Computer-aided diagnosis of lossy microwave coupled resonators filters

Computer-Aided Diagnosis of Lossy MicrowaveCoupled Resonators Filters

Rui Wang, Jun Xu

School of Physics and Electronics, University of Electronic Science and Technology of China,Chengdu, China

Received 20 January 2011; accepted 2 April 2011

ABSTRACT: A method is presented for extracting the coupling matrix (CM) and the unloaded

Q from the measured (or electromagnetic simulated) scattering parameters of a lossy coupled

resonators bandpass filter. The method can be used for computer-aided tuning of a microwave

filter. The method consists of two elements: 1) a three-parameter optimization method is pro-

posed to obtain the unloaded Q (assuming all the resonators with the same unloaded Q) and to

remove the phase shift of the measured S-parameters caused by the phase loading and the

transmission lines at the input/output ports of a filter; 2) the Cauchy method is used for deter-

mining characteristic polynomial models of the S-parameters of a microwave filter in the nor-

malized low-pass frequency domain. Once the characteristic polynomials of the S-parameters

without phase-shift effects are determined, the CM of a filter with a given topology can be

extracted using well-established techniques. Three diagnosis examples illustrate the validity of

the proposed method. VC 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE 21:519–525, 2011.

Keywords: coupling matrix; unloaded Q; diagnosis; extraction; bandpass filter

I. INTRODUCTION

Microwave filters with general Chebyshev response have

found wide applications such as wireless base stations and

satellite communication systems. A great deal of effort

has been made in analytically synthesizing the filter cou-

pling matrix (CM) according to a given topology. The

most recent representative work in this subject would be

Cameron’s methods [1, 2].

For a given CM and filter topology, physical realization

of a filter would largely depend on a tuning process due to

manufacturing and material tolerances. The core task in fil-

ter tuning is a filter diagnosis (also called CM extraction)

that reveals the differences with the designed one and then

guide technologists during the tuning process. Computer-

aided diagnosis and tuning has a significant impact on the

overall filter production cost and project schedules.

In recent years, the interest is growing on methods for

diagnosis of a microwave filters from measurements (or

simulations) including losses. The existing diagnosis techni-

ques mainly include nonlinear optimization methods [3–5]

and analytical methods [6–13]. These optimization methods

either require more computer time (for global optimization)

or rely greatly on the initial values of the variables (for gra-

dient-based local optimization). These analytical diagnosis

methods can be divided into three categories: 1) polynomial

models match the measured admittance parameters [6, 7];

2) analytical models based on the locations of system zeros

and poles [8, 9]; and 3) polynomial models match the

measured S-parameters (Cauchy method) [10–14]. The

methods [6, 10, 11, 14] only deal with a lossless or low-loss

filter, which restrict its practical uses. The methods [8, 9] are

only suitable for cascaded and symmetrically coupled filters.

The diagnosis method can be used to a general coupled reso-

nator filter with losses [7], but it requires complicated de-

embedding techniques and dealing with the degenerate poles

of admittance parameters. The diagnosis method [12] can

also deal with a microwave filter with large losses, but the

characteristic polynomials used for the CM extraction must

be solved in two steps (a Feldkeller’s equation solved in the

second step). Polynomials can be solved in one step from

loss or lossless filter response [13], but these polynomials

are not suitable for the CM extraction by well-known estab-

lished techniques [1, 2], as they include phase shift, when

raw measured S11 and S21 are used directly.

The phase-shift effects of the measured S-parameters

caused by the phase loading and the transmission lines at

the input/output (I/O) ports of a physical filter model is

Correspondence to: R. Wang; e-mail: [email protected]

VC 2011 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20537Published online 27 July 2011 in Wiley Online Library

(wileyonlinelibrary.com).

519

difficult to measure because of the higher order mode

effect. The concept of the phase loading is revealed for

the first time in the community of computer-aided diagno-

sis [7]. Some techniques have also been proposed for

removing phase-shift effects from S-parameters [7–9].

