCLUSION AND FUTURE WORK ............................................................................ 54
BIBLIOGRAPHY .............................................................................................................................. 55
v
LIST OF TABLES
Table 1.1 Filter Polynomials for an 8th Order Chebyshev Filter. ................................................... 17
Table 2.1 Extracted TX Filter Polynomials..................................................................................... 25
Table 2.2 Extracted RX Filter Polynomials .................................................................................... 25
Table 2.3 Duplexer Polynomials.................................................................................................... 26
Table 5.1 Low Pass Polynomial Coefficients ................................................................................. 50
vi
LIST OF FIGURES
Fig. 1.1 (a) Equivalent circuit of n-coupled resonators for loop equation formation (b) Its
network representation. ......................................................................................................... 4
Fig. 1.2 A general coupling arrangement ........................................................................................ 8
Fig. 1.3 Filter with NRN – dotted circles represent frequency invariant susceptance; Solid circles
represent capacitance in parallel to frequency invariant susceptance (normalized
resonator); Black lines represent admittance inverters. ...................................................... 10
Fig. 1.4 Canonical transversal array. (a) N resonators including direct source-load coupling M SL .
(b) Equivalent circuit of the k th low-pass resonator in the array. ......................................... 13
Fig. 1.5 Series type low pass prototype with inter-resonator couplings. ..................................... 13
Fig. 1.6 N + 2 canonical coupling matrix for the transversal array. The matrix is symmetric with
respect to the principal diagonal. ......................................................................................... 16
Fig. 1.7 Polynomial response of the 8th order filter in normalized frequency domain. ............... 18
Fig. 1.8 Polynomial response of the 8th order filter in real frequency domain. ........................... 19
Fig. 1.9 Filter response generated from coupling matrix ............................................................. 19
Fig. 2.1 Duplexer configuration..................................................................................................... 20
Fig. 2.2 Frequency mapping: f 0 =
, B
f1, RX f 2, RX =
f 2,TX − f1, RX ................................................ 21
Fig. 2.3 Synthesized Duplexer polynomial response .................................................................... 24
Fig. 3.1 Effect of narrowing of passbands on return loss levels. .................................................. 30
Fig. 3.2 Narrowing done around upper passband. ....................................................................... 30
Fig. 3.3 Narrowing done around lower passband edges. ............................................................. 31
Fig. 3.4 Effect of additional transmission zero at w = 1.6 ............................................................. 32
vii
Fig. 3.5 A fifth order filter with three real transmission zeros (solid lines). The return loss levels
are equalized with complex transmission zero pair at 0.825 ±1.6j (dashed lines)............... 33
Fig. 4.1 Frequency transformation for dual band duplexers. ....................................................... 35
Fig. 4.2 Proposed dual-band duplexer: (a) topology with non-resonating nodes, (b) topology
with cavity resonators........................................................................................................... 39
Fig. 4.3 Synthesized response of the dual-band duplexer with non-resonating node topology
(S21 for RX channel, S31 for TX channel).............................................................................. 41
Fig. 4.4 Insertion loss at RX channel ( S12 ). .................................................................................... 43
Fig. 4.5 Insertion loss at TX channel ( S13 )...................................................................................... 44
Fig. 4.6 Return loss at Input ( S11 ). ................................................................................................. 44
Fig. 5.1 SNR = 5 dB. ....................................................................................................................... 48
Fig. 5.2 SNR = 20 dB. ..................................................................................................................... 48
Fig. 5.3 SNR = 50 dB. ..................................................................................................................... 49
Fig. 5.4 Example of polynomial estimation for a 6th order filter. ................................................. 51
Fig. 5.5 Exact and estimated polynomial response for duplexers. ............................................... 52
viii
CHAPTER 1
SYNTHESIS OF NARROW BAND MICROWAVE FILTERS
1.1
Introduction
Microwave filters and duplexers are essential components of communication systems.
Frequency spectrum is the most expensive resource among all and to optimize its use filters
and duplexers must be designed to be compact and capable of handling high power at the same
time.
Filters may be classified into categories in several ways. One typical way is to classify
them based on different classes of response functions, defined in terms of the location of the
poles, the insertion-loss function, and the zeros within the passband. The zeros are usually
spaced throughout the passband to give an equiripple or Chebyshev response since this is far
more optimum and superior to the maximally flat or Butterworth response, which is rarely
used. As far as the poles are concerned, the most common type of filter response has all these
poles located at dc or infinity and is often described as an all-pole Chebyshev filter, or simply as
a Chebyshev filter [1]. When one or more poles are introduced into the stopbands at finite
frequencies, the filter is known as a generalized Chebyshev filter or as a pseudo-elliptic filter.
The special case where the maximum number of poles are located at finite frequencies such
that the stopbands have equal rejection level is the well-known elliptic function filter. This is
now rarely used since it has problems in practical realization and is not optimum when specific
stopbands are required—one seldom needs rejection up to infinite frequency. It is almost
always better to place the poles where they are most needed, and also to minimize their
1
number, since each additional finite frequency pole may increase the implementation
complexity and expense.
The above discussion relates equally to the main categories of filters defined in terms of
the general response types of low-pass, bandpass, high-pass, and bandstop.
In general, pseudoelliptic filters (or generalized Chebyshev filters) as the most useful
and powerful filter types are best designed using exact synthesis techniques. Several
techniques were developed to reduce computational complexities [2]. The typical procedure is
to synthesize a low-pass prototype, which is then resonated to form a bandpass filter. The
categories considered are combline, interdigital, parallel-coupled-line bandpass and bandstop,
ring and patch filters [3], and stepped-impedance filters [4]. The several media for
implementation include waveguide, dielectric resonators, coaxial lines, evanescent-mode
filters, and various printed circuit filters using microstrip, stripline, and suspended substrate.
Also frequency tuning is another very important aspect of filter designs [5].
In this chapter the synthesis technique for cross coupled resonator bandpass filters
exhibiting pseudo-elliptic filter response and the concept of non-resonant nodes has been
described, which also lays foundation for the discussion in the next a few chapters. A history of
early filter researches can be found in [6], [7]. A large number of filter concepts discussed in this
thesis can be found in great detail in the classic book on microwave filters by Matthaei, Young
and Jones [8]. Also extensive details on the newer filter synthesis techniques can be found in
[9].
1.2
Original Contributions
There are two significant contributions in this thesis:
2
•
Comprehensive study on synthesis of dual band filters, control of return loss
levels using stopband modifications and additional transmission zeros.
•
Proposing a method for the synthesis of dual band duplexers, which is based on
duplexer synthesis and dual band filter synthesis techniques.
1.3
Coupled Resonator Filters
Extensive literature is available on the theory of coupled resonator filters [10], [11], [12],
[13], [14], [15], [16], [17], [15], [18], [19], [20]. The equivalent circuit of an
n coupled resonator
filter network is shown in Fig. 1.1. The loop equations can be written as:

1 
es
 R1 + jwL1 +
 i1 − jwL12 i2  − jwL1n in =
jwC1 


1 
0
− jwL21i1 +  jwL2 +
 i2  − jwL2 n in =
jwC
2 


(1.1)

1 
0
− jwLn1i1 − jwLn 2 i2  +  Rn + jwLn +
 in =
jwCn 

where Lij = L ji represents the mutual inductance between resonators i and j , and all
the loop currents are supposed to have the same direction as shown in Fig. 1.1.
3
Fig. 1.1 (a) Equivalent circuit of n-coupled resonators for loop equation formation (b) Its
network representation.
In matrix form, the set of equations can be represented as:
1

 R1 + jwL1 + jwC
1


− jwL21






− jwLn1

where the
n×n
− jwL12
jwL2 +
1
jwC2

− jwLn 2


  i1  es 
   

− jwL2 n
 i2  =  0 
    
   


