Chapter 3 Development and Optimization of Compact Filters Microwave

Chapter III

Development and Optimization of

Compact Filters Microwave Filtering Technique

3.1 Introduction

Microwave filters are essential components in a microwave system. Thus, much

literature is available regarding the designs and implementations of microwave filters in
various wireless communication systems. Combine bandpass filters are the most compact
microwave filters [1]. This work presents a sizing method and a global optimization for
diagnosing bandpass filters with coupled resonators. The process can extract the coupling
matrix of measured or simulated admittance parameters EM from a lossy narrowband
coupled resonator bandpass filter. The optimization method is used to eliminate phase shift
effects of EM-simulated parameters caused by the transmission lines loaded at the ports of a
filter. [1] This chapter proposes several combined bandpass filters designed and optimized
using the extraction of the admittance technique. HFSS and AWR simulators create a rough
electromagnetic filter model based on a Chebyshev lowpass prototype with frequency
transformation. The method is based on introducing additional internal ports in the filter
design to extract the multiport admittance matrix. The filter polynomial characteristics are
generated from the simulated scattering parameters using the Coupling matrix. After several
iterations, an exemplary filter model with optimal physical dimensions is developed,
providing an ideal frequency response. The optimized filter is tuned using an admittance
technique, which extracts the system quickly compared to manual experience tuning.
3.2 Coupling Matrix Theory

Following the determination of the design polynomials, an association is established

between the implementation of the circuit and the approximation theory. Matrixes come into
play at this juncture. The utilization of matrix operations, including inversion, similarity
transformation, and partitioning, becomes feasible when the circuit is represented in matrix
form, thereby augmenting its practicality. These operations streamline the processes of
synthesizing, reconfiguring the topology, and simulating the performance of intricate circuits.
However, it should be noted that the coupling matrix can incorporate specific practical
characteristics of the filter elements. Every element within the matrix can be distinctly
associated with a particular component of the final microwave device. This allows for the

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attributions of the electrical properties of individual elements, including the quality factor Q
values associated with each resonator cavity, to be accounted for. Accomplishing this using a
polynomial representation of the filter's characteristics may be challenging if attainable.

3.3 General NxN coupling matrix of N-coupled resonator circuit

Atia and Williams [8] introduced a coupling matrix-based design method for a band-
pass waveguide cavity filter in the early 1970s. They implemented the (n×n) coupling matrix,
which is derived from a bandpass circuit prototype depicted in Fig 3.1.

Fig 3. 1 Multi coupled series-resonator bandpass prototype network.

Transistors or magnetic couplings are utilized to connect the cascaded Nth-order filter.
Each resonator comprises an inductor L = 1H and a capacitor C = 1F. Consequently, each
resonator operates at a frequency of 1 Hz and represents the source and load resistances. (It is
important to note that lessness is presumed in the equivalent lumped circuit; resistance and
conductance are solely present in the source and load.) reflects the current through the cycle
of every resonator. The denotation for the coupling between resonators p and q is a natural
number that remains constant concerning frequency. It is possible to apply the coupling
matrix theory to circuits that contain asynchronously tuned resonators or the general (nn)
coupling matrix. The formulation of the general (nn) coupling matrix is elaborated in [1]. For
filters featuring resonators that are coupled magnetically and electrically, distinct schematics
are created.

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Fig 3. 2 Equivalent filter circuit of n-coupled resonators for (a) loop-

Equation formulation and (b) node-equation formulation [8].

The equivalent circuit featuring magnetically coupled resonators is illustrated in Figure 3.2
(a). The coupling matrix is obtained by deriving an impedance matrix from a set of loop
equations using Kirchhoff's voltage law. The alternative electrically coupled circuit is
illustrated in Figure 2(b). According to Kirchhoff's current law, the coupling matrix is
obtained using an admittance matrix formulated as a set of node equations. A comprehensive
matrix [A] composed of coupling coefficients and external quality factors, irrespective of the
nature of the coupling, is introduced by the researchers in [10].
[ ] [ ] [ ] [ ]

(3.1)
[ ] [ ]

[ ]

( )

The identity matrix [U], the complex lowpass frequency variable p, the center frequency of
the filter [FBW], and p represent the center frequency and fractional bandwidth, respectively.
(i = 1 and n) represent the resonator's scaled external quality parameters. are the normalized
coupling coefficients between resonator q and p, respectively. The S-parameters can be

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computed utilizing the scaled external (i = 1 and n) and matrix [A], as described in reference
[11]:

( [ ] )
(3.2)
[ ]
√

3.4 NxN and N+2 Coupling Matrixes

The n+2 coupling matrix, which is an extension of the general (nn) coupling matrix, expresses
a two-portccircuit. Figure 3 presents a general n+2 couplingmmatrix.