However, the method [7] requires carefully select fre-

quency samples far below or above the center frequency

because of some features of the response such as the pres-

ence of spurious passbands and the frequency-dependent

coupling. The methods [8, 9] require additional transmis-

sion lines at a filter I/O ports, which lead to the inconven-

ience and difficulties in practical uses.

In this article, a simple three-parameter optimization

method is proposed to obtain the unloaded Q and to remove

the phase shift of the measured S-parameters. A modified

frequency transformation [12] is adopted to remove loss of

a filter. Characteristic polynomials are solved in one step by

the method in Ref. [13], after the phase shift of the meas-

ured S-parameters are removed. Finally, the CM is extracted

from the polynomials by established techniques [1, 2]. Dif-

ferent from direct optimization of the CM elements [3–5],

the proposed method only includes three optimized parame-

ters, which is independent on the order and the topology of

a filter. With respect to the method in Ref. 7, the proposed

method is simple and the degenerate poles of admittance

parameters are not required to be dealed with here. In addi-

tion, with respect to the methods in Refs. 8 and 9, the one

here proposed allows no restricted filter topologies and does

not require additional transmission lines. This technique can

be applied to the CM extraction of a general coupled reso-

nators filter with losses.

II. CALCULATION OF CHARACTERISTIC POLYNOMIALS

S21 and S11 can be approximated by two rational functions

with a common denominator [10–14]

S11ðsÞ ¼ FðsÞEðsÞ ¼

PNk¼0

að1Þk sk

PNk¼0

bksk; S21ðsÞ ¼ PðsÞ

EðsÞ ¼Pnzk¼0

að2Þk sk

PNk¼0

bksk

(1)

where, N is the filter order or the number of the resonator

and nz is the number of finite-location transmission zeros.

F, P, and E are three characteristic polynomials. A modi-

fied frequency transformation in Ref. [12] used for convert-

ing the measured S-parameters from the bandpass domain fto the normalized lowpass domain s is adopted here as

s ¼ jX ¼ f0BW

1

Quþ j

f0BW

f

f0� f0

f

8>>: 9>>;: (2)

Here, the unloaded quality factors Qu of all resonators are

assumed to be the same, and BW and f0 are bandwidth

and center frequency of the filter, respectively.

The formulation of the Cauchy method allows the

evaluation of the complex coefficients að1Þk , a

ð2Þk , and bk

(and then of the polynomials F, P, and E) in one step by

solving the following the (over determined) system [13]:

VN 0Ns�ðnzþ1Þ �S11VN

0Ns�ðNþ1Þ Vnz �S21VN

" # að1Þ

að2Þ

b

264

375¼ X

að1Þ

að2Þ

b

264

375¼ 0

(3)

where a(1) ¼ [að1Þ0 , K, a

ð1ÞN ]T, a(2) ¼ [a

ð2Þ0 , K, að2Þnz ]

T, b ¼[b0,K,bN]

T, S21 ¼ diag{S21(si)}i¼1,K,Ns, S11 ¼ dia-g{S11(si)}i¼1,K,Ns, and Vr [ CNs � (r þ 1) is a Vandermonde

matrix with elements Vi,k ¼ (si)k � 1, k ¼ 1, K, r þ 1. The

S21(si) and S11(si) are measured or simulated S-parameters at

frequency points si (i ¼ 1,2,...,Ns). Ns is the number of fre-

quency points. Solution of Eq. (3) can be obtained by apply-

ing singular value decomposition (in essence, least square

fitting) to the matrix X. Each evaluation requires at least (Nþ 1 þ nz þ 1 þ N) frequency samples of the S-parameters.

Cauchy method has proved to be a robust technique for

extracting the characteristic polynomials in Refs. [10–14].

It must be observed that the polynomials F, P, and Esolved in one step in Ref. [13] are not suitable for the CM

extraction by well-known techniques [1, 2], before the

phase shift of the measured S-parameters are removed. To

make the characteristic polynomials to satisfy the circuit

model in Refs. [1, 2], the phase shift of the measured

S-parameters should first be removed. Failing to remove the

phase-shift effect will lead to an incorrect CM extraction.

