# The Scope And Limitation Biology Essay

Published:

This essay has been submitted by a student. This is not an example of the work written by our professional essay writers.

Nowadays, various applications of modern communication system which broadly used in wireless and mobile communication services has demand high performance in microwave filter and RF subsystem. The demand of microwave filter performance is being emphasized more and more, thus become as an important domain in the development of communication system. Thereby, the electrical performances of the microwave filter become greater and exhibits better selectivity. In addition, due to the state-of-the-art of the communication components and systems, the compact size of the filter is being emphasized in order to reduce the expense while maintaining their performance. However, due to the simplicity of fabrication, low cost in fabrication process and compatibility with various circuit components, the comprehension of planar transmission line filters appears as a great deal of interest and becomes one of the media in the step of designing a filter.

In the development of microwave filters, planar transmission line offer a great advantages which very selective, lower loss and not too bulky which convenient for many applications. In the process of design filters, the degree of selectivity is one of the most important parameters which lead to the better electrical characteristics.

In fact, various topologies have been invented based on this requirement. Most of these topologies, concerned on the filter which introduced a transmission zero at both sides of the passband response. This type of filter called as an Elliptic or pseudo-elliptic filters which function to control the characteristic of the attenuation in the stopband by positioning the transmission zero at a certain place in order to satisfy the specifications.

In addition, by introducing the transmission zero in the passband response, the level of losses slightly can be reduced. Furthermore, it is also convenient for implementation a filter of higher order degree. Therefore, this concept is effective to improve the electrical performance of the filter.

In microstrip technology, many filters based on the coupled line have been proposed and introduced numerous advantages. Some of the advantages of this coupled line filter are the possibility of achieving a wide range of filter fractional bandwidth. Therefore, in this job of study, we introduced a new topology which generates two transmission zeros and two poles in the response that have lower losses and wide bandwidth. This structure is applicable for wideband applications.

## Problem Statement

High demand in wireless communication system requires new specifications for microwave components to cater larger bandwidth for larger data transfer with lower cost of production and smaller size. This scenario led to the development of microwave filters which requires better performance in term of bandwidth, selectivity, high rejection level and so on. In this circumstance, various types of wideband bandpass filters based on the coupled line have been developed and emerged as respond to new demand of wireless communication systems such interdigital filter, combline filter, ring resonator filter, parallel coupled line filter and ect.

In general, numerous filter designs have different advantages and disadvantages. For instant, interdigital filter is quite popular because the compactness of the circuit size and having the nearest spurious response at 3f0. This filter is also convenient for wideband bandwidth. However, this type of filter requires vias for microstrip realization and quite sensitive to the element variation and more lossy as compared to other coupled line filter. While, combline filter provides high rejection capabilities in its performance. Besides, this combline filter is more practically for narrow bandwidth.

In addition, the ease of fabrication is also required in realization of filter. In practically, to design a wide bandwidth which more than 40 percent, the small strip width and the gap spacing are required to ensure the tight coupling. However, for parallel coupled line filter, this factor makes the designed difficult to realize in technology fabrication.

Therefore, this study is focused on the dual-path coupled line filter which are satisfying the requirements that need in the application of communication system. In the recent report, the existing of transmission zeros at the both side of the passband leads to the better selectivity. Besides, in the design of dual-path coupled line filter, the appearance of both transmission zeros make the passband response look very sharp. This topology provides wide bandwidth which more than 40 percent and easy to fabricate. The synthesis of this topology was developed which help to fixing the whole filter characteristics (central frequency, transmission zeros frequencies and bandwidth).

## Scope and Limitation

The scope of this study is divided into three parts. The first part is to explore all the possible configurations of identical dual-path coupled lines filters with very high performance in order to respond to the need of optimizing frequency spectrum usage while the size and cost of devices become important issues.

The second part is focusing on the arrangement of this resonator in obtaining the simpler configuration in order to define its core synthesis equations. This new topology and its synthesis lead to better performance in the frequency response and design simplicity.

The third part is to enhance the overall performance of the resonator using coupled-lines-based filter by increasing the number of order in the passband responses. The improvement in terms of highly selectivity of frequency response with wider bandwidth is aimed to be achieved.

For limitation of this study; in the design the size of the coupling gap between coupled lines is the critical parameter that needs to be considered. The narrower coupling gap can provide low insertion losses but difficult to realized in technology fabrication. For ease fabrication the minimum optimal gap is set to 100-Âµm.

Another limitation is the edge-gap between the successive coupled lines allows the parasitic effect in their frequency response and affects the electrical performances of the resonator. However, it can be controlled by maintaining the separation between the coupled lines.

## Objectives

This study embarks on the following objectives:

To introduce a new topology of wideband bandpass filter using dual-path coupled lines.