0
 i
1  n  
 Rn + jwLn +
jwCn 

− jwL1n
(1.2)
matrix is the impedance matrix [ Z ] . For simplicity, let us first consider a
synchronously tuned filter. In this case, all resonators resonate at the same frequency, namely
the
midband
frequency
of
filter w0 = 1
LC ,
where
L= L=
L=

= Ln and
1
2
C= C=
C=
= Cn . The impedance matrix may be expressed by
1
2
[ Z ] =w0 L ⋅ FBW ⋅  Z 
4
(1.3)
where FBW = ∆w w is the fractional bandwidth and  Z  is the normalized impedance matrix
which, for synchronously tuned filter is given by:
R1

 w L ⋅ FBW + p
 0

1
w L21
⋅
− j
 Z  =  w0 L FBW



 − j w Ln1 ⋅ 1
 w0 L FBW
−j
1
w L12
⋅
w0 L FBW
p

−j
1
w Ln 2
⋅
w0 L FBW
w L1n 1 
w0 L FBW 

1 
w L2 n
 −j
⋅

w0 L FBW 




Rn
+p 


w0 L ⋅ FBW

−j
(1.4)
with
=
p
j
1  w w0 
−


FBW  w0
w
(1.5)
as the complex lowpass frequency variable. Also,
Ri
1
=
w0 L Qei
(1.6)
Qe1 and Qen are the external quality factors of the input and output resonators,
respectively. The coupling coefficients are defined as:
M ij =
Lij
L
(1.7)
The normalized coupling coefficients are given by:
mij =
M ij
FBW
(1.8)
In case of an asynchronously tuned filter the resonant frequency of each resonator is
different and may be given by w0i = 1
Li Ci , the coupling coefficient of asynchronously tuned
filter is defined as:
5
Lij
M ij =
(1.9)
Li L j
From the above relations the normalized  Z  can be derived as:
 1
 q + p − jm11
 e1
− jm21

 Z  = 



− jmn1


Each
− jm12
p − jm22

− jmn 2




− jm2 n






1

+ p − jmnn 
qen

− jm1n

(1.10)
mii account for asynchronous tuning for each resonator, i.e., frequency shifts from
the center frequency. Then the scattering parameters are obtained as:
=
S 21 2
1
⋅ [ A]n1
−1
qe1 ⋅ qen

2
−1 
S11 =
± 1 −
⋅ [ A]11 
 qe1

(1.11)
with
[ A] =
[ q ] + p [U ] − j [ m]
where [U ] is the
n×n
identity matrix, [ q ] is
n×n
matrix with all elements zero except the
q11 = 1 qe1 and qnn = 1 qen , [ m] is the general n × n coupling matrix. The n + 2 coupling
matrix also includes the external couplings from source and load to each resonator. It is very
easy to transform one form of coupling matrix to another [12], [21], [22]. Till now only inductive
coupling has been considered. Coupling can be capacitive too, and the corresponding coupling
coefficients are called electrical coupling coefficients detailed in [11].
6
1.4
Filters using Non-Resonant Nodes
Using non-resonant nodes with admittance inverters provides a useful approach to filter
design, which is often convenient for microstrip implementation. Several literatures are
available on non-resonant nodes [23], [24], [25], [26], [27], [28]. Overall, four types of
components are used in the low pass prototype in this approach.
1) Resonators: These are represented by unit capacitors in parallel with the
frequency-invariant reactances
jbi which account for the frequency shifts in
their resonant frequencies from the center frequency.
2) Admittance Inverters
J i : These are identical to the coupling coefficients
between the nodes.
3) Non-resonating Nodes: These are internal nodes connected to ground by
frequency-invariant reactance
jBi . These are not in parallel with any capacitor.
4) Input (source) and Output (load): These are normalized conductances,
G=
G=
1.
S
L
A resonator that is responsible for an attenuation pole at a normalized frequency
si = jwi is represented by a unit capacitor in parallel with a constant reactance jbi = − jwi .
Such a dangling resonator is only connected to an NRN. For a filter of order N with
attenuation poles at finite real frequencies, there are
N z number of dangling resonators and N
- N z resonators along the inline path between the input and the output. When
choice of arrangement is not unique, but flexible and for the designer to decide.
7
Nz
N z < N the
The general topology of a filter with non-resonating node is shown in Fig. 1.2. There are
three kinds of possible couplings here: Resonant-Resonant ( J 34 ), Resonant-Non Resonant ( J13 )
and Nonresonant-Nonresonant ( J12 ). A suitable topology is shown in Fig. 1.3.
Fig. 1.2 A general coupling arrangement
The generalized coupling coefficient is defined as:
ki , j =
Ji, j
Bi ⋅ B j
(1.12)
where

 BNR ,i non-resonant susceptance
Bi = 

 Beq ,i resonant susceptance
(1.13)
where BNR ,i is the non-resonant susceptance and Beq,i is the resonant susceptance. Beq,i is
defined as
Beq ,i =
1 ∂Bris ,i
2 ∂w
8
(1.14)
w = w0
Bris ,i ( w ) represents the total susceptance of the i -th resonator. In case of coupling
with the external loads, a generalized external Q is similarly defined:
QEXT ,i =
where
Bi
J
2
0,i
(1.15)
G0
Bi is still given by (1.13) and G0 is the external conductance. The parameters k and QEXT
have the same dimensions. The generalized coupling coefficients are tabulated in [29]. Let M i , j
be the admittance inverter parameters,
bk the frequency-invariant susceptances, and ck be
the capacitances of the filter prototype. Then a resonant node is defined by the parameters
( bk , ck ) and
NRN is associated with ( bk , 0 ) . Assuming
Bn = B f 0
is the filter fractional
bandwidth, the novel generalized parameters are evaluated as follows:
•
Resonant-resonant coupling:
ki , j = B n
•
ci , c j
(1.16)
Resonant-nonresonant coupling:
ki , j = Bn
•
M i, j
M i, j
ci ⋅ b j
(1.17)
Nonresonant-nonresonant Coupling:
ki , j =
M i, j
bi ⋅ b j
QEXT ,i can be evaluated as:
9
(1.18)
QEXT ,i =
QEXT ,i =
Also, the resonant frequencies
ci Bn
, node i resonant
M 0,2 i
bi
M 0,2 i
, node i nonresonant
(1.19)
(1.20)
f k and sign of susceptance bk of NRN are related by:
2


 Bn ⋅ bk 
f 0  Bn ⋅ bk
fk = −
+ 
 + 4

2 
ck
c
 k 


(1.21)
Once these are obtained, the microstrip implementation can be done with procedure
outlined in [30].
Fig. 1.3 Filter with NRN – dotted circles represent frequency invariant susceptance; Solid circles
represent capacitance in parallel to frequency invariant susceptance (normalized resonator);
Black lines represent admittance inverters.
Synthesis procedures involving non-resonating nodes were found in several literatures
such as [31], [32], [33].
10
1.5
Transfer and Reflection Polynomial Synthesis
Synthesis of these polynomials is outlined in [9], [34], [35]. For any two-port lossless
filter network composed of a series of N intercoupled resonators, the transfer and reflection
functions (scattering parameters) (definitions can be found in [36]) may be expressed as a ratio
of two N th degree polynomials:
S11 ( w ) =
FN ( w ) ε R
EN ( w )
(1.22)
PN ( w) ε
EN ( w)
(1.23)
S 21 ( w ) =
where
s by
w
is the real frequency variable related to the more familiar complex frequency variable
s = jw . For a Chebyshev Filtering Function, ε is a constant normalizing
S21 ( w) to the
equiripple level at w = ±1 as follows:
=
ε
PN ( w )
10 RL /10 − 1 FN ( w ) w=1
1
⋅
(1.24)
where RL is the prescribed return loss level in decibels and is assumed that all the
polynomials have been normalized such that their highest degree coefficients are unity.
or ε R =
ε
ε −1
2
if the function is fully canonical.
εR =1
S11 ( w ) and S21 ( w ) share a common
denominator EN ( w ) and the polynomial PN ( w ) contains n fz transfer function finite position
transmission zeros.
2
2
1 , along with
Using the law of energy conservation for a lossless network, S11 + S 21 =
(1.22) and (1.23) we have,
11
=
S 212 ( w )
1
=
2 2
1 + ε CN ( w )
1
(1 + jε C ( w) ) (1 − jε C ( w) )
N
(1.25)
N
where
CN ( w ) =
FN ( w )
PN ( w )
(1.26)
CN ( w ) is known as filtering function of degree N and has a form for the general Chebyshev
characteristic:
N