Fig 3. 3 the n+2 Coupling Matrix

Resonator i is self-coupling, where the subscripts (s) and (l) denote the source and load,
respectively. Asynchronous tuning is achieved when a subset of the filter's entries take on
non-zero values. The n+2×n coupling matrix comprises supplementary columns and rows to
accommodate the source and load, respectively, in contrast to the general (n) coupling matrix.
In addition to establishing the coupling between the load and resonator, they also establish the
self-coupling between the source and the load. The following are the benefits of the n+2
coupling matrix due to the additional columns and rows [12]:
A single resonator may be coupled to multiple ports, whereas one port may be connected with
various resonators.
Coupling is a feasible process between the burden and the source. Thus, the (n×n) coupling
matrix lacks generality compared to the n+2 coupling matrix.
3.5 Physical realization of coupling matrix
Once the normalized coupling matrix [m] for a coupled resonator topology has been
determined, the prototype-normalization of the matrix [m] at the desired bandwidth can be
used to derive the actual coupling matrix [M] of a coupled-resonator device with the given

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specification [3]: Calculated by prototype de-normalization of the matrix [m] at a desired

bandwidth, as follows [3]:
g 0 g1 g n g n 1 (3.3)
QeS  , QeL 
FBW FBW

FBW (3.4)
M i ,i 1  , i  1 to n  1
gi gi 1

M i ,i 1 (3.5)
mi ,i 1  , i  1, ...n
FBW

3.6 Band pass filter based on an electronic circuit

The process of developing bandpass filters through electronic circuit design consists of
the subsequent stages:
 Specifications include the cutoff frequency, center frequency, filter type, filter order n,
filter ripple level, filter bandwidth BW, and relative bandwidth FBW.
 The process of ascertaining the values of the elements.
 The computation of the external quality factors (Qext and Qen), which are associated
with the elements gi and the coupling elements M(i,i+1).
 The combined element R0L0C0 of resonators is computed.
 Utilising lumped elements to compute impedances in a series of equivalent circuits.

3.7 Design Specifications and Initial Microstrip Filter: Model One

Table 1 shows the specifications imposed for this filter. The circuit to be synthesized is
defined in Environment as a distributed element bandpass filter.

Table 1 The following specifications of the filter.

Function Chebyshev
Order N=4
Center 1.2 GHz
Frequency
LAr (passband 0.04321 dB
ripple)
Bandwidth at -3 BW=154.1MHz
dB
Return loss: RL < - 20 dB
TFZ fzt1=1.08 GHz et fzt2=1.37
GHz

3.7.1 Frequency response of the ideal bandpass filter

Once the order filter is determined, knowing the maximum ripple of 0.04321 dB and the
specifications defined in the specifications, we obtain the coefficients gi (i = 1-4) of the band-
pass prototype of the type filter Chebyshev:

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g0 g1 g2 g3 g4 g5

1 0.9314 1.2920 1.5775 0.7628 1.2210

Using the specifications defined in the specifications, to calculate the relative bandwidth,
quality factor and coupling coefficients, respectively:
(3.6)

(3.7)

Coupling coefficients

0.1099 0.1229 0.1099 -0.0282

Fig 3.3 Coupling diagram of Four-poleecross-coupled microstrip resonator filter

Design the four-polemmicrostripccross-coupled filter based onnthe prescribeddgeneral

coupling matrix:

[ ] [ ]

3.7.2 Equivalentccircuit of theefilter:

The filter specifications translate into the desired coupling matrix elements M(i,i+1), Qen , and
Qext. Figure 5 shows the circuit diagram for this filter, where the lumped elements R 0L0C0
represent the four synchronously tuned resonators and the quarter-wave transmission lines.
These transmission lines have an electrical length EL = ±90° at the central frequency 0. The
corresponding design parameters for the bandpass filter are:
Lumped elements R0L0C0 of resonators:
(3.8)

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(3.9)

(3.10)

The impedances of the resonators:

(3.11)

When Z0=50Ω is the supplyiimpedance at the I/O ports:

,
,

The band-pass filter prototype operates using the characteristic impedance of the

positive quarter-wave resonator lines and a parallel R0L0C0 resonant circuit. The filter's

equivalent circuit is shown in Fig. 5 after calculating the elements of the series and parallel

branches.

Fig 3. 3 Band pass filter with lumped elements of order 4 with circuit resonant R0L0C0
parallel

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0
S21(dB)
S11(dB)
-5

-10

-15
S-parameters (dB)

-20

-25

-30

-35

-40

-45

-50
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
Frequency (GHz)

Fig 3. 4 Iideal response for the 4th order bandpass filter

As determined by AWR, Fig. 6 illustrates the ideal reflection and transmission response
of the equivalent circuit in lumped elements. Our bandpass filter's frequency response
indicates that its center occurs at 1.20 GHz. At 154.1 MHz, there are reflection losses of -20
dB; this is a narrow frequency band.