To develop the mathematical modeling and creation of the synthesis of new topology.

To implement the basic concept of dual-path coupled line bandpass filter for implementation high-order filter.

To realize the implementation of dual-path coupled line filter on technology fabrication.

## The work, Methodology and Contribution

The work which we introduced in this study focused on three main points which is the implementation of a new topology of coupled line filter, the development of the mathematical modeling or its synthesis and the working of this topology for the development of new structures for high order of filters.

To achieve the objectives of this study, there are three main stages will be adopted. At first stage we emphasizes on the state of the art of the planar microwave filter. In this stage, we represent topologies and review on the basic synthesis method of the coupled line. This includes the investigation of all the fundamental concepts of coupled line filter in wide band. We also represent their implementation in different classical topologies. Finally, we represent the resolutions for filter realization in technology fabrication.

In the second stage, we concerns on our intention on the implementation of filter based on the line coupled. The topology of dual-path filter consists of coupled line which is connected in series. The circumference of this topology is equal to the length of wave of the coupled line and introduced dual resonances in the passband. The frequency of zeros is placed at both sides of the passband. The even- and odd-mode impedance of the coupled line, Z0e and Z0o, is use to control the matching level of the topology filter. Therefore, the development of the total synthesis is introduced for the case of second order of filter. At this point, the elements of control of the electrical characteristics are able to identify. All steps that performed in building of synthesis allow determining the impedances of the essential elements of filter according to the electrical characteristics to be attained.

The last stage of this work of thesis will focus on the implementation of high order filters based on the basic concept dual-path line coupled. The characteristic of filter order is increase by increasing the number of quarter of wave of coupled line. Then it will focus on Electromagnetic (EM) simulation to optimize and demonstrate the characteristics and performances of the filter. At this stage, the final design will be fabricate on planar technologies such as microstrip, coplanar waveguide and multilayer environment. It is then followed by the measurement of the characteristic and response to validate the theory.

Finally, this work of thesis will be concludes followed by recommendations for future work.

## CHAPTER 2: STATE OF THE ART OF MICROWAVE PLANAR FILTER

## Introduction

Microwave filter is an essential component in a huge variety of electronic systems including the telecommunication system such as in mobile radio, satellite communication and radar. This component used to select or reject signal at different frequencies. The microwave filter contained some components or a part that functions are depends on the specific applications. Besides, the desired frequency response can be obtained by the use of filters component. Filters can be designed at low cost with precise frequency response as desired and also can be fabricated either from lumped element or distributed element or combination of both elements. Thus, these all component of microwave filter is required for filter realization. In addition, the choices of topologies is depends on the characteristic of the filters such as chebyshev or elliptic, size and power handling.

For filter realization, there are two general steps are required; synthesis and technological implementation. The synthesis of filter plays important roles because it allows to identify the essential elements in the topology circuit and to define the electrical characteristics of the filter. The circuit elements of the topology include their electrical length and the value of impedances. While, the primary parameters of interest in the electrical characteristic are the bandwidth, level of rejection, attenuation, and the frequency range. At this stage, the circuit parameters are defined and the electrical characteristics can be controlled by these parameters for example, the characteristic impedance of coupled lines. In the technological implementation, the choices of substrates are very important. In practically, choosing the substrates are depends on the few factors such as size, higher-order modes, surface wave effects, dielectric loss, power handling and ect.

Therefore, in this chapter will discuss the components that contribute in the synthesis process on the development of new topologies of coupled lines filter which introduced filter with better performance in terms of selectivity and their bandwidth.

## Microwave Planar Filter Design, Topology and Technology

This section will begin with a general discussion of the microwave filter theory and design which consist of synthesis method, filter implementation using transformation tools, coupled lines topologies, and planar technologies for filter implementation.

## Microwave Filter Theory

Microwave filter is broadly use in many applications. It consists of a components or parts where the function is depends on the require specifications of the application. Microwave filter mostly used to control the frequency response at a certain point in microwave system by providing transmissions at frequencies within the passband of the filter and attenuation in the stopband of the filter [1]-[5]. The most common filter can be categorized into four main types which are:

Low Pass Filter

High Pass Filter

Band Stop Filter

Band Pass Filter

The frequency responses of these types of filter are illustrates in Fig. 2.1. In addition, an ideal characteristic of these filters shows zero insertion loss, constant group delay over the desired passband and infinite rejection. In practically, these characteristics are only achieved in high frequency limit for any given practical filter structure where its characteristics will degenerate due to the junction effects and resonances within the elements [2]-[3].