CN ( w ) = cosh  ∑ cosh −1 ( xn ) 
 n =1

(1.27)
where
xn =
and
w − 1 wn
1 − w wn
jwn = sn is the position of the n fz number of finite position transmission zeros in the
complex
s plane and the remaining N − n fz
transmission zeros at w = ±∞ . For a prescribed set
of TZs that make up the polynomial P ( w ) and a given equiripple return loss level, the reflection
numerator polynomial F ( w ) may be built using efficient recursive technique [1] and then E ( w )
may be found using the Conservation of Energy principle for lossless networks.
1.6
Evaluation of Coupling Matrix
The second step of the synthesis procedure is to calculate the values of coupling
elements of a canonical coupling matrix from the transfer and reflection polynomials. The
coupling matrix is a very special matrix that is extremely common in the literature on
microwave filters. A coupling matrix, all by itself, can characterize a low pass prototype filter
12
network. Also, a coupling matrix can be modified using similarity transform, which is a purely
mathematical technique, to obtain different configurations which are easy to realize with a
practical circuit.
Fig. 1.4 Canonical transversal array. (a) N resonators including direct source-load coupling M SL .
(b) Equivalent circuit of the k th low-pass resonator in the array.
Fig. 1.5 Series type low pass prototype with inter-resonator couplings.
13
A general coupling matrix is called transversal coupling matrix, which is shown in Fig.
1.4. The transversal coupling matrix comprises a series of N individual first-degree low pass
sections, connected in parallel between the source and load terminations but not to each
other. The direct source load coupling inverter M SL is included to allow fully canonical transfer
functions to be realized (according to the minimum path rule, i.e., n fz max , the maximum number
of finite position transmission zeros that may be realized by the network = N − nmin where
nmin
is the number of resonator nodes in the shortest route through the couplings in the network
between the source and load terminations). In a fully canonical network,
nmin = 0 and so
n fz max = N , which is the degree of the network.
Each of the N low-pass sections comprises one parallel-connected capacitor Ck and one
frequency invariant susceptance Bk , connected through admittance inverters of characteristic
admittances M Sk and
M Lk to the source and load terminations, respectively (the values of all
these parameters will be extracted through the synthesis procedure). The circuit of the k th lowpass section is shown in Fig. 1.4(b).
Now the admittance parameter matrix [YN ] is derived in two ways. One is from the
scattering parameters and the other is from the circuit elements of the transversal array
network. By comparing them, elements of the coupling matrix can be derived in terms of the
coefficients of the S11 ( w ) and S 21 ( w ) polynomials.
14
From the derived coefficients the eigenvalues λk and the associated residues
r22k and
r21k for k = 1, 2,..., N can be calculated using partial fraction expansion. Thus the following
expression for [YN ] is obtained:
[YN ] =
 0
j
 K∞
K∞  N
r
1
+∑
⋅  11k

0  k =1 ( s − jλk )  r21k
r12 k 
r22 k 
(1.28)
Now using ABCD matrices, converting the elements of low-pass resonator prototypes to
individual y - parameter matrices, and then adding them together to form the complete [YN ]
matrix the second expression is obtained:
 0
[YN ] = j  M
 SL
M SL  N
 M Sk2
1
+
⋅

∑
0  k =1 ( sCk + jBk )  M Sk M Lk
M Sk M Lk 

2
M Lk

(1.29)
This leads to the following relations:
Ck = 1 , Bk = M kk = −λk , M SL = K ∞ , M Lk2 = r22 k and M Sk M Lk = r21k
Therefore,
=
M Lk
r21=
k , M Sk
r21k
=
, k 1, 2,..., N
r22 k
(1.30)
The capacitors Ck are all unity and the frequency-invariant susceptances Bk ( =
representing the self-couplings
−λk ,
M 11 → M NN ), the input couplings M Sk , the output couplings M Lk
, and the direct source-load couplings M SL are all defined, thus completing the reciprocal N + 2
transversal coupling matrix M representing the network. With this coupling matrix,
M Sk are
the N input couplings and they occupy the first row and column of the matrix from positions 1
15
to N . Similarly,
M Lk are the N output couplings and they occupy the last row and column of
M . All other entries are zero. The resulting coupling matrix is illustrated in Fig. 1.6.
Fig. 1.6 N + 2 canonical coupling matrix for the transversal array. The matrix is symmetric with
respect to the principal diagonal.
1.7
Coupling Matrix Reconfiguration
Once the coupling matrix is obtained, series of similarity transformations can be applied
to it to obtain different filter topologies, without affecting the filter response. This is extremely
useful because this allows filter designers to conveniently change the topology to fit with
practical realization. In [35], [37], several configurations such as folded form, arrow canonical
form, wheel form and cul-de-sac form and ways to convert from one form to another have
been explained.
16
1.8
Example of Synthesis
To demonstrate, an 8th order bandpass filter with cutoff frequencies at 885MHz and
934MHz is synthesized. The synthesized filter polynomials (normalized to the unit frequency)
are listed in Table 1.1 with decreasing order of frequency.
Table 1.1 Filter Polynomials for an 8th Order Chebyshev Filter.
ε
5.5793
FN ( w )
PN ( w )
EN ( w )
1.0000
1.0000
1.0000
-2.5508
-4.6509
-2.5508 + 1.9856i
0.8225
8.0518
-1.1489 - 5.1480i
2.6956
-6.1556
7.8891 + 1.2723i
-2.2980
1.7547
-5.1496 + 6.8045i
-0.1900
-3.1443 - 6.0988i
0.6775
4.1799 + 0.2094i
-0.1495
-1.0236 + 1.2853i
-0.0073
-0.0517 - 0.3103i
The polynomial response is shown in Fig. 1.7 (note: this is the response at the
normalized low-pass frequency range). The response in bandpass real frequency is obtained by
the following transformation:
f =
wB +
( wB )
2
2
17
+ 4 f 02
(1.31)
where f is the real frequency and B is the bandwidth. The polynomial response in real
frequency is shown in Fig. 1.8. Also the filter response generated from synthesized coupling
matrix is shown in Fig. 1.9.
Correspondingly, the N + 2 coupling matrix is obtained as the following:
-0.3546 0.5022 -0.4767 0.3871 -0.2372 0.0858 -0.2952 0.2949
 0
-0.3546 1.2494
0
0
0
0
0
0
0