III. THREE-PARAMETER OPTIMIZATION METHOD

In a physical filter model, there is always a section of

transmission line at a filter I/O ports, which shifts the ref-

erence planes. A phase offset u connected to each port

can be very well approximated by the following function

in a wide frequency range [7]

u ¼ u0 þ bDl (4)

where the frequency invariant constant term u0 is called the

phase loading, and b and Dl are the propagation constant and

an equivalent length of the transmission line, respectively.

For a typical transmission line, bDl can be expressed as

bDl ¼ 2p f Dlffiffiffiffiffiffiffiffiffieeffl

p(5)

where eeff is the effective permittivity. Assuming eeff is

frequency invariant constant within the range of frequency

samples. So, bDl can be derived as

bDl ¼ f h0�f0 (6)

where h0 ¼ 2pf0Dlffiffiffiffiffiffiffiffiffieeffl

pis equivalent electrical length of

the transmission line in radian at f0.The following phase shift caused by the phase loading

and the transmission lines should be removed from the

measured S11 and S21

D/ ¼ �2ðu0 þ fh0�f0Þ: (7)

An (N þ 2) normalized CM [M0] for loss case can be

expressed as

½M0� ¼ ½M� � j½G�: (8)

520 Wang and Xu

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 5, September 2011

Here, [M] represents the coupling between coupled reso-

nators for lossless case, and [G] is the diagonal matrix

[G] ¼ diag[0,G1,…,GN,0], which represents the loss of

the filter. The loss factor Gi (i ¼ 1,2,…,N) for the ith res-

onator can be evaluated by Gi ¼ BW/(f0Qu). Once [M0] is

extracted, the filter response for loss case can be obtained

via the following equations

S21 ¼ �2j½A�1�Nþ2;1; S11 ¼ 1þ 2j½A�1�1;1: (9)

Here, A ¼ [XU � jR þ M0, X, [R], and [U] can refer to

Ref. [15]. Equation (9) allows calculation of a loss filter

response, which is different from that given in Ref. [15].

Also, as Qu approach infinity, [M0] degenerates to [M],

and Eq. (9) is exactly the same as that given in Ref. [15],

which is suitable for the lossless filter response.

u0, y0, and Qu are the unknown parameters to be opti-

mized. Once they are known, the phase shift of the meas-

ured S-parameters can be removed using (7), and then the

characteristic polynomials F, P, and E are solved in one

step by the method in Ref. [3]; the next step consists of

extracting the CM [M] from the characteristic polynomials

by established techniques [1, 2] and corresponding [M0].

Unknown parameters u0, y0, and Qu are obtained by

minimizing the following objective error function using

genetic algorithm (GA)

F ¼XNsi¼1

½jSext21 ðsiÞj � jSmea21 ðsiÞj�2 þ ½jSext11 ðsiÞj � jSmea

11 ðsiÞj�2

(10)

where Sext21 (si) and Sext11 (si) are the extracted S-parameters

calculated from [M0] using (9), and Smea

21 (si) and Smea11 (si)

are the measured S-parameters.

GA can be used to solve the global minimum value of a

multivariate function. In this article, the GA toolbox for

Matlab provided by the University of Sheffield [16] is cho-

sen to minimize the error function in (10). A GA starts with

an initial set of random configurations and uses a process

similar to biological evolution to improve upon them. The

set of configurations is called the population. Each configura-

tion in the population will be a set of designable parameters.

A GA is based on some genetic operators such as selection,

crossover, mutation, and inversion to emulate an evolution-

ary process. Main steps to solve the minimum value of the

objective error function using GA are given below.

Step 1, define parameters for GA as follows:

• Nind ¼ 100; % Number of individuals per populations.

• Maxgen ¼ 80; % Max Number of generations.

• Nvar ¼ 3; % Number of variables of the objective error

function.

• Preci ¼ 20; % Precisicion of binary representation.

• Gap ¼ 0.9; % Generation gap, how many new individu-

als are created.

• gen ¼ 0;% reset count variables.