(b)

(c) (d)

Figure 2.1: Four types of filter characteristics (a) Low pass, (b) High Pass, (c) Band Stop, (d) Band Pass [3]

Generally, to design a filter, the following parameters are defined to characterise its frequency characteristics:

Pass bandwidth

Stop band attenuation and frequencies

Input and output impedances

Return loss

Insertion loss

Group delay

The amplitude of the filter response is the most important parameter in designing a filter which concentrates on the insertion loss and frequency characteristics.

## Synthesis Method Of Filter Design

In microwave filter design, the most popular techniques of synthesis were used which is using parameter method and the insertion loss method. Based on these two methods, insertion loss method is more preferable and suitable for filter that is going to synthesize because it gives complete specifications of frequency characteristic which over the entire pass and stop bands [2]- [14]. Basically, the basic design of microwave filter such as low pass, high-pass, band-pass and band-stop operated at arbitrary frequency bands and between arbitrary resistive loads. Such basic filters are designed based on the prototype low pass filter through frequency transformation and element normalization. Normally, the element values are determined based on the low pass response approximation such as Butterworth (called as Maximally flat or Binomial) and Chebyshev or equal ripple passband response. The response shapes of the filter are controlled by the values of the element coefficients (g0, g1â€¦ gn+1) as defined in Fig. 2.2 low pass filter prototype.

Figure 2.2: Low pass filter prototype [3].

The Butterworth filter utilizes a maximally flat frequency response which is no ripple in the passband. Using this approximation the attenuation in the stopband constantly increased. The equation of insertion loss for Butterworth low pass filter is given by:

The passband range is from = 0 to = and the maximum insertion loss in the passband is 3 dB at where equal to 1. The angular frequency of the passband edge, Ï‰`c and resistance r are normalized equal to unity, respectively where Ï‰`c is measured in radians/second. The characteristic response of Butterworth is illustrates in Fig. 2.3.

Figure 2.3: The characteristic response of insertion loss for Butterworth low pass filter [3]

Chebyshev filters have more passband ripple or stopband ripple compared to Butterworth filter. It is also called as equal ripple or minimax which have a steeper passband edge which can minimize the error between idealized and actual filter characteristic over the range of the filter. However, it exhibits a ripple in the passband. The approximation of insertion loss for Chebyshev low pass filter is given by [2]-[5]:

where:

n = the degree of approximation which called as number of reactive elements

am = the ripple factor

Tn () = Chebyshev polynomial of degree n

In this case, the insertion loss oscillates between 1 and in the passband response. So that becomes at the cut off frequency and increases monotonically beyond stopband . The characteristic response of Chebyshev are illustrates in Fig. 2.4.

Figure 2.4: The characteristic response of insertion loss for Chebyshev low pass filter [3]

The element values of low pass ladder network can be derived using both type of filter Butterworth and Chebyshev, respectively. The normalization values can be calculated based on the following equation.

Prototype element values for Butterworth [2]:

for all n

The response of n order for Butterworth function is depicted in Fig. 2.5.

Figure 2.5: The characteristic response of n order for Butterworth function [3].

Prototype element values for Chebyshev [2]:

for all n odd

for all even n

where;

The response of n order for Chebyshev function is illustrates in Fig. 2.6.

Figure 2.6: The characteristic response of n order for Chebyshev function [3].

## Filter Implementation Using Transformation Tools

In section 2.2.1 discussed the types of filter response that commonly used in microwave filter. This several type of filter gives some general solutions for low pass filter transformation element. However, they generally work at low frequency. Based on the concept of these types of filter, many topologies have been proposed. The use of these concept helps to design a filter according to the specifications.

In this section, some of transformation tools are discussed where the possibility to improve the filter response at high frequencies is high. At microwave frequencies, lumped elements are generally difficult to implement due to the limited range of values and the distances between the components. Therefore, Richard's transformation and Kuroda's identities are used to convert the lumped element to transmission line and separate the filter element using the transmission line sections, respectively [3]-[4].

## Richard's Transformation

This transformation was proposed in order to synthesize an LC network using open and short circuit transmission line stubs. The reactance of lumped element such inductor and capacitor basically have different mathematical form to that of transmission line stubs. The equation is given as follows [2];

For inductor;

For Capacitor;

In addition, the impedances of transmission line stubs and lumped element are different functions according to the chosen frequency. Some of the equivalent circuits derived using Richard's transformation at different length of line stubs are illustrates in Table 2.1.

Table 2.1: Equivalent Circuit of the Transmission Line at Open and Short Circuit with Different Length.

l =

l =

l =

## Kuroda's Identities

The four Kuroda's Identities are used in the implementation of microwave filter in order to separate the transmission line stubs, to transform the series stubs into shunt stubs or vice versa and also to change the impractical characteristic impedances into more realizable one. The four identities are illustrates in Table 2.2 where each box represent a unit element or transmission line that indicates their characteristic impedance [2].