3.7.3 EM design of the band pass filter in planar technology

3.7.3.1 Cross-coupled filter

Figure 7 illustrates the schematic representation of our order 4 bandpass filter. This filter
operates within the frequency range of L [14] GHz and comprises four folded quarter-wave
resonators, each with its own folded wavelength. Consequently, the dielectric substrate has a
thickness denoted as h, and the filter topology is compact.
One end of the filter is short-circuited (via-hole grounding), while the other is open-circuited.
The input and output (I/O) resonators, numbered 1 and 4, are inter-coupled, with cross-
coupling occurring between them.

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Fig 3. 5 3D Structure. Four-poleecross-coupled microstrip resonatorrfilter

The simulation is performed on a "RO4003" substrate with the following properties: a

thickness of 0.813mm, a dielectric loss tangent of 0.0027, and a relative permittivity of
εr = 3.55.

Table 2 Dimensions of the band pass filter

Parameters x1 y1 H Hs L1 L2 L3

Values 25 52 0.017 0.813 10.9 11 22

Parameters L4 s1 s2 s3 T Wfeed w
Values 10.5 0.9 0.25 1.13 2.3 2 2

3.7.3.2 The filter dimensions

In the context of full-wave electromagnetic (EM) simulations for the design of a

microstrip filter, it is more efficient computationally to decompose the filter into distinct
components and have each component simulated by the EM simulator to derive the desired
design parameters by a general coupling matrix. They are subsequently combined to produce
the global filter's response. Demonstrating the efficacy of This CAD methodology for
developing narrowband filters is a four-pole cross-coupled filter, as illustrated in Figure 7.

3.7.3.3 Quality factor extraction

The external quality factor of the I/O resonator is determined in Figure 8. In the simulation,
the resonator is presumed to operate without any loss. A 50-ohm line is connected to port 1 at
a specific location denoted as t. To obtain the external quality factor Q en, port 2 is coupled
weakly to the resonator to determine the bandwidth of the response magnitude S21, which is -
3 dB.

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Fig 3. 6 Implementation for extraction of theeexternalqquality factor

3.7.3.4 Extraction of Qen from a volume resonator by group delay

The quality factor was established by analyzing the reflection coefficient group delay 11.
The result of the group delay outcome is shown in Fig. 9.

Fig 3. 6 Group delay of a volume resonator for quality factor sizing

The group delay values and quality coefficients at the resonant frequency have been
meticulously derived from the data presented in Figure 9, resulting in the following
estimations:
(3.12)

The table below presents the values of several iterations to calculate the quality factor by
varying the port location indicated by Table 3:

Table 3 Dimensions and Parameters Extraction

T f0(GHz)
0.2 0.9609 1.9 2.8678

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0.4 1.0672 2.72 4.5597

0.6 1.0852 2.78 4.7389
0.8 1.1273 2.83 5.0112
1 1.119 4.15 7.2965

Fig 3. 7 Design of the curve of Qen against t

Figure 10 illustrates the findings presented in Table 3. The design of the Qen-versus-t curve
can be deduced, as illustrated in Figure 10. In this instance, as t increases, the branch line is
redirected towards the resonator short-circuit (via-hole grounding); consequently, the source's
coupling becomes weakened, resulting in an increase in Qen.

3.7.3.5 Coupling coefficient extraction

The interaction between two resonators is contingent upon the distance that separates them.
Proximity between two resonators leads to disruption in their resonances due to the coupling
that links them. The HFSS simulator can determine the resonant frequency of both even and
odd modes. The coefficient for inter-resonator coupling, denoted as (k), can be calculated
using the following formula:
(3.13)

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Fig 3. 8 Arrangement for extracting coupling coefficient M14

An electromagnetic simulation layout of the figure is employed to derive the cross-coupling

between resonators 1 and 4 from the filter configuration depicted in Fig 3.9. The coupling
between them, in this instance, is governed by the electric field and is therefore referred to as
electric coupling. To attain a pair of transmission zeros at finite frequencies, M 14 and M23
must have opposite signatures. Consequently, this implementation of cross-coupling is
essential. As illustrated in simulated Fig3.10, the structure's frequency response comprises
two resonance peaks utilized to assess the coupling coefficient between the initial and fourth
resonators.
0.00
Curve Inf o

-5.00 dB(S(1,1))
dB(S(2,1))

-10.00
S-parameters (dB)

-15.00

-20.00

-25.00

-30.00

-35.00
0.75 0.88 1.00 1.13 1.25 1.38 1.50
Freq [GHz]

Fig. 12
Fig 3. 9 Response Electromagnetic of the 2nd order bandpass filter

3.7.3.6 Effect of the spacing Soon, the frequency response of the

filter

The first parametric study shows the influence of the S1 spacing on the simulation
results. When the space S1 increases (space between the first and second resonators), we can

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see that the bandwidth decreases, which implies a narrower bandwidth. We can also see that
the best result we obtained is S1=0.9 mm of reflection losses below -25 dB and bandwidth
between 1.13GHz and 1.28GHz. The variation of the coefficients of the filter matrix [S] as a
function of frequency for these different values is shown in the graphs of Fig. 13.