 0.5022
0
0.8472
0
0
0
0
0
0

0
0
0.0426
0
0
0
0
0
 -0.4767
 0.3871
0
0
0
-0.6102
0
0
0
0

0
0
0
0
-0.9184
0
0
0
 -0.2372
 0.0858
0
0
0
0
0
0
0
-0.9966

0
0
0
0
0
0
-1.0819
0
 -0.2952
 0.2949
0
0
0
0
0
0
0
-1.0830

 0
0
0
0
0
0
0
0
0
10
0 
0.3546 
0.5022 

0.4767 
0.3871

0.2372 
0.0858 

0.2952 
0.2949 

0 
S21
0
S11
Reflection, Rejection Loss
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-3
-2
-1
0
Frequency
1
2
3
Fig. 1.7 Polynomial response of the 8th order filter in normalized frequency domain.
18
10
S21
0
S11
-10
Reflection, Rejection Loss
-20
-30
-40
-50
-60
-70
-80
-90
-100
8
8.5
9
Frequency
9.5
10
8
x 10
Fig. 1.8 Polynomial response of the 8th order filter in real frequency domain.
0
S21
S11
Reflection, Rejection Loss
-20
-40
-60
-80
-100
-120
-3
-2
-1
0
Frequency
1
2
Fig. 1.9 Filter response generated from coupling matrix
19
3
CHAPTER 2
SYNTHESIS OF MICROWAVE DUPLEXERS
2.1
Introduction
In addition to microwave filters, microwave duplexers are indispensable microwave
components for communication systems. They are typically used to connect the RX and TX
filters of a transceiver to a single antenna through a suitable three-port junction. The rapid
development of mobile communication systems over the past decade has stimulated the need
for duplexers with compact size as well as high selectivity. To meet these stringent
requirements, different synthesis techniques have been proposed. Specifically a very
convenient synthesis method was discussed in [38]. Here the general procedure of it is
described.
2.2
General Structure
A common duplexer configuration is shown below. A TX and an RX filter are connected
via a transformer with a turn ratio of n :1 and a susceptance b0 .
Fig. 2.1 Duplexer configuration
20
2.3
Polynomial Synthesis
A duplexer is a lossless three port network and four polynomials are required to define
its scattering parameters in the low pass normalized domain.
=
S11
n0 N ( s )
=
, S 21
D (s)
p0t Pt ( s )
=
, S31
D (s)
In (2.1), the highest degree coefficients of
together with constants
p0 r Pr ( s )
D (s)
(2.1)
N , D, Pt , Pr are imposed to be equal to 1
no , p0t , p0 r respectively. The roots of D ( s ) represent poles of the
network and roots of N ( s ) represent the transmission zeros in the complex plane.
The synthesis of the duplexer is carried out in a normalized frequency domain defined
=
Ω
by the usual low pass to band pass frequency transformation
Fig. 2.2 Frequency mapping: f 0 =
, B
f1, RX f 2, RX =
The frequency mapping and definitions of
( f0 B ) ( f
f 0 − f0 f ) .
f 2,TX − f1, RX .
f 0 and B are shown in Fig. 2.2. The passband
limits of RX filter are f1,RX and f 2,RX while those of the TX filter are represented by f1,TX , f 2,TX
respectively. The two low pass prototype RX and TX filters are characterized through their
characteristic polynomials which are related to their scattering parameters, which are given in
(2.2).
21
S11TX =
PTX ( s )
=
ETX ( s )
TX
=
S 21
p0TX PTXn ( s )
ETX ( s )
F (s)
= RX
ERX ( s )
RX
11
S
S 21RX
=
In (2.2), the polynomials
FTX ( s )
ETX ( s )
PRX ( s )
=
ERX ( s )
(2.2)
p0 RX PRXn ( s )
ERX ( s )
FTX and ETX have degree npTX (order of TX filter) and FRX ,
ERX have degree npRX (order of RX filter). The highest degree coefficients of these
polynomials are equal to 1. The polynomials
given by
PTX and PRX have the highest degree coefficients
p0TX and p0RX which determine the return loss level at passband limits. The TX and RX
transmission zeros then define the normalized polynomials
PTXn and PRXn .
By analyzing the three port network and computing the admittances,
Ny
jb S S + DTX DRX + DRX STX
2
=
yin n=
n 2 0 TX RX
Dy
STX S RX
where
(2.3)
STX , S RX , DTX , DRX are given by
ETX + FTX
2
E + FRX
S RX = RX
2
E − FTX
DTX = TX
2
E − FRX
DRX = RX
2
STX =
Finally we get the following expressions:
22
(2.4)

D D +S S 
=
N ( s )  STX S RX − n 2 TX RX 2 TX RX 
1 − jn b0


2
1 − jn b0
n0 =
1 + jn 2b0
=
D ( s ) STX S RX + n 2
DTX DRX + STX S RX
1 + jn 2b0
(2.5)