Table 2.2: Kuroda Identities (n2 = 1 + Z2 / Z1)

(a)

(b)

(c)

(d)

The definition of unit element with respective to characteristic impedance is illustrates in Fig. 2.7.

Figure 2.7: Unit element

## Impedance and Admittance Inverter

These inverters essentially form the inverse of the load impedance or admittance where they can be used to transform series element to shunt element or vice versa. J and K inverter can be constructed using quarter-wave transformer of the characteristic impedance [2]. The concept of impedance and admittance inverter is illustrated in Fig. 2.8 where this transformation is useful for bandpass and bandstop filter with narrow bandwidth (< 10%).

(a)

(b)

Figure 2.8: (a) Operations of Impedance and admittance inverter. (b) Implementation as quarter-wave transformer [2].

## Coupled Line Theory

Coupled line is known as a coupled transmission line and largely used in microwave circuits. The coupled line consists of two unshielded transmission line where the lines are closed to each other. The interaction of electromagnetic field of each line presents a fractional of power between the lines [1]. In general, coupled transmission line usually operates in TEM mode and it's suitable for stripline and microstrip structure. Examples of the stripline and microstrip structure are shown in Fig. 2.9.

(b)

Figure 2.9: Example of coupled transmission line [1]; (a) Stripline structure (b) Microstrip structure

The structure of this coupled transmission line consist of three-wire line which can support the propagation modes where it can be use for implementation of filters and directional coupler. Fig. 2.10 shows the structure of three-wire line of coupled transmission and its equivalent capacitance network.

Figure 2.10: A three-wire coupled transmission line and its equivalent capacitance network [1].

There are two types of line which is symmetrical (where both conductors have same dimension) and asymmetrical (have different dimension). The configurations for symmetrical coupled line, both conductors use equal width and having constant gap spacing between the conductors. This structure also called as symmetric and uniformly coupled. For asymmetrical coupled microstrip line, the spacing between the line conductors also constant same as symmetric but the different is the width of the line [1], [4]-[14]. This configuration use unequal width of the line conductors. This structure also called as a uniformly coupled asymmetric line. The structure of asymmetric coupled line is shown in Fig. 2.11.

Figure 2.11: Microstrip Coupled line with unequal width (asymmetrical) [1].

For microstrip coupled line, the separation gap between the lines can be variable depends on the applications. If the separation between the lines is large, the coupling effect will reduce thus improve the electrical performances according to specifications. Generally, the fabrication for large separation is easy to fabricate. However, the filter bandwidth can only be achieving less than 20%. Hence, to design a wider bandwidth filter, the separation between the lines require tight coupling gap which are difficult to fabricate.

Next section will discuss the properties of the characteristics single quarter-wave coupled line section where it can be used to design bandpass filter such parallel coupled line filter.

## Properties of Quarter-Wave Coupled Line Section

The electromagnetic coupling that interferes between the two transmission lines can be used to design a several filters. The arrangement of some cases of symmetrical coupled line are illustrates in Table 2.3. As indicated in the table, the schematic diagrams of each type coupled line section are shown together with their formula and equivalent circuit. This various circuit has different frequency responses such as low pass, bandpass, all pass and all stop [2].

Table 2.3: The Canonical Coupled Line Circuit [3]

Coupled Line

Equivalent Circuit

Circuit Parameter

Band Pass

Band Pass

All Stop

All Stop

All Stop

All Pass

Band Pass

## Various Filter Topologies

Various topologies have been proposed and invented based on the theory of the coupled lines. Basically, the filter based on the coupled lines more particularly work on the narrow band bandpass filter. As an example, Fig. 2.12 illustrates some of the topologies based on the coupled lines that are used; parallel coupled lines filter, interdigital filter, combline filter and ring resonator filter where the lines are coupled laterally with the ring.

Parallel coupled line filter [3]

Interdigital filter [3]

Combline filter [3]

Ring resonator filter [39]

Figure 2.12: Example of coupled line topologies. (a) Parallel coupled line filter (using lines of quarter of wave and resonator half wave), (b) Interdigital filter, (c) Combline filter, (d) Ring resonator filter

## Parallel Coupled Line Filter

The relationship between the immittance inverter and coupling between the lines are very important in designing a parallel couple line filter. As discus in previous section, immittance inverters, J and K inverter can be constructed using quarter wave transformer or using lumped element [1, 2, 14]-[27]. However, for the case of parallel coupled line, the resonator width and the separation gap between the lines are controller for the immittance inversion. The basic coupled line section and admittance inverter are illustrates in Fig. 2.13. It seen that, two transmission line resonator length Î¸ are coupled together by an admittance inverter.

Figure 2.13: Equivalent circuit of coupled line section [3].