Step 2, Iterate population:

• Call function ‘‘rep’’ to build field description matrix of

the parameters to be optimized.

• Call function ‘‘crtbp’’ to create initial population.

• Call function ‘‘bs2rv’’ to decode binary chromosomes

of population into vectors of reals.

• Call function ‘‘objfun’’ to evaluate objective error function

for initial population, where ‘‘objfun’’ is the name of the

objective function programmed based on eq. (10).

• while gen < Maxgen,

• Call function ‘‘ranking’’ to assign fitness values to

whole population.

• Call function ‘‘select’’ to select individuals from population.

• Call function ‘‘recombin’’ to recombine selected indi-

viduals (crossover).

• Call function ‘‘mut’’ to mutate offspring.


of offspring into vectors of reals.

• Call function ‘‘objfun’’ to evaluate objective function

for offspring.

• Call function ‘‘reins’’ to insert best offspring in popula-

tion replacing worst parents.

• gen ¼ gen þ 1;

• Call function ‘‘min’’ to obtain minimum of objective

function for offspring.

• end

Step 3, Results:


of the optimum offspring into vectors of reals;

• Obtain solution.

IV. DIAGNOSIS EXAMPLES

A. Filter 1 (Fourth-Order Filter)The technique presented here is first applied to the simu-

lated S-parameters of fourth-order filter with f0 ¼ 2.51

GHz and BW ¼ 90 MHz (filter 1). The structure (see

Fig. 1a), presented in Ref. [17], is designed on a Rogers

RO3010 substrate with a relative dielectric constant er ¼10.2, a thickness h ¼ 1.27 mm, and a loss tangent d ¼0.0023. The filter has been simulated using a full-wave

simulator IE3D. The loss factors (conductor loss and

dielectric loss) are included in the simulated response.

The proposed method is applied with N ¼ 4, nz ¼ 2,

and Ns ¼ 76 (frequency interval 2.43–2.58 GHz). u0 ¼0.3756, y0 ¼ 1.5589 and Qu ¼ 225.4065 are obtained by

optimization. Phase shift of simulated S-parameters can be

removed using (7), and then characteristic polynomials F,P, and E are solved in one step as

P ¼ ½0:0106� j0:1742 0:0603� j0:0106 0:0171� j0:7126�;F ¼ ½1 � 0:0273þ j0:1017 0:8085� j0:0175

� 0:0205þ j0:2433 0:0927� j0:0055 �;E ¼ ½0:9857þ j0:0260 1:7593þ j0:1322

2:3143� j0:0061 1:7499þ j0:010 0:7123� j0:0297 �:(11)

From polynomials in (11), the denormalized coupling

coefficients and external quality factors are extracted by

well-known established techniques [1, 2] as

Diagnosis of Lossy Resonators Filters 521

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

K ¼

0:0045 0:0270 0 �0:0036

0:0270 �0:0009 0:0231 0:0024

0 0:0231 �0:0049 0:0267

�0:0036 0:0024 0:0267 0:0056

26664

37775

Qes ¼ 31:6539; QeL ¼ 31:9161:

(12)

In Figure 1b, the extracted S-parameters are compared

with the original simulated S-parameters samples. Very

good agreement can be observed.

Note that, the measured (or simulated) S21 and S11samples in (3) should be chosen around the passband in

Cauchy method; in fact it is not convenient to consider

frequency points too much distant from the passband

because the accuracy of the model may be reduced by

second-order effects such as the frequency-dependent cou-

plings. For this example, the data set here employed refers

to frequency interval 2.43–2.58 GHz. Moreover, some fea-

tures of the response in a practical filter, such as the pres-

ence of spurious couplings, spurious passbands, and fre-

quency-dependent couplings out of the pass band, may

cause spurious coupling elements in the extracted CM.

The parameters, BW and f0, in Eq. (2) are required for

evaluating the polynomials; their values can be easily

obtained from the measured response or the design values.

However, the parameter nz must be obtained from the

design value, because some transmission zeros cannot be

observed from the filter amplitude response such as a self-

equalized filter; in addition, loss factors can also cause

some transmission zeros unable to be observed.