n
=
Pt ( s ) P=
p0TX 

TXn S RX , p0 t
2
 1 + jn b0 


n
=
Pr ( s ) P=
p0 RX 

RXn STX , p0 r
2
 1 + jn b0 
Using the transmission zeros of TX and RX filters and the reflection zeros at the input
port of the duplexer, the duplexer characteristic polynomials are generated. Then using an
iterative method by using lossless conditions, the polynomials PTX ,
coefficients
PRX and also their
p0TX and p0RX are obtained. In this iterative method the goal is to satisfy lossless
condition preserving real transmission zeros and reflection zeros. Once these polynomials are
obtained, the TX and RX filters polynomials are extracted using final values of STX ,
S RX , DTX ,
DRX and polynomial fitting.
Once the characteristic polynomials are obtained the individual TX and RX filters and the
junction can be realized using Waveguides, Cavity resonators or Non-Resonant nodes. For the
cavity type design, coupling matrices need to be obtained. While for the non-resonant node
realization, admittance parameters and dangling resonator frequencies are to be determined.
2.4
Example of Synthesis
A duplexer with RX passband from 880MHz to 915MHz and TX passband from 925MHz
to 960MHz is synthesized. RX Filter is of order seven and TX Filter is of order eight. RX finite
23
transmission zeros are taken at 850, 925 and 913 MHz while TX finite transmission zeros have
been selected at 898, 911.4, 915 and 969MHz. The response of the synthesized duplexer
polynomial is shown in Fig. 2.3. Synthesized duplexer polynomials are listed in Table 2.3. Also,
the extracted TX and RX Filter polynomials are given in Table 2.1 and Table 2.2.
20
0
S11
Scattering Parameters in dB
-20
S21
S31
-40
-60
-80
-100
-120
-140
-160
700
750
800
850
900
Frequency
950
1000
Fig. 2.3 Synthesized Duplexer polynomial response
24
1050
1100
Table 2.1 Extracted TX Filter Polynomials
F (s)
p0
2.5729e-004
E (s)
1.0000
1.0000
P (s)
1.0000
0.7476 - 4.4213i
0.1086 - 4.4213i
0 - 0.3826i
-7.8915 - 2.8652i
-8.1651 - 0.4299i
0.8315
-4.3972 + 7.2898i
-0.6960 + 8.1757i
0 + 0.2068i
3.7094 + 3.4598i
4.8206 + 0.5930i
-0.0130
1.4865 - 1.0193i
0.2845 - 1.7009i
-0.1355 - 0.3441i
-0.3481 - 0.0760i
-0.0394 + 0.0059i
-0.0103 + 0.0376i
-0.0001 + 0.0017i
0.0016 + 0.0005i
Table 2.2 Extracted RX Filter Polynomials
7.3187e-004
E (s)
F (s)
p0
P (s)
1.0000
1.0000
1.0000
0.1186 + 3.6918i
0.7922 + 3.6918i
0 + 1.3574i
-5.4723 + 0.3845i
-5.1655 + 2.4742i
0.7503
-0.4868 - 4.1685i
-2.9047 - 3.3865i
0 - 0.0776i
1.7339 - 0.3034i
1.0221 - 1.5925i
0.0960 + 0.3859i
0.4132 + 0.1065i
-0.0416 + 0.0141i
0.0038 + 0.0459i
-0.0007 - 0.0017i
-0.0017 + 0.0007i
25
Table 2.3 Duplexer Polynomials
N (s)
D (s)
Pt ( s )
Pr ( s )
1.0000
1.0000
1.0000
1.0000
-1.5000 + 0.7267i
3.2670 + 0.7267i
0.4554 - 3.3091i
0.4281 + 3.0639i
2.6305 - 1.0900i
6.8423 + 2.3874i
-3.0748 - 1.2551i
-1.2767 + 1.0664i
-3.9458 + 1.8946i
11.1661 + 4.9989i
-0.7701 - 1.5344i
0.0110 + 6.5595i
2.5074 - 2.8418i
13.1200 + 8.1490i
-5.2668 - 0.9836i
-12.5978 + 2.6996i
-3.7611 + 1.7976i
13.2342 + 9.5834i
-1.8198 + 4.5704i
-3.9035 -10.8536i
1.0509 - 2.6964i
10.0267 + 9.6647i
2.0718 + 1.2250i
5.4041 - 2.7098i
-1.5763 + 0.7615i
6.4497 + 7.3658i
0.4402 - 0.5458i
1.0818 + 1.6577i
0.1869 - 1.1422i
3.0628 + 4.7883i
-0.0848 - 0.0905i
-0.3157 + 0.2589i
-0.2804 + 0.1442i
1.1545 + 2.3448i
-0.0105 + 0.0075i
-0.0365 - 0.0361i
0.0113 - 0.2163i
0.2804 + 0.9428i
0.0003 + 0.0006i
0.0023 - 0.0028i
-0.0169 + 0.0112i
0.0391 + 0.2720i
0.0000 - 0.0000i
0.0001 + 0.0001i
0.0001 - 0.0169i
-0.0025 + 0.0611i
-0.0001 + 0.0003i
-0.0019 + 0.0090i
-0.0000 - 0.0005i
-0.0004 + 0.0010i
0.0000 + 0.0000i
-0.0000 + 0.0001i
-0.0000 - 0.0000i
-0.0000 + 0.0000i
Once the extracted filter polynomials are obtained, coupling matrices can be evaluated
and the duplexer can be implemented with cavity resonators or waveguides [38], [39], [40],
[41].
26
CHAPTER 3
SYNTHESIS OF DUAL BAND FILTERS
3.1
Introduction
In modern telecommunication systems, the need for devices that can work at multiple
frequency bands simultaneously is becoming increasingly important in order to reduce size and
power requirement. For example, the employment of a dual band filter eliminates the
requirement of using two filters working at different bands by taking care of both bands, which
leads to significant cost reductions. Also, being one unit they are generally smaller than two
filters combined together. Several literatures are available on dual band filter synthesis and
design such as [42], [43], [44], [45], [46], [47]. In this chapter, the procedure to synthesize a
dual band filter will be discussed.
3.2
Polynomial Synthesis
The characteristic polynomials for a dual band filter are defined similar to that of a
single passband filter. However, there exists no direct iterative method to determine the
polynomials given the transmission zeros, passbands and return loss levels in either bands.
Therefore, an alternative method is initially described in [48] and is further studied in this
chapter. The procedure is summarized below:
•
•
First an initial set of values of poles and zeros is assigned in the passbands and
stopbands.
Next initial filter function is constructed:
N
A ( w)
C=
( w) =
B ( w)
∏(w − p )
i =1
M
i
∏(w − z )
i =1
27
i
(3.1)
where p and
•
z
denote the initial poles and zeros while N , M denote their numbers.
The roots of the following expression are obtained:
dC ( w )
= B ( w ) A′ ( w ) − A ( w ) B′ ( w )
dw
•
(3.2)
The complex roots are discarded and the real roots are arranged including the passband
and stopband edges. Let the roots in the passbands be
α
and the roots in the stopband
be β . Each zero (pole) should lie between two roots or stopband (passband) edge.
•
The zeros and poles are now updated using the following equations:
pl′ =
pl α l −1C0 (α l −1 ) + α l C0 (α l )  − α l −1α l C0 (α l −1 ) + C0 (α l ) 
pl C0 (α l −1 ) + C0 (α l )  − α l C0 (α l −1 ) + α l −1C0 (α l ) 
zl  β l −1C0 ( β l ) + β l C0 ( β l −1 )  − β l −1β l C0 ( β l −1 ) + C0 ( β l ) 
zl′ = 
zl C0 ( β l −1 ) + C0 ( β l )  −  β l −1C0 ( β l −1 ) + β l C0 ( β l ) 
(3.3)
•
This procedure is continued until the values of poles and zeros converge.
•
Stopband edges are moved in order to bring equal return loss in lower and higher
passbands (this will be explained later).
•
Also complex transmission-zeros can be introduced to control return loss levels.
•
Once the poles and zeros are found,
F=
(s)
N
∏ ( s − jp )
i
i =1
P=
(s)
M
∏ ( s − jz )
k
k =1
•
(3.4)
E ( s ) is obtained by taking the roots of ε R P ( s ) + ε F ( s ) and mapping right half plane
roots to the left half plane.
28
Once the filter coefficients are obtained, the coupling matrix synthesis is similar to that
of a single band filter. It is noted that the return loss level in the passband depends heavily on
the width of the passbands. Any dual band filter with unequal passbands tends to have a
different return loss level in the two bands. Through our study, this challenge is overcome by
placing suitable transmission zeros in the stopbands or by using complex transmission zeros in
the passband if necessary. These techniques are illustrated below.
Narrowing of Passbands
Effect of narrowing transmission zeros towards passbands on return loss levels is
illustrated in Fig. 3.1. It is to be noted here that in only one of the band edge, value of
ε
[34] is
computed. Any asymmetry in the filter causes the other band to have a return loss other than
the specified one. The band in which
ε
is computed is not affected since
ε
automatically
compensates it. Each of the stopband edge has its own effect (not equal, generally) on how
much it changes the return loss level, and the positions of stopband edges (and transmission
zeros) are not unique in order to get equal return loss responses. Therefore this process is