For parallel coupled line filter of nth section, the admittance inverter implemented at each coupling section where the value of J01, J12, and Jn+1 for each coupling are different based on the specification. The arrangement of nth section parallel coupled filter is shown in Fig. 2.14.

Figure 2.14: Parallel coupled line of nth section [3].

Based on this arrangement, the formula of the characteristic admittance of J-inverter can be calculated using ABCD matrix. The ideal admittance inverter can be obtained by substituted Î¸ = -90 degree and Z0 = J in the ABCD matrix of the transmission line of electrical length and characteristic impedance Z0. Hence, the ABCD parameter of the ideal admittance inverter is computed as follows;

The ABCD parameters of this admittance inverter were calculated by considering it as quarter wave length of transmission of characteristic impedance, 1/J. At this point, the J inverter for the various section are refer to the low pass normalized elements values, g0, g1, . . . , gn+1 given as follows:

where Î” = (Ï‰2-Ï‰1)/Ï‰0 is equal the fractional bandwidth of the filter. To determine the overall microstrip layout dimensions (width and spacing) of the parallel coupled line, the formula of the characteristic impedance even and odd mode are computed as follows:

For example, let consider a design parallel coupled line bandpass filter of order three (n=3) which centered at 1 GHz. The filter performance can be obtained by full wave electromagnetic simulation (EM) and an example of final layout of this design is presented in Fig. 2.15 with its EM simulated passband performance.

(a)

(b)

Figure 2.15: Overall layout with its EM simulated passband performance [1].

## Interdigital Filter

In the past few years, interdigital line structures are commonly used as slow wave structures [29, 30,], [36-38]. However, interdigital lines also have very interesting interdigital bandpass filter properties. As an example, the typical interdigital line filter with short and open-circuited line is shown in Fig. 2.16.

(a)

(b)

Figure 2.16: Interdigital filter (a) short-circuited lines at the ends, (b) open-circuited lines at the ends [36]

The structure of this filter consist of arrays of TEM-mode transmission line resonator between parallel ground plane. In the Fig. 2.16(a), each resonator line is a quarter-wavelength long at the mid-band frequency. The lines are short-circuited at one end and open-circuited at the other end. This resonator element are arrange in parallel array with the positions of the short-circuited ends alternating. While in Fig. 2.16(b), the terminating lines are open circuited and particularly work for the filter with moderate to wide bandwidth which is around 30 percent or greater.

In the study, the interdigital filter is one of the most popular structures that have very attractive features. The structure of this filter is very compact and uses the available space efficiently. It can be designed either narrow or wide bandwidth (30 to 70 percent) depending on the applications. In addition, the tolerances required in their manufacturer are relatively relaxed since the spacing between the resonator elements is large. In this filter, there is no possibility of spurious response exist because the second passband is centered at three times the center frequency, 3f0, of the first passband and the rates of cutoff and strength of the stopbands can be enhanced by multiple poles of attenuation at dc and at even multiples of the center frequency of the first passband. This filter also can be fabricated without dielectrics, thereby eliminating the dielectric losses.

## Combline Filter

The typical combline filter schematic in strip-line form is shown in Fig. 2.17. This resonator filter consist of TEM mode transmission line elements that are short-circuited at one end and consist of lumped capacitance Csj between the other end of the resonator line element and ground. In the schematic diagram, the lines 1 to n along with the capacitances element Cs1 to Csn comprise resonator, while lines 0 to n+1 are not comprise as a resonator since it were some part of impedance-transforming sections at the ends [3, 30]-[35]. In this filter, the coupling between the resonators is achieved by means of fringing field between the resonator lines.

Figure 2.17: Schematic of combline filter [3].

The lumped capacitances Csj allow the resonator lines to be less than Î»0/4 long at resonance. In this case without these capacitances, the resonator line would be fully Î»0/4 long at resonance, so that the passband would not have in the structure. This is because without some kind of reactive loading at the end of the resonator lines, the magnetic and electric coupling effects would cancel out each other so that the structure of the combline will become an all stop structure. Therefore, to achieve resonance the resonator lines with capacitively loaded the length of the lines must be less than 90 degrees long at center frequency. As an example, if the capacitance are made relatively large and the length lines are 45 degrees or less, the structure would be very compact and efficient coupling structure. In order to make the loading capacitor in this filter large so that the resonator will be Î»0/8 or less. However, in this filter, if the length of the line resonator is Î»0/8 long at the center frequency, the second passband will be located at slightly over four times the center frequency, 4fo.

In the theory, the attenuation through the filter will be infinite at the frequency for which the resonator lines are Î»0/4 wavelength long. This is because the attenuation above the primary passband is very high and depends on the electrical length of the resonator lines. In other words, the closer to Î»0/4 long the resonators are at the passband center, the steeper the rate of cutoff will be above the passband. This type of filter can be fabricated without dielectric support material and if desired the dielectric losses also can be eliminated.