Fig 3. 10 Frequency response of the CCM filter against the design goals after optimization

Fig 3. 11 HFSS simulation results proposed with different values of S3

Fig 3.12 illustrates the simulation results in 11 and 21 proposed with different S3 values. This
parametric study shows the influence of the ‗S3‘ spacing between the first and fourth
resonators. As the ‗S3‘ gap increases, the upper cutoff frequency and the band gap point are
shifted slightly downwards, which means a narrower bandwidth.

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Fig 3. 12 HFSS simulation results proposed with different values of S2.

Fig3.13 shows the results of the HFSS simulation proposed with different values of S2. This
second parametric study shows the spacing 'S2' influence between the second and third
resonators. When 'S2' increases, the lower cutoff frequency stays the same, and the upper
cutoff frequency moves lower. Therefore, 'S2' is the second parameter affecting the HFSS
result: when 'S2' increases, the bandwidth is reduced.

0
CM S21(dB)
CM S11(dB)
HFSS S21(dB)
-10 HFSS S11(dB)

-20
|S11| & |S21| [dB]

-30

-40

-50

-60
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
Frequency [GHz]

Fig 3. 13 Final response of the cross-coupled bandpass four-pole filter after optimization
compared with the ideal response

In Figure 3.14, it is observed that the proposed filter has attained a filtered bandwidth ranging
from 1.13 GHz to 1.28 GHz, centered at 1.2 GHz. Within the passband, the insertion loss
measures approximately -1.1 dB, while the return loss registers below -24.7 dB, achieving a
value of -44.9 dB at the frequency of 1.18 GHz.

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Fig 3. 14 Distribution of the electric field at 1.2 GHz

Exceptional rejection levels are achieved beyond the filtered band region, with
attenuation measuring below -39.3 dB at the 1.08 GHz frequency and hitting a peak rejection
of -55.8 dB at 1.37 GHz. The designed filter, featuring two finite transmission zeros,
demonstrates the anticipated frequency response. Figure 3.16 compares the simulated
response of the proposed four-pole microstrip bandpass filter and the ideal coupling matrix
response, showcasing a remarkable alignment between the two. Additionally, Figure 17
displays the electric field distribution of the proposed bandpass filter.
Table 3.4 Comparison of characteristics between the Microstrip filter and the recently
introduced Microstrip filters.
Refs. Ordre Center frequency Bandwidth in MHz Retun loss (RL) dB
in GHz
[13] 04 1.26 75.6 18 dB
Work in HFSS 04 1.20 153.9 24.7dB
Work in AWR 04 1.20 154.1 25 dB

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Table 3.4 presents a performance comparison between the proposed work and recently
published works. The table indicates that the microstrip filter under consideration possesses
favorable parameters, including minimal insertion loss and substantial isolation between two
channels at the operational frequencies. This characteristic is critical in contemporary
communication systems. As a result, the overall performance of the proposed circuit is
comparatively superior to that of the alternative filters.

3.7.4 Fabrication and Experimental Results

A cross-coupled filter has been fabricated on a 30 × 20 mm2 substrate. A substrate with a

relative permittivity of 3.55 and thickness of 0.508 mm is used.

(a) (b)

(c) (d)

Fig 3. 15 Photograph of the proposed filter. (a).(c) Top view, (b) layer view.(d) the
experimental measurement.

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0
-5 S11
-10 S21
-15
-20
S Parameters (dB)

-25
-30
-35
-40
-45
-50
-55
-60
-65
-70
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Frequency (GHz)

Fig 3. 16 Measured S-parameter response.

Figure 3.16 illustrates the simulated and measured outcomes of the cross-coupled filter.
Figure 3.17 shows that the suppression of out-of-band frequencies in the measured data
closely mirrors the results obtained through simulation. Comparable responses in the pass-
band region are observed in simulations and actual measurements. The simulation data
reveals:
 Bandwidth: 0.151 GHz
 Centre Frequency (fC): 1.2 GHz
This comparison of simulated and measured findings holds significant importance in
confirming the efficacy of the filter design. The alignment between simulation and
measurement results, particularly in out-of-band suppression and pass-band response,
indicates the filter's functionality as intended.

3.8 Design Specifications and Initial Microstrip Filter: Microwave

filter design theory Dual-Mode Open-Loop Resonators

3.8.1 Introduction:

Microwave devices are increasingly prevalent today, making precise simulation before
implementation essential. The second part of the chapter presents the design results of
microwave filters achieved using the HFSS (High-Frequency Structure Simulator) software.

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This comprehensive tool facilitates the design of filters and the plotting of their frequency
responses. In the second part of this chapter, we explore a new type of miniature microstrip
dual-mode resonator for filter applications. The open-loop resonator is renowned for its
versatility in designing cross-coupled resonator filters. Several filter examples demonstrate
the applications of this innovative dual-mode resonator.