B. Filter 2 (Sixth-Order Self-Equalized Filter)As a second example, the diagnosis technique will be applied

in the measured S-parameters of a sixth-order coaxial cavity

resonator self-equalized filter with f0 ¼ 910 MHz and BW ¼40 MHz (filter 2). The structure of filter 2 is shown in Figure

2a. A coupling tuning screw and a frequency tuning screw

were changed to demonstrate the applicability of the proposed

method to a severely detuned filter. The proposed method is

applied with N ¼ 6, nz ¼ 4 (including two unobserved com-

plex transmission zeros for group delay equalization), Ns ¼ 31

(frequency interval 880–940 MHz). u0 ¼ 1.3183, y0 ¼ 1.8006

and Qu ¼ 1845.97 are obtained by optimization.

The characteristic polynomials F, P, and E are

obtained as

P ¼ ½�0:0002þ j0:0683 � 0:0334þ j0:0034 0:0002

þ j0:2291 � 0:0557þ j0:0119 0:0034� j0:5576 �;F ¼ ½1 0:0085þ j0:5024 2:0846þ j0:1173 � 0:0674

þ j0:6259 0:6919� j0:1604 � 0:1605

� j0:2017 0:2786� j0:2489�;E ¼ ½1:0293þ j0:0612 2:1073þ j0:6263 4:3677

þ j1:2102 5:1464þ j1:8203 4:4707

þ j1:8524 2:2496þ j1:0873 0:5807þ j0:3361�:(13)

From these polynomials, the denormalized coupling coef-

ficients and external quality factors are extracted as

Figure 1 (a) Physical dimensions and (b) the simulated S-pa-

rameters samples and the extracted S-parameters of filter 1.

[Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]

Figure 2 (a) Photograph and (b) the measured and the

extracted S-parameters of filter 2. [Color figure can be viewed in

the online issue, which is available at wileyonlinelibrary.com.]

522 Wang and Xu


K ¼

0:0058 0:0382 0 0 0 �0:0014

0:0382 0:0185 0:0303 0 0:0030 0:0001

0 0:0303 �0:0002 0:0390 �0:0006 0

0 0 0:0390 �0:0008 0:0283 0

0 0:0030 �0:0006 0:0283 0 0:0392

�0:0014 0:0001 0 0 0:0392 0:0013

2666666664

3777777775

Qes ¼ 21:5262; QeL ¼ 21:4742:

(14)

In Figure 2b, the original measured S-parameters are

compared with those calculated by the extracted CM. Very

good agreement between the simulated and extracted

response can be observed.

C. Filter 3 (Eighth-Order Dual-Passband Filter)As a third example, the diagnosis technique will be

applied in the simulated S-parameters of an eighth-order

dual-passband filter with passbands at 3.90–3.95 and

4.05–4.10 GHz (filter 3). A stripline structure is used for

this filter [18]. Figure 3a shows its conductor layer,

which is positioned in the middle of two metal-backed

dielectric layers (er ¼ 2.2, h ¼ 1.574 mm, and d ¼0.0009). The filter has been simulated using IE3D. The

loss factors (conductor loss and dielectric loss) are

included in the simulated response.

Special attention should be paid to the determination

of BW and f0 in (2) for dual-passband filter; here, f0 is

taken as the arithmetic mean of the two passband center

frequencies and BW is the difference between the lower

edge of the first passband and the upper edge of the sec-

ond passband. So f0 ¼ 4 GHz and BW ¼ 0.2 GHz are

determined for this example.

The proposed method is applied with N ¼ 8, nz ¼ 6, and

Ns ¼ 51 (frequency interval 3.88–4.13 GHz). u0 ¼ 0.9765,

y0 ¼ 1.8084 and Qu ¼ 334.2751 are obtained by optimization.