## Ring Resonator Filter

The ring resonator is a transmission line which form in closed loop function. The topology of ring resonator was introduced by Woff and Knoppik for microwave substrate measurement [3]. The basic circuit of ring resonator is very simple which consist of the feed lines, coupling gap and the resonator [39]. Fig. 2.18 shows an examples circuit arrangement of ring resonator.

(b)

Figure 2.18: Examples of ring resonator (a) ring resonator with asymmetrical feed lines and notch, (b) Ring resonator side coupled via quarter wavelength lines [4, 39]

In this filter, the power is coupled into and out of the resonator through coupling gap and feed lines. The distance between the feed line and the resonator give an impact to the coupling gaps; thereby affect the resonant frequencies of the ring. One of the advantages of this resonator is it can support two degenerate modes which are orthogonal and have identical resonant frequencies [4]. The uniform ring resonator is fed by an asymmetric arrangement of feeding lines or by perturbation along the ring thereby the degenerate modes become coupled and resulting in a narrow band bandpass response. Besides, there exists two transmission zeros near the fundamental frequency that located at both side of the passband.

In addition, ring resonator either end-coupled or side-coupled to the microstrip transmission line has very interesting features [39]-[46]. The use of quarter wavelength side coupled lines to feed the ring introduced the two-tier resonance in the passband. Besides, the electric characteristics of the resonator such as matching level in the passband, bandwidth and transmission zeros frequencies can be controlled by varying the characteristic impedance of the coupled line (even- and odd-mode) and also line impedance of the ring resonator. The examples of the application ring resonator are shown in Fig. 2.19.

Figure 2.19: Application of side coupled ring resonator [39].

## Various Planar Technologies For Filter Implementation

In the microwave systems, the transmission line media such as coaxial lines, waveguide and planar is very important elements for high frequency realization. The development of this transmission line media is characterized for low loss transmission of microwave power. Choosing the right physical elements of transmission line media is depends on the several factors such as frequency range, physical size, power handling capability and production cost [2]-[7].

In the early of microwave systems, waveguide have their own advantages in term of capability of power handling and losses. However this type of transmission line media having a bulky size and high cost during the production. High bandwidth is one of the require specification in electrical characteristic of filter. So that, coaxial line is convenient for the application that having high bandwidth. However, it is not suitable to fabricate filter that having complex microwave components [48].

Planar transmission line structures are mostly employed for microwave integrated circuits and monolithic microwave integrated circuits (MICs). The geometry of planar configuration implies that characteristics of the element can be determined by dimensions in a single plane. Basically, the complete transmission line circuit can be fabricated in one step using thin film technology and photolithography techniques. There are several transmission structures that satisfy the requirement of being planar. The most common of these configurations are:

Striplines

Microstrip Lines

Coplanar Waveguide (CPW)

## Stripline

The symmetric stripline is reliable method for creating a transmission line. The stripline is aÂ TEM (transverse electromagnetic) transmission line.Â Stripline is well known as a planar type of transmission that lends itself to microwave integrated circuit and photolithographic fabrication. The geometry of a stripline is illustrated in Fig. 2.20 where it consists of three signal layer to accommodate a single signal carrying conductor.

Figure 2.20: The geometry of a stripline [2].

It is constructed with a flat conductor suspended between two ground planes where the conductor and ground planes are separated by a dielectric [2, 4]. The electric field in such lines propagated perpendicular to the center and its ground conductors and also concentrated over the width of the center conductor. The propagation characteristic in such line is nearly TEM mode where Fig. 2.21 shows theÂ fringing fields lines at the edges of the center strip.

Figure 2.21: Electric and magnetic field lines [2]

A major advantage using strip transmission line is that the conductor is practically self-shielded. The possibility of the radiation loss would be from the sides of the stripline structure. At this point, the structure of stripline behaves in a very predictable way and the characteristic impedance of the conductor is solely controlled by width of the conductor stripline. In addition, the uses of stripline in realization of filter offer better bandwidth in their characteristics performance.

However, like other transmission line media, stripline also have some disadvantages. There are two major points that has been listed. At first, it is much harder and difficult to fabricate than other type of planar transmission line. It is noted that, the structure of stripline having some number of layers. Therefore, this type of transmission line will cause difficulties in fabrication using PCB. This is because the signal conductor must be sandwiched between two layers of dielectric. Thus, it will costly in the production process.

The second point is a stripline transmission line also requires three separate layers to be dedicated to a single transmission line. The strip width and the board thickness at the second layer ground plane are much narrower for a given impedance such 50 ohm. This can be a problem if other components need to be attached to the line [2, 4, 48]. Fig. 2.22 illustrates the stripline fabricated transmission line.