3.8.1 Bandpass filter based on an electronic circuit:

The design of bandpass filters based on electronic circuits is based on the following steps:

Specification: filter type, filter order n, filter ripple level, filter bandwidth BW, relative
bandwidth FBW, cutoff frequency, and centre frequency determination of the values of the
elements gi.

 Calculation of the coupling elements Mi,i+1 and the external quality factors (Qe1 and Qen),

which are linked to the elements gi.

 Calculation of localised elements R0L0C0 of resonators.

 Calculation of series impedances of equivalent circuits with lumped elements.

3.8.2 Dual-Mode Open-Loop Resonators :

3.8.2.1 two-pole Dual-Mode Open-Loop Resonators

 example A

 Filter specifications:
Table 3.1: Specifications

Settings Values
Filter order (Number of poles) 02
Type of approximation Tchebychev
Central frequency f0 1.063GHz
Bandwidth (BW) à –3 dB 0.120 GHz
Amplitude of l‘ondulation LAr (passband ripple) 0.1 dB
Attenuation dB
TFZ fzt1=1.1667 GHz

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 Frequency response of the ideal bandpass filter:

Once the order of the filter is determined, knowing the maximum ripple of 0.1 dB and the
specifications defined in the specifications, we obtain the coefficients gi (i=1-2) of the
bandpass prototype of the type filter Chebyshev:

1 0.8431 0.6620 1.3554

Using the specifications defined in the specifications to calculate:

Relative bandwidth:
(3.14)

Quality Factor:
(3.15)
(3.16)

The resonant frequency :

f01= f0 ( ) ( ) (3.17)

f02= f0 ( ) ( ) (3.18)

Coupling coefficients:
(3.19)
√

0.1578 0.1578

Fig 3. 17 Coupling structure for a two-pole dual-mode open-loop resonator filter [1]

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 Coupling matrix
 Normalized:
Where node S denotes the source or input, node L denotes the load or output, and
nodes 1 and 2 represent the odd and even modes [1], respectively. For this coupling
structure, an n + 2 coupling matrix is given by:

m=[ ] (3.20)

m= [ ] (3.21)

it should be noted that mL1 =−mS1 for the odd mode and mL2 = mS2 for the even mode. Using
the formulations introduced in [1] .
 Dénormalized:
After determining the normalized coupling matrix [m] for a coupled resonator topology, the
actual coupling matrix [M] of a coupled resonator device with given specification can be
calculated by prototype de-normalization of the matrix [m] at a desired bandwidth, as follows:
M i,i+1  m i,i+1 .FBW For i=1 to n-1

3.8.2.2 Filter equivalent circuit

The filter specifications result in desired coupling matrix elements Mi,i+1,Qe and Qs .
The circuit diagram for this filter is shown in Figure 3.19 , where the lumped elements R0L0C0
represent the four synchronously tuned resonators and quarter-wave transmission lines, which
have an electrical length EL = ±90° at frequency central 0, The corresponding design
parameters for the bandpass filter are:
The localized elements R0L0C0 of resonators:
(3.22)

(3.23)

Resonator impedances :

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(3.24)

Lorsque est l'impédance d‘alimentation au niveau des ports d'E/S :

3.8.2.3 Equivalent scheme under AWR:

The bandpass filter prototype operates using the characteristic impedance of the
positive two-wave resonator lines and a parallel R0L0C0 resonant circuit. After calculating the
elements of the series and parallel branches, the equivalent circuit of the filter is illustrated in
Figure 3.19:

Fig 3. 18 bandpass filter with localized elements of order 2 with circuit

resonant R0L0C0 parallel.

Fig 3. 19 Ideal response of the Chebyshev bandpass filter of order 2

The ideal response in transmission and reflection of the equivalent circuit in lumped elements,
analyzed with AWR is shown in Figure (3.20)
3.8.2.4 EM design of the bandpass filter in planar technology

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Figure 3.21 shows the basic topology of a dual-mode microstrip open-loop resonator.

The dual-mode resonator consists of an open loop with an open gap, g.

The open loop has a line width of W1 and assize of L1a × L1b. A loading element with an

open stepped impedance stub is tapped from inside onto the open loop. The loading element

has dimensions of L2a and W2a for the narrow line section and L2b and W2b for the wide-line

section. The resonator is coupled to the input and output (I/O) ports with a feed structure with

a W line width and coupling spacings. The port terminal impedance is W feed. The first two

resonating modes, existing in the resonator of Figure 3.21, are referred to as the odd and even

modes.