analogous to a tuning process to get the desired return loss level as well as the steepness of
passband roll-off. Fig. 3.2 and Fig. 3.3 further explain this effect. The positions of stopband
edges (and transmission zeros) are not unique in order to get equal return loss levels.
29
Fig. 3.1 Effect of narrowing of passbands on return loss levels.
Fig. 3.2 Narrowing done around upper passband.
30
Fig. 3.3 Narrowing done around lower passband edges.
Effect of narrowing towards the passband on the higher frequency band is shown
above. In Fig. 3.2 the stopband edges have been narrowed towards the upper passband, which
pushes the return loss levels up in the upper passband. On the other hand, in Fig. 3.3 the
stopband edges have been pushed towards lower passband, which should push up the return
loss levels in the lower passband. However it is to be noted that during the synthesis procedure
ε
has been evaluated in the lower passband and hence return loss level is fixed in lower
passband. Therefore, the change is again observed in the upper passband return loss level,
which is not restricted by the value of
ε
and is pushed down.
Additional Transmission Zeros
Addition of a finite transmission zero to a dual band filter with equal return loss levels in
the two bands will make it unequal. In Fig. 3.4 an addition transmission zero at w = 1.6 has
been imposed. The two filtering bands have the same order. As we can see, the higher filter
31
band having transmission zero at 1.6 has higher level of return loss in the passband. It can be
concluded the ability to place transmission zeros at finite frequencies gives designers great
flexibility to design filters that have certain rejection level at a pre-specified frequency band.
Fig. 3.4 Effect of additional transmission zero at w = 1.6
Complex Transmission Zeros
Finally we find that a transmission zero at any point brings down the value of the
polynomial P as given in (3.4) at that point. Therefore a complex conjugate transmission zero
pair can be introduced whose real part lies in the passband to adjust the return loss level of
that passband. A smaller imaginary part has more effect while a large imaginary part of the
complex transmission zero has little effect on it. This is illustrated in Fig. 3.5.
32
Fig. 3.5 A fifth order filter with three real transmission zeros (solid lines). The return loss levels
are equalized with complex transmission zero pair at 0.825 ±1.6j (dashed lines).
33
CHAPTER 4
DESIGN OF DUAL BAND DUPLEXERS
4.1
Introduction
In this chapter we propose a method to design a Microwave Duplexer that can work in
two different frequency bands simultaneously [44]. The discussion will involve design of dual
band filters followed by the combination of the dual band filters to form a Dual Band Duplexer.
4.2
Synthesis Procedure
The proposed dual-band microwave duplexer is composed of two dual-band filters with
the two input ports connected through a three-port junction. The two dual-band filters (RX and
TX) can be characterized separately from the duplexer through several techniques [48] [43]
[42]. In general, the dual-band filter employed in the analysis can be seen as a single-band filter
with some of its transmission zeros falling in the passband, separating the passband into two
bands. In our analysis the passband limits of the RX filter are represented by
f1RX , f 2RX , f3RX ,
f 4RX while those of the TX filter are f1TX , f 2TX , f3TX , f 4TX .
The duplexer is synthesized in a normalized frequency domain with the suitable lowpass
=
Ω
↔ bandpass frequency transformation
( f0 B )( f
f 0 − f 0 f ) as shown in Fig. 4.1, where
f1RX f 4TX
(4.1)
f 0 and B are defined as follows:
f0 =
=
B
f 4TX − f1RX
34
Fig. 4.1 Frequency transformation for dual band duplexers.
Once the lowpass band limits are obtained, the procedure described in [3] was used to
obtain the filter polynomials for both the TX and RX filters. The two lowpass prototype filters in
the normalized frequency domain can be characterized separately from the duplexer through
their characteristic polynomials. The characteristic polynomials of the TX and RX filters are
related to their scattering parameters as follows:
=
S11TX
FTX ( s ) TX PTX ( s ) p0TX PTXn ( s )
=
, S 21
=
ETX ( s )
ETX ( s )
ETX ( s )
FRX ( s ) RX PRX ( s ) p0 RX PRXn ( s )
S
=
, S 21
=
=
ERX ( s )
ERX ( s )
ERX ( s )
(4.2)
RX
11
where ETX ( s ) , ERX ( s ) are polynomials of degree
npTX , npRX respectively. All of these
polynomials have unity coefficient for the highest degree.
The major difficulty in evaluating the polynomials of dual-band duplexer is to make the
responses of the dual-band RX/TX filters equal ripple in both of their passbands. As we know,
the return loss level in the passbands depends heavily on the bandwidth of the passbands. A
narrower bandwidth generally gives lower return loss level, and vice versa. Therefore, the dual
band filters with unequal passbands tend to have different return loss level in their two bands.
This challenge can be overcome by adjusting the transmission zeros positions located in the
35
stopbands or adding complex transmission zeros within the passbands as discussed in Chapter
3.
•
Evaluation of Duplexer Polynomials
The derivation of duplexer polynomials N ( s ) , D ( s ) , Pt ( s ) , Pr ( s ) is done using the
reflection zeros at the input port of the duplexer and the transmission zeros of the TX and RX
Filters. Assuming lossless overall duplexer and unitary condition of the scattering matrix we
have,
D (=
s ) D* ( − s ) n0 N ( s ) N * ( − s ) + p0 r Pr ( s ) Pr * ( − s ) + p0t Pt ( s ) Pt * ( − s )
2
2
2
(4.3)
N ( s ) N * ( − s ) depends only on the imposed reflection zeros with n0 = 1 .
Now the evaluation is done in the following iterative steps:
1) Initialization: The RX and TX filters are synthesized independently of the duplexer
0
0
0
with general Chebyshev characteristics.( i.e., the polynomials FTX , ETX , PTX
and FRX , ERX , PRX are generated given the number of poles ( npTX ,
0
0
0
npRX ), the
return loss in the two channels, and the transmission zeros of the two filters. For
junctions causing additional zero, an approximate zero has to be added. An initial
estimate of
STX and S RX are available from the above polynomials.
2) Iteration begin:
Pt and Pr are evaluated by polynomial convolution
Pt = conv ( PTXn , S RX ) , Pr = conv ( PRXn , STX ) .
36
3) Evaluation of p0t and p0r : The required return loss in the two channels ( RLTX
and
RLRX ) is imposed at the normalized frequencies,
N ( j)
2
2
2
N ( j ) + p0 r ⋅ Pr ( j ) + p0t ⋅ Pt ( j )
2
N (− j)
2
2
2
= 10− RLTX
(4.4)
10
2
2
N ( − j ) + p0 r ⋅ Pr ( − j ) + p0t ⋅ Pt ( − j )
2
s = ±j.
2
2
= 10− RLRX
10
(4.5)
p0t and p0r are obtained by solving the above system of linear equations. Applying
spectral factorization technique to (4.3) D ( s ) is evaluated from its roots.
4) New
estimation
( aN ( s ) + bD ( s ) )
of
STX and S RX : The roots of the polynomial
1 − jn 2 ⋅ b0 and
2=
STX ( s ) ⋅ S RX ( s ) are computed with a =
b=
1 + jn 2 ⋅ b0 . The roots are arranged in ascending order of imaginary parts. Let
npRX _ LB , npRX _ UB be the number of poles in lower and upper bands of RX filter
respectively. Similarly, let npTX _ LB , npTX _ UB be the number of poles in lower and
upper bands of TX filter. Then First npRX _ LB roots are assigned to
S RX . Next
npTX _ LB roots are assigned to STX . Next npRX _ UB roots are assigned to S RX and
next npTX _ UB roots are assigned to
5) Once the new
STX .
STX and S RX are obtained, step 2 to step 4 are iterated until
convergence is achieve to a high degree.
37
Computation of RX and TX Filter Polynomials
p0TX
p0 RX
Now the polynomials
[38]. Let
 1 + jn 2 b0 
= p0t 