Figure 2.22: Stripline Fabricated Transmission Line

## Microstrip

Microstrip is a type of electrical transmission line which is one of the most popular types of planar. Microstrip can be fabricated by photolithographic process and is used to conveyÂ microwave frequency signals. It is easily integrated with other passive and active microwave devices. In the general structure of this planar transmission line, microstrip consists of a single ground plane and a thin strip conductor on a low loss dielectric substrate above the ground plate. A conductor of width W is printed on a thin, and grounded with dielectric of thickness d and relative permittivity Îµr. The general microstrip structure is illustrates in the Fig. 2.23. While in Fig. 2.24 illustrates the field lines that consist on the microstrip conductor.

Figure 2.23: General microstrip structure [1]

Figure 2.24: Electric and magnetic field lines [4]

Like other planar transmission line, microstrip also has disadvantages. One of the disadvantages of this microstrip is radiation loss. Microstrip lines basically suffer more radiation loss compared to the stripline transmission line. It is note that a stripline has a minimal radiation loss. This is because the signal conductor is surrounded by a uniform dielectric material which is, in turn, confined by ground planes [4]. In fact, microstrip lines have a dielectric interface between dielectric material and free space at the signal conductor which causes the radiation loss.

One major advantage of this microstrip line is the size of the circuit can be reduced because the signal conductor for microstrip line is exposed to free space of uniform dielectric constant Îµ0 (typically the surrounding environment is air and Îµ0 = 1). In addition, microstrip also easy to fabricated using standard PCB techniques because microstrip topology only requires two signal layers. The characteristic impedance (Zo) of the signal conductor is controlled by the various geometries as defined in Fig. 2.25.

Figure 2.25: Microstrip geometry definition

where;

T = conductor thickness

W = conductor width

H = dielectric material thickness

Îµ0 = dielectric constant of free space

Îµ = dielectric constant of material

Figure 2.26: 3-Dimension of microstrip line

The ground plane and the signal conductor residing on separate layers is the types of coupling that can be utilized. Fig. 2.26a represents a 3-Dimension of a microstrip line and shows some of the possible coupling techniques that can be employed. While, Fig. 2.26b illustrates end coupled between the microstrip lines. At this point, the signal is capacitively coupled across the gap on the signal conductor. Fig. 2.27c illustrates the parallel coupled between the two microstrip lines where the signal is capacitively coupled between the overlapping parallel lines.

## Coplanar Waveguide

CPW transmission lines also have advantages and disadvantages as compared to microstrip lines. CPW lines require only a single layer on which both signal conductor and ground planes reside. Typical CPW geometries are defined in Figure 2.27.

Figure 2.27: Coplanar waveguide geometry definition

Where,

S = spacing between conductor and ground plane

W = conductor width

Îµ0 = dielectric constant of free space

Îµ = dielectric constant of material

The geometry of the CPW line allows for convenient connection of the signal conductor to the ground layer. Although this connection is simpler to achieve as compared to the microstrip topology, the problem of parasitic inductance still persists albeit on a smaller scale. This smaller parasitic inductance is still present because any connection will still add distance to the path the signal must travel [48]. As in the microstrip case, the ease of fabrication using standard PCB practices and accessibility of the signal conductor advantages still apply. A disadvantage of CPW lines from a theoretical standpoint is the area of the ground planes. According to CPW theory, these ground planes should extend to infinity as illustrated in Figure 2.28. In a practical situation, if the ground plane is greater than three times the signal width (W) the performance discrepancy is negligible.

Figure 2.28: Cross section of theoretical infinite CPW ground planes

Overall, the stripline, microstrip, and CPW transmission line technologies have many advantages and disadvantages. Table 2.4 outlines a few general differences between the three transmission line types.

Table 2.4: The Outlines of General Differences between the Three Transmission Line Types.

Transmission Line Type

Attribute

Stripline

Microstrip

CPW

Conductor layer Required

3

2

1

Signal Conductor Access

No

Yes

Yes

Construct using standard PCB Techniques

No

Yes

Yes

In the table 2.4, stripline is a type of transmission line that suffers from lack of access to signal conductor. It is also complicated to fabricate and consist of a high layer count as compared to the two other types. While, using microstrip line, the flexibility to adapt in many different coupling schemes such as end coupled, parallel coupled, and variations on parallel coupling are easier. This type of planar transmission line also has been well established, and it is a convenient form of transmission line structure for probe measurements of voltage, current and waves. For ease of fabrication and coupling versatility, the microstrip transmission line is the best choice of the three transmission lines discussed.

## Conclusion

In this chapter we discussed about a short state of the art of microwave planar filter. The parameters involved in development of filter synthesis are necessary in order to develop a new concept of coupled line filter topology. The topologies that proposed in this chapter introduced numerous advantages. The concept of parallel coupled lines and ring resonator that proposed in this chapter is use in designing a topology of dual-path coupled line filter.