Fig 3. 20 :topology of a dual-mode microstrip open-loop resonator

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0.00

-5.00 Curve Info

dB(S(1,1))
Setup1 : Sw eep
dB(S(2,1))
-10.00 Setup1 : Sw eep

-15.00
S-parameters (dB)

-20.00

-25.00

-30.00

-35.00

-40.00

-45.00
0.50 0.75 1.00 1.25 1.50
Freq [GHz]

Fig 3. 21 : EM-simulated responses of the filter with HFSS

We present in Figure 4.5, the simulation results of the filter ( two pole dual mode open-
loop) and transmission obtained with the HFSS software.
The central frequency of this filter is of order 3 and f = 1.063 and the bandwidth
is approximately 0.120 GHz. The frequency response shows that|S11| is less than -20dB
between 0.89 GHz and 1.02 GHz.
3.8.2.5 The Electromagnetic design of the applied filter

This geometric configuration is simulated on a ―RO3010‖ substrate having a relative

permittivity εr=10.2, a dielectric loss tangent 0.017 and a thickness of h= 1.27 mm.

Table 3.2: planar bandpass filter dimensions for example A

Settings x1 y1 H Hs L1a L1b L2a L2b W2a W2b s g Wfeed W1

Values 20 30 0.017 1.27 15.1 15 5.5 5.5 0.7 8.1 0.2 0.9 2 1.5

 Example B

 Filter specifications:

Tqble 3.3: Specifications B

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Settings Values
Filter order (Number of poles) 02
Type of approximation Tchebychev
Central frequency f0 1.0155GHz
Bandwidth (BW) à –3 dB 0.120 GHz
Amplitude of l‘ondulation LAr (passband ripple) 0.1 dB
Attenuation dB
TFZ fzt1=0.8409 GHz

Frequency response of the ideal bandpass filter:

Once the order of the filter is determined, knowing the maximum ripple of Ar= 0.1 dB
and the specifications defined in the specifications, we obtain the coefficients gi (i=1-2) of the
bandpass prototype of the type filter Chebyshev:

1 0.8431 0.6620 1.3554

Using the specifications defined in the specifications to calculate:

Relative bandwidth :
(3.25)

Quality Factor:
(3.26)
(3.27)

The resonant frequency :

(3.28)
( ) ( )

( ) ( )

Coupling coefficients:

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Chapter 3 Development and Optimization of Compact Filters Microwave

(3.29)
√
Qe1 M22 Qe2
10.398 0.1744 0.1744 25.9340

Fig 3. 22 Coupling structure for a two-pole dual-mode

open-loop resonator filter [1].

 Coupling matrix
 Normalized:
where the node S denotes the source or input, the node L denotes the load or output, and
the nodes 1 and 2 represent the odd and even modes [1] , respectively. For this coupling
structure, an n + 2 coupling matrix is given by:

m= [ ] (3.30)

it should be noted that mL1 =−mS1 for the odd mode and mL2 = mS2 for the even mode. Using
the formulations introduced in [1]
 Filter equivalent circuit:
The localized elements R0L0C0 of resonators:
(3.31)

(3.32)

Resonator impedances :

(3.33)

Lorsque est l'impédance d‘alimentation au niveau des ports d'E/S :

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3.8.2.6 Equivalent scheme under AWR

The bandpass filter prototype operates using the characteristic impedance of the

positive two-wave resonator lines, and a parallel R0L0C0 resonant circuit. After calculating the

elements of the series and parallel branches, the equivalent circuit of the filter is illustrated

in Figure 3.23:

Fig 3. 22 : bandpass filter with localized elements of order 2 with circuit

resonant R0L0C0 parallel.

Fig 3. 23 Ideal response of the Chebyshev bandpass filter of order 2 .

The ideal response in transmission and reflection of the equivalent circuit in lumped
elements, analyzed with AWR is shown in Figure (3.24).

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3.8.2.7 EM design of the bandpass filter in planar technology

Figure 3.25 displays the layout of the designed filter. The two small open stubs attached
to the loading element inside the open loop are deployed for additional control of the even-
mode characteristics, which also allow efficient utilization of the circuit area inside the open
loop

Fig 3. 24 topology of a two-port microstrip dual-mode open loop filter

0.00

-10.00
S-parameters (dB)

-20.00
Curve Info
dB(S(1,1))
Setup1 : Sw eep
dB(S(2,1))
-30.00 Setup1 : Sw eep

-40.00

-50.00
0.50 0.75 1.00 1.25 1.50
Freq [GHz]

Fig 3. 25 EM-simulated responses of the filter with HFSS

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We present in Figure 3.26, the simulation results of the filter ( two pole dual mode
open-loop) and transmission obtained with the HFSS software. The central frequency of this
filter is of order 3 and f = 1.0155 and the bandwidth is approximately 0.120 GHz. The
frequency response shows that|S11| is less than -20dB between 0.89 GHz and 1.02 GHz.
3.8.2.8 The Electromagnetic design of the applied filter:
This geometric configuration is simulated on a ―RO3010‖ substrate having a relative
permittivity εr=10.2, a dielectric loss tangent 0.017 and a thickness of h= 1.27 mm.
Table 3.4 :the filter dimensions for example B
Settings x1 y1 H Hs L1a L1b L2a L2b W2a W2b s g Wfeed W1 W3b