The characteristic polynomials F, P, and E are

obtained as

P ¼ ½�0:0044� j0:1266 � 0:0659� j0:0043

� 0:0104� j0:2618 � 0:1915� j0:0060

� 0:0032þ j0:0252 � 0:0126� j0:0008 j0:0039 �;F ¼ ½1 0:0035� j0:9217 2:1583� j0:0227 � 0:0049

� j1:6595 1:6958 � j0:0304 � 0:0081 � j0:9332

0:5608� j0:0072 � 0:0013� j0:154 0:0731�;E ¼ ½1:0100þ j0:0268 1:1610� j0:8986

2:5823� j1:0702 1:9023� j2:3252

2:2493� j1:6839 0:7297� j1:5135

0:5785� j0:5694 0:0746� j0:2287

0:0622� j0:0388 �:(15)

From these polynomials, the N�N normalized CM and

normalized source and load resistors are extracted as

M ¼

0:0296 0:8870 0 0 0 0 0 0:1108

0:8870 �0:1825 0:4776 0 0 0 �0:1177 �0:0658

0 0:4776 0:1147 0:4727 0 �0:4301 �0:1421 0

0 0 0:4727 �0:2664 0:1823 0:1788 0 0

0 0 0 0:1823 �0:2583 0:5137 0 0

0 0 �0:4301 0:1788 0:5137 �0:2371 0:4518 0

0 �0:1177 �0:1421 0 0 0:4518 �0:2171 0:8633

0:1108 �0:0658 0 0 0 0 0:8633 0:1116

266666666666664

377777777777775

RS ¼ 0:5696; RL ¼ 0:5673:

(16)

In Figure 3b, the original simulated S-parameters are

compared with those calculated by the extracted CM. Very

good agreement between the simulated and extracted

response can be observed. The simulated frequency

response has somewhat lower attenuation at both sides

out-of-passband than the extraction one, which is due to

second order effects of a physical filter.

V. CONCLUSIONS

A method for the accurate diagnosis (CM extraction) of

lossy coupled resonator filters is presented. To make the

characteristic polynomials (solved in one step by Cauchy

method) suitable for the CM extraction, a three-parameter

optimization method is proposed to obtain the unloaded Q

and to remove the phase-shift effects from a given filter

response. Three diagnosis examples are provided, includ-

ing one measured filter and two electromagnetic simulated

filters, to show the validation of the proposed method.

The diagnosis of a dual-passband filter composed of mul-

tiple-coupled resonators is investigated for the first time in

the field. Some parameter settings (such as BW, f0, nz, andS-parameters frequency samples) for evaluating the



characteristic polynomials are discussed in detail in practi-

cal examples. This diagnosis tool will find many practical

applications for the computer-aided tuning of microwave

coupled resonators filters.

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Figure 3 (a) Conductor layer of stripline structure (b) the simu-

lated and the extracted S-parameters for the dual-passband filter

3. [Color figure can be viewed in the online issue, which is avail-

able at wileyonlinelibrary.com.]

524 Wang and Xu


BIOGRAPHIES

Rui Wang was born in Taian, China,

in January 1980. He received the

B.S. degree in Physics from Ludong

University in 2004, and the M.S.

degrees in Radio Physics from

Xidian University in 2007. From

2007 to 2009, he was a Research

Engineer with China air to air missile

academy, where he was involved with the design of anten-

nas and components for transmitters and receivers. Since

2009, he has been working toward the Ph.D. degrees in

Radio Physics at the University of Electronic Science and

Technology of China (UESTC). His current research inter-

ests include computer-aided millimeter-wave circuit, pas-

sive component design, electromagnetic field theories, and

numerical analysis.

Jun Xu was born in Chengdu, China,

in March 1963. He received the B.S.

and M.S. degrees in Electronics and

Engineering from the University of

Electronic Science and Technology

of China (UESTC), Chengdu, China,

in 1984 and 1990, respectively. Since

1984, he has been with the School of

Physics and Electronics at the UESTC, China, where he is

currently a Professor. His research interests include milli-

meter-wave circuits and systems and transmitters, and

receivers for millimeter-wave radar system.



Documents

Computer-aided diagnosis of lossy microwave coupled resonators filters