n


2
 1 + jn b0 
= p0 r 

n


(4.6)
DTX and DRX defined similarly as in (2.4) are evaluated same as
zSTX and zS RX be the roots of STX and S RX respectively. The values of DTX ( zSTX )
and DRX ( zS RX ) are computed from the following expressions:
D ( zS )
D ( zSTX ) =
A ⋅ DTX ( zSTX ) S RX ( zSTX ) ⇒ DTX ( zSTX ) =TX
A ⋅ S RX ( zSTX )
D ( zS )
D ( zS RX ) =
A ⋅ DRX ( zS RX ) STX ( zS RX ) ⇒ DRX ( zS RX ) =RX
A ⋅ STX ( zS RX )
(4.7)
where A is given by n 2 (1 + jn 2 b0 ) . Now from the knowledge of the degree of
DRX and DTX
one can find
DRX and DTX using polynomial interpolation.
=
DTX ( s ) polyfit ( zSTX , DTX ( zSTX ) , npTX − 1)
=
DRX ( s ) polyfit ( zS RX , DRX ( zS RX ) , npRX − 1)
(4.8)
npRX npRX , LB + npRX ,UB and=
npTX npTX , LB + npTX ,UB .
where=
Finally, the filter polynomials are obtained as:
F=
STX ( s ) − DTX ( s )
TX ( s )
E=
STX ( s ) + DTX ( s )
TX ( s )
F=
S RX ( s ) − DRX ( s )
RX ( s )
E=
S RX ( s ) + DRX ( s )
TX ( s )
38
(4.9)
4.3
Implementation in Cavity type and Non-Resonating node type
To verify the dual-band duplexer synthesis technique presented above, a prototype
duplexer is synthesized using both the non-resonating node topology and the cavity-resonator
topology [38]. The specifications of the duplexer are listed as the following:
RX Passbands: 800MHz to 807MHz, 850MHz to 857MHz
TX Passbands: 815MHz to 822MHz, 865MHz to 872MHz
Center Frequency: 835.22 MHz
Fig. 4.2 Proposed dual-band duplexer: (a) topology with non-resonating nodes, (b) topology
with cavity resonators.
39
•
Dual-band Duplexer with non-resonating node topology
The topology of the dual-band duplexer with non-resonating nodes is shown in Fig.
4.2(a). Both the RX and TX filters are fourth-orders with two attenuation poles. Following the
proposed synthesis procedure, the duplexer characteristic polynomials are given below. The
polynomial coefficients are reported in descending order.
N = [1, -0.08029i, 2.0807, -0.1405i, 1.4401, -0.07284i, 0.3786, 0.01046i, 0.03211];
D = [1, 1.1634-0.0803i, 2.7574-0.1083i, 1.9599-0.2122i, 2.1502-0.1511i, 0.90040.1306i, 0.5544-0.0504i, 0.0999-0.0204i, 0.0327-0.0029i];
Pr = [1, 0.2866 - 0.6026i, 1.1758 -0.1140i, 0.1921-0.3167i, 0.4208-0.0278i, 0.0192 +
0.0047i, 0.0214 - 0.0034i]
Pt = [1, 0.2950+0.4282i, 1.2800+0.0602i, 0.2089+0.1365i, 0.4556-0.0048i, 0.01770.0543i, 0.0159+0.0018i];
p0r = 0.2439; p0r = 0.2712
The S-parameters of the synthesized duplexer are shown in Fig. 4.3. The corresponding
extracted J (admittance inverter parameter) and B (susceptance) values are listed below.
A) RX Filter:
Inverter J values: 0.5429, 1.0000, 0.5277, 1.0000, 1.6988, 3.4385 and 0.5431.
Inline Susceptance: 1.4937, 0.7449, -8.6256, -0.9618
Susceptance of dangling resonators: 0.5918, -0.2178
Resonator Frequency (in MHz): 783.2, 841.9, 843.1, 870.06
B) TX Filter:
40
Inverter J values: 0.5357, 1.0000, 0.4973, 1.0000, 1.9835, 3.9943 and 0.5354.
Inline Susceptance: -1.5686, -0.7128, 11.7735, 1.0060
Susceptance of dangling resonators: -0.6052, -0.1371
Resonator Frequency (in MHz): 893.6, 857.3, 830.3, 799.79
With the above extracted parameters, the duplexer can be easily implemented using
transmission lines.
Scattering Parameters (dB)
20
0
-20
-40
S11
-60
S31
-80
-100
700
S21
750
800
850
900
Frequency
950
1000
Fig. 4.3 Synthesized response of the dual-band duplexer with non-resonating node topology
(S21 for RX channel, S31 for TX channel).
Dual-band Duplexer with cavity resonator topology
Similarly, the dual-band duplexer with cavity resonator as given in Fig. 4.2 (b) is also
synthesized following the procedure described previously. The duplexer characteristic
polynomials are given below. Junction capacitance is chosen as 0.7.
41
N = [1, -0.7000 - 0.0803i, 2.0807 + 0.0562i, -1.4565 - 0.1405i, 1.4401 + 0.0983i, 1.0081 - 0.0728i, 0.3786 + 0.0510i, -0.2650 - 0.0105i, 0.0321 + 0.0073i, -0.0225 - 0.0000i];
D = [1, 1.8199 - 0.0803i, 3.4917 - 0.1615i, 3.7966 - 0.2817i, 3.3984 - 0.2985i,
2.3483 - 0.2298i, 1.1349 - 0.1422i, 0.4870 - 0.0541i, 0.1005 - 0.0178i, 0.0232 - 0.0021i];
Pr = [1, 0.2764 - 0.5418i, 1.0522 - 0.0820i, 0.1614 - 0.2621i, 0.3347 - 0.0193i,
0.0135 + 0.0001i, 0.0161 - 0.0021i];
Pt = [1, 0.2836 + 0.3613i, 1.1371 + 0.0271i, 0.1702 + 0.0974i, 0.3582 - 0.0081i,
0.0117 - 0.0390i, 0.0119 + 0.0010i];
p0t = 0.2780; p0r = 0.3134
The extracted TX and RX Filters have the following N+2 (including the source and the
load) coupling matrices respectively.
M TX
0.6574
0
0
0
0 
 0
0.6574 − 0.1630 0.5744 − 0.0259 0.4022
0 

 0
0.5744 − 0.3174 − 0.1296
0
0 
=

0.0259 − 0.1296 − 0.1903 0.6158
0 
 0
 0
0.4022
0
0.6158 − 0.2450 0.5257


0
0
0
0.5257
0 
 0
M RX
 0
0.6716

 0
=
 0
 0

 0
0.6716
0
0
0
0.1723 0.5720
0.0124 0.4381
0.5720 0.2630 − 0.1648
0
0.0124 − 0.1648 0.1283 0.6086
0.4381
0
0
0
0.6086
0
42
0.2658
0.5325
0

0 
0 

0 
0.5325

0 
To validate the proposed synthesis method for dual-band duplexers, the cavityresonator based duplexer presented above is implemented. As shown in Fig. 4.4 and Fig. 4.5
both the RX and TX filters of the duplexer are fourth-orders. The transmission zeros in the
RX/TX channels are obtained with a quadruplet. The RX/TX junction at the input is realized by
adding an extra resonator.
Fig. 4.4, Fig. 4.5, Fig. 4.6 show the simulated insertion losses ( S12 for the RX channel and
S13 for the TX channel) and return loss ( S11 at the input) of the cavity resonator based duplexer
with optimized coupling coefficients. For comparison, the synthesized responses of the
duplexer are also plotted in this figure. The simulated results agree well with the synthesis as
expected, confirming the feasibility of the proposed synthesis technique for dual-band
duplexers.
0
Syntehsized
Simulated
-20
S12 (dB)
-40
-60
-80
-100
700
750
800
850
900
Frequency (MHz)
950
Fig. 4.4 Insertion loss at RX channel ( S12 ).
43
1000
0
Synthesized
Simulated
S13 (dB)
-20
-40
-60
-80
700
750
900
800
850
Frequency (MHz)
950
1000
Fig. 4.5 Insertion loss at TX channel ( S13 ).
0
Synthesized
Simulated
S11 (dB)
-20
-40
-60
-80
-100
700
750
950
900
850
800
Frequency (MHz)
Fig. 4.6 Return loss at Input ( S11 ).
44
1000
CHAPTER 5
MODEL ORDER REDUCTION OF MICROWAVE DUPLEXERS
5.1
Introduction
Often to meet specific requirements and lack of flexibilities in microwave duplexer
designs lead to very high order of microwave filters. In practice, it is convenient to reduce the
order of such duplexers that matches the original high order duplexer to a high degree in a
specific band of concern. One method to obtain coupling matrix elements from filter response
by genetic algorithm was described in [49].
The Cauchy method deals with estimation of a function by a ratio of two polynomials.
Given the values of the function and its derivatives at a few points the order of the polynomials
and their coefficients can be evaluated. Once the coefficients of the two polynomials have been
estimated, they can be used to generate the parameter over the entire band of interest. It is
particularly useful when the values of the function have been measured and a mathematical
model for the function has to be estimated.
However any measurement is accompanied by addition of noise. It is shown that for low
level of noise, performance of Cauchy method is very good over the entire sample space.
5.2
Formulation of Cauchy Method
The Cauchy method approximated a system function H(s) with a ratio of two
polynomials. The procedure was described in [50] and is discussed below.
P
H (s) ≈
A( s)
∑a s
k
k
k =0
=
Q
B (s)
∑ bk s k
k =0
45
(5.1)
Here the information is assumed to be N measured values of the function (H) at
frequency points s j , j = 1, 2,3...N .
In such case, the Cauchy Problem is,
Given H ( s j ) for j=1, 2, 3…N, find P, Q, {ak, k = 0, 1, 2,..P} and {bk, k=0, 1, 2..Q}
Then from (5.1),
A(s j ) = H (s j ) B (s j )
A(s j ) − H (s j ) B (s j ) =
0
(5.2)
Now using the polynomial expansions for A ( s ) and B ( s ) ,
a0 + a1 s j + a2 s 2j +  + aP s Pj − H j b0 − H j b1 s1j −  − H j bQ s Qj =
0
(5.3)
for j = 1, 2, 3 N .
Writing in matrix form,
a 
=0
b 
[C ] 
(5.4)
where
1 s1