Values 20 30 0.017 1.27 15.1 15 5.5 5.5 0.7 8.1 0.2 0.9 2 1.5 4.9

3.8.3 The dimensions:

The first two resonating modes, existing in the resonator of Figure 3.25 , are referred to
as the odd and even modes,Depending on the dimensions of the resonator these two modes
can have the same or different modal frequencies and, for the latter, the modal resonant
frequency of one mode can be either higher or lower than that of the other one. The
characteristics of the dual-mode open-loop resonator are investigated by full-wave
electromagnetic (EM) simulation. To excite the resonator, two ports are weakly coupled to the
resonator with a large spacing s

4.3.5 Field distribution:

The HFSS software was used to perform a frequency analysis in the (1and 2) GHz band L of

this structure. Figures 3.27 illustrate the distribution of the field lines. electrical bandpass

filter and electric field mapping of the filter

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Fig 3. 26 Distribution of the electric field of the 3th order bandpass filter.

3.8.3 Four-pole Dual-Mode Open-Loop Resonators :

 Filter specifications:
Settings Values
Filter order (Number of poles) 03
Type of approximation Tchebychev
Central frequency f0 1.05GHz
Bandwidth (BW) à –3 dB 0.115 GHz
Amplitude of l‘ondulation LAr (passband ripple) 0.1 dB
Attenuation dB
TFZ fzt1=1.0955 GHz
Table 3.6: Specifications C
3.8.3.1 Frequency response of the ideal bandpass filter
Once the order of the filter is determined, knowing the maximum ripple of Ar= 0.1 dB and the
specifications defined in the specifications, we obtain the coefficients gi (i=1-2) of the
bandpass prototype of the type filter Chebyshev:
N g1 g2 g3 g4
3 1.0316 1.1474 1.0316 1.0000
Using the specifications defined in the specifications to calculate:
Relative bandwidth:
(3.34)

Quality Factor:
(3.35)

Coupling coefficients:

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(3.36)
√
Qe1 M22 M33 Qe2
12.64 0.1744 0.1744 0.1744 27.1475

Fig 3. 27 . Order 3 coupling graph of the filter [1]

Coupling matrix
 Normalized:
The extracted coupling matrix is given by :

m= (3.37)

[ ]
3.8.4.2 Filter equivalent circuit
The localized elements R0L0C0 of resonators:
(3.38)

(3.39)

Resonator impedances:
(3.40)

Lorsque est l'impédance d‘alimentation au niveau des ports d'E/S :

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Equivalent scheme under AWR

The bandpass filter prototype operates using the characteristic impedance of the positive

two-wave resonator lines and a parallel R0L0C0 resonant circuit. After calculating the

elements of the series and parallel branches, the equivalent circuit of the filter is illustrated in

Figure 3.29 .

Fig 3. 28 : bandpass filter with localized elements of order 3 with circuit

$resonant R0L0C0 parallel.

Fig 3. 29 Ideal response of the Chebyshev bandpass filter of order 3.

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Figure (3.30) shows the ideal transmission and reflection response of the equivalent circuit
in lumped elements analyzed with AWR. Our bandpass filter's frequency response appears to
be centred on 1.05 GHz.
3.8.4.3 design of the bandpass filter in planar technology
Figure 3.31 a displays the layout of the designed filter. The two small open stubs attached
to the loading element inside the open loop are deployed for an additional control of the even-
mode characteristics, which also allow an efficient utilization of the circuit area inside the
open loop

Fig 3. 30 Structure of filter with three resonators.

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0.00

Curve Info
dB(S(1,1))
-10.00 Setup1 : Sw eep
dB(S(2,1))
Setup1 : Sw eep

-20.00
S-parameters (dB)

-30.00

-40.00

-50.00

-60.00
0.50 0.75 1.00 1.25 1.50
Freq [GHz]

Fig 3. 31 EM-simulated responses of the filter with HFSS

We present in Figure 3.32, the simulation results of the filter ( four pole dual mode open-

loop) and, transmission obtained with the HFSS software. The central frequency of this filter

is of order 3 and f = 1.05 and the bandwidth is approximately 0.115 GHz. The

frequency response shows that|S11| is less than -20dB between 0.9277 GHz and 1.0559 GHz.

3.8.4.5 Influence of geometric parameters

a) Variation of via hole diameter “W1”:

The following curves present the influence of some geometric parameters on the

frequency response of the parameters S. Figure 3.32 shows the variations of the coefficient

reflection and transmission as a function of frequency, taking the diameter of the via (W1) as

set to vary. In our simulation, we chose three diameter values: W1= 1.51

mm, W1 = 1.71 mm (this is the reference model), W1= 1.91 mm.

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Fig 3. 32 Simulation results of the planar technology cell proposed with different -W1
values

b) The effect of the spacing S on the frequency response of the

filter:
The first parametric study shows the influence of the S1 spacing on the simulation
results (Figure 3.33). When the S1 space increases (space between the first resonator and the
second resonator), we can see that the bandwidth decreases, which implies a narrower
bandwidth. We can also see that the best result we obtained is at the value S 1=0.62 mm of
reflection losses below -20 dB and bandwidth between 0.925 GHz and 1.075 GHz.