The global synthesis of dual-path coupled lines will be discussed in the next chapter. Besides, the microstrip planar technology is used for the filter implementation.

## References

[1] R. K. Mongia, I. J. Bahl, P. Bhartia, J. Hong, "RF and Microwave Coupled-Line Circuites second edition," Artech House, Inc., London, 2007.

[2] David M. Pozar, "Microwave Engineering," John Wiley & Sons, Inc., USA, 2005.

[3] G. Matthaei, L. Young, E. M. T. Jones "Microwave Impedance-matching Networks, and Coupling Structures" Artech House, USA, 1985.

[4] J. S. Hong, M. J. Lancaster, "Microstrip Filters For RF/Microwave Applications" John Wiley & Sons, Inc., USA, 2001.

[5] R. Collin, "Foundations for Microwave Engineering (2nd edition)" New York: John Wiley & Sons, 2001

[6] I. Z. Anatol, "Handbook Of Filters Synthesis" John Wiley & Sons, Inc., USA, 2005.

[7] K. Chang, "Encyclopedia of RF and Microwave Engineering" New York: John Wiley & Sons, 2005.

[8] R. Levy and I. Whiteley, "Synthesis of Distributed Elliptic-Function Filters from Lumped-Constant Prototypes" Microwave Theory and Techniques, IEEE Transactions on, vol. 14, pp. 506-517, 1966.

[9] Levy, R., Snyder, R.V., Matthaei, G. "Design of Microwave Filters" IEEE Transactions on Microwave Theory and Techniques, Vol. 50, No. 3, pp. 783-793, 2002

[10] C. Kuo-Sheng and K. Jen-Tsai, "Insertion loss function synthesis of maximally flat parallel-coupled line bandpass filters," Microwave Theory and Techniques, IEEE Transactions on, vol. 53, pp. 3161-3168, 2005.

[11] T. Zhi-hong and C. Qing-Xin, "Determining Transmission Zeros of General Chebyshev Lowpass Prototype Filter," in Microwave and Millimeter Wave Technology, 2007. ICMMT '07. International Conference on, 2007, pp. 1-4.

[12] J. Zdunek, "The network synthesis on the insertion-loss basis," Proceedings of the IEE - Part C: Monographs, vol. 105, pp. 259-291, 1958.

[13] C. Chang, "Maximally Flat Amplitude Low-Pass Filter with an Arbitrary Number of Pairs of Real Frequency Transmission Zeros," Circuit Theory, IEEE Transactions on, vol. 15, pp. 465-467, 1968.

[14] S. B. Cohn, "Parallel-Coupled Transmission-Line-Resonator Filters," Microwave Theory and Techniques, IRE Transactions on, vol. 6, pp. 223-231, 1958.

[15] A. Dal, et al., "The design of parallel coupled line filter with arbitrary image impedance," in Microwave Symposium Digest, 1998 IEEE MTT-S International, 1998, pp. 909-912 vol.2.

[16] A. Riddle, "High performance parallel coupled microstrip filters," in Microwave Symposium Digest, 1988., IEEE MTT-S International, 1988, pp. 427-430 vol.1.

[17] P. Kwangsik, et al., "A study on the open end effect of coupled transmission lines," in Radio-Frequency Integration Technology, 2009. RFIT 2009. IEEE International Symposium on, 2009, pp. 296-298.

[18] R. Ma, et al., "Method of determining feed positions in coupled half-wavelength stepped impedance resonator bandpass filter," Microwaves, Antennas & Propagation, IET, vol. 3, pp. 856-862, 2009.

[19] C. Peng, et al., "Synthesis and realization of novel ultra-wideband bandpass filters using 3/4 wavelength parallel-coupled line resonators," in Microwave Conference, 2006. APMC 2006. Asia-Pacific, 2006, pp. 159-162.

[20] T. G. Bryant and J. A. Weiss, "Normal Mode Impedances of a Coupled Pair of Microstrip Transmission Lines," in Microwave Symposium, 1968 G-MTT International, 1968, pp. 117-122.

[21] E. Hanna, et al., "Synthesis and design of a suspended substrate capacitive gap-parallel coupled line bandpass filter with one transmission zero," in Electronics, Circuits and Systems, 2003. ICECS 2003. Proceedings of the 2003 10th IEEE International Conference on, 2003, pp. 547-550 Vol.2.

[22] A. Kian Sen, et al., "Analysis and design of coupled line impedance transformers," in Microwave Symposium Digest, 2004 IEEE MTT-S International, 2004, pp. 1951-1954 Vol.3.

[23] C. Kuo-Sheng, et al., "New Synthesis of Parallel-Coupled Line