1 s
C
=
[ ]  2
 

1 sN
 s1P
 s2P
 
 sNP
− H1
−H2

−H N
− H1 s1
− H 2 s2

− H N sN
 − H1 s1Q 

 − H 2 s2Q 

 

 − H N sNQ 
(5.5)
where
[ a ] = [ a1 , a2 , a3 , , aP ]
T
(5.6)
T
(5.7)
and
[b] = b1 , b2 , b3 , , bQ 
46
The order of matrix C is N × ( P + Q + 2 ) .
A singular value decomposition (SVD) gives us the required coefficients for
[U ][Σ][V ]
T
0
=
a and b .
(5.8)
According to the Theory of Total Least Squares,
a 
 b  = ( const.) * [V ]P +Q + 2
 
(5.9)
That is, the elements in the last column of V give us the solution.
5.3
Application in General Polynomial Function:
Cauchy method has been applied on a function as below:
4
H (s) =
∑ ks
k
(5.10)
k =0
5
∑ (k + 1)s
k
k =0
This ratio is evaluated to find the exact values at 21 points in the range of s=2 to s=4.
Different levels of noise are added to this exact value to see its effect on the estimation. The
estimation accuracy is observed in the following figures. It can be seen that with low noise
levels Cauchy method gives excellent estimation of this kind of functions.
47
0.5
Reference H
Estimated H without noise
Estimated H with Noise
0.45
0.4
Value of H
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
2
2.2
2.4
2.6
2.8
3
3.2
Frequency
3.4
3.6
3.8
4
Fig. 5.1 SNR = 5
0.5
Reference H
Estimated H without noise
Estimated H with Noise
0.45
0.4
Value of H
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
2
2.2
2.4
2.6
2.8
3
3.2
Frequency
Fig. 5.2 SNR = 20
48
3.4
3.6
3.8
4
0.5
Reference H
Estimated H without noise
Estimated H with Noise
0.45
0.4
Value of H
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
2
2.2
2.6
2.4
2.8
3.2
3
Frequency
3.4
3.6
3.8
4
Fig. 5.3 SNR = 50 dB.
5.4
Application in General Microwave Filter function
As mentioned in previous chapters, a general two port network can be characterized by
a set of two parameters of its scattering matrix, namely the transmission and reflection
coefficients,
S11 and S21 . These parameter share the same poles. Therefore to generate the
polynomial models for this case it is necessary to ensure the use of a common denominator.
The filter response is then modeled as,
 n
  n

S11 ( s )  ∑ a1k s k   ∑ bk s k  ≈ S11 ( s )
=
=
 k 0=
 k 0

(5.11)
 nZ
  n
=
S21 ( s )  ∑ a2 k s k   ∑ bk s k
=
 k 0=
 k 0
(5.12)
where
n
is the filter order and

 ≈ S 21 ( s )

nZ is the number of finite transmission zeros.
This can be represented in a matrix equation:
49
a 
− S11Vn   1 
=
 a2
VnZ
− S 21Vn   
 b 
S11 = diag {S11 ( si )}
 Vn

0 N ×n
0 N ×nZ
a 
b 
X ]   [ 0]
[=
(5.13)
S 21 = diag {S 21 ( si )}
V is a Vandermonde matrix defined as,
 s1m
 m
s
[Vm ] =  2

 m
 sN
s1m −1  s12
s1
s2m −1  s22

 
s2

sNm −1  sN2
sN
1

1


1
(5.14)
The Exact polynomial model used for simulation is tabulated below and the exact
polynomial response and estimated response are shown in Fig. 5.4. More details on the
procedure can be found in [51].
Table 5.1 Low Pass Polynomial Coefficients
ak(
2)
k
ak(1)
0
- 0.0244 – 0.0110j
0.1123 – 0.1158j
1
- 0.0453 – 0.2532j
0.6002 – 0.5472j
2
- 0.0745 – 0.1796j
1.3800 – 1.3009j
3
-0.1619 – 1.5168j
4
0.6856 – 0.2242j
2.4082 – 1.5359j
5
-0.0139 – 1.3188j
1.9356 – 1.3007j
6
1.0000
1.0000
0.1133 + 0.0545j
50
bk
2.4933 – 1.9912j
Filter Scattering Parameter Extraction
10
|S |
11
Estimated |S |
11
0
|S |
Scattering Parameters in dB
21
Estimated |S21|
-10
-20
-30
-40
-50
-60
-2
-1.5
-1
-0.5
0
Frequency
0.5
1
1.5
2
Fig. 5.4 Example of polynomial estimation for a 6th order filter.
5.5
Application in Microwave Duplexers
Application of the Cauchy Method on Microwave Duplexer was detailed in [52]. The
parameters
S11 , S21 and S31 can be measured from Lab Setups for a diplexer. After
measurement of the scattering parameters, the filters can be approximated using Cauchy
Method.
∑
∑
S (s)
∑
K=
(s) =
S (s) ∑
nTX + nRX
S11 ( s )
K=
s
=
TX ( )
S 21 ( s )
RX
k =0
mTX + mRX
k =0
11
nTX + nRX
k =0
mTX + mRX
31
k =0
Equation (5.15) can be written in matrix form as:
51
ak s k
ck(1) s k
ak s k
ck( 2) s k
(5.15)
 S21VnTX + nRX

 S31VnTX + nRX
− S11VmTX + nRX
0
 a 
  (1) 
=
 c
− S11VmRX + nTX   ( 2) 
c 
 a 
 
M ]  c(1)  0
[=
c( 2) 
 
0
(5.16)
where V is the Vandermonde Matrix defined as:
 s1m
 m
s
[Vm ] =  2

 m
 sN
s1m −1
s2m −1

sNm −1
 s12
 s22
 
 sN2
s1
s2

sN
1

1


1
(5.17)
As an example, the exact and estimated polynomial responses are shown in Fig. 5.5,
which agree with each other very well.
Scattering Parameters in dB
20
Diplexer Scattering Parameter Extraction
0
-20
-40
-60
-80
|S |
11
Estimated |S |
11
|S21|
-100
Estimated |S21|
|S31|
Estimated |S31|
-120
-2
-1.5
-1
-0.5
0
Frequency
0.5
1
Fig. 5.5 Exact and estimated polynomial response for duplexers.
52
1.5
2
5.6
Conclusion
As we can see, Cauchy Method is an excellent mathematical procedure that uses the
concept of Total Least Squares to estimate the same or reduced order polynomial models of
measured parameters. Even with noisy data, the models obtained do not deviate far from exact
models.
53
CHAPTER 6
CONCLUSION AND FUTURE WORK
In this work methods to synthesize microwave filters and duplexers have been
discussed. Especially, a method to synthesize a dual band microwave duplexers is proposed. In
addition to using dual band filters to synthesize dual band duplexer, a multiband filter can be
used to synthesize a multi band duplexer. However this will add the complexity of the synthesis
and require high degree of optimization at later stages of design, which can be studied in future
work.
Also, waveguide is another way to design microwave filters and duplexers. Synthesized
dual band duplexers can be designed using waveguides which can sustain high power.
Once the designs of filters and duplexers are complete, measurements can be done on
the prototypes and the scattering parameters can be measured with vector network analyzer at
different frequencies. These measurements can be used to obtain a lower order polynomial
model using Cauchy method in order to obtain filters of desired orders that roughly behave the
same as the original ones.
54
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