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Fig 3. 33 : Results of proposed HFSS simulation with different values of S1

Figure 3.34 shows the proposed HFSS simulation results with different values of S2.
This second parametric study shows the influence of the ‗S2‘ spacing between the second and
third resonators. When 'S2' increases, the lower cutoff frequency remains the same and the
upper cutoff frequency moves lower. Therefore, 'S2' is the second parameter that affects the
HFSS result: when 'S2' increases, the reduced bandwidth

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Fig 3. 34 Résultats de simulation HFSS proposée avec différentes valeurs de S2

The electric field distribution of the proposed bandpass filter is shown in Figure (3.36)

Fig 3. 35 Distribution of the electric field of the 3rd-order bandpass filter.

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Figure (3.36) illustrates the electric field distribution of the 3rd-order bandpass
filter. We notice that the maximum of the E fields is represented by dark colors (high
intensity)

In all cases, only one modal resonant frequency is affected, while the other is hardly
changed. It can be shown that the mode whose resonant frequency is being affected is
an even mode.

3.9 Conclusion

This chapter concludes with an analytical method using Chebyshev polynomials to calculate
the synthesis parameters of a bandpass microstrip cross-coupled filter. The technique involves
the following steps: determining the coupling matrix, establishing the quality factor (Q-factor)
coefficients crucial for filter performance, and defining the initial geometric dimensions.
Additionally, a novel miniature microstrip dual-mode resonator for filter applications is
explored. This open-loop resonator stands out for its versatility in designing cross-coupled
resonator filters. Various examples highlight the innovative applications of this dual-mode
resonator, demonstrating its potential to advance filter design technology.

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3.10 References

[1] Hunter IC,BillonetL ,Jarry B, Gullion P. Microwave filters—applications and

technology. IEEE Trans Microwave Theory Tech 2002; 50 (3) :794–805.
[2] Kuo JT,Chen SP,Jiang M.Parallel-coupled microstrip filters withover-coupled end stages
for suppression of spurious responses. IEEE Microwave Wireless Components Lett
2003;13(10):440–2.
[3] Lopetgi T,Laso MAG,Hemandez J, Bacicoa M, BenitoD, Garde MJ,Sorolla M,
Guglielmi M. New microstrip wiggly- line filters with spurious passband suppression.
IEEE Trans Microwave Theory Tech2001;49(9):1593–8.
[4] Castillo MD,Ahumada V,Marte lJ, Medina F. Parallel coupled microstrip filters with
ground-plane aperture for spurious band suppression and enhanced coupling. IEEE Trans
Microwave Theory Tech2004;52(3):1082–6.
[5] Special Issue on RF and Microwave Filters, Modeling and Design, Int. J. RF Microwave
Computer-Aided Eng. 17, 2007.
[6] Castillo MD, Ahumada V,Martel J,Medina F.Parallel coupled microstrip filters with
floating ground-plane conductor for spurious band suppression. IEEE Trans Microwave
Theory Tech 2005;53(8):1823–8.
[7] Kuo JT, Jiang M, Chang HJ. Design of parallel-coupled microstrip filters with
suppression of spurious resonances using substrate suspension. IEEE Trans Microwave
Theory Tech 2004;52(1):83–9.
[8] Rezaei, A., Noori, L., & Jamaluddin, M. H. (2019). Novel microstrip lowpass-bandpass
diplexer with low loss and compact size for wireless applications. AEU-International
Journal of Electronics and Communications., 1(101), 152–9. https:// doi. org/ 10. 1016/j.
aeue. 2019. 02. 005
[9] Pozar, D. M. (1998). Microwave engineering handbook (2nd ed., p. 1998). New York:
Wiley.
[10] Lee SY, Tsai CM. New cross-coupled filter design using improved hairpin resonators.
IEEE Trans Microwave Theory Tech 2000;48(12):2482–90.
[11] Mouloud Challal, Kenza Hocine, Ali Mermoul (2019) A Novel Design of Compact
Dual‑Band Bandpass Filterfor Wireless Communication Systems. Wireless Personal
Communications https://doi.org/10.1007/s11277-019-06648-9.
[12] Mohammad Moein Shirkhar, Sobhan Roshani (2021). Design and Implementation of a
Bandpass–Bandpass Diplexer Using Coupled Structures. Wireless Personal
Communications. https://doi.org/10.1007/s11277-021-09002-0
[13] Hong JS, Lancaster MJ. Design of highly selective microstrip bandpass filters with a
single pair of attenuation poles at finite frequencies. IEEE Trans Microwave Theory and
Tech 2000;48(7):1098–107.
[14] Hong JS, Lancaster MJ. Coupling of microstrip square open- loop resonators for cross-
coupled planar microwave filters. IEEE Trans Microwave Theory Tech
1996;44(12):2099–109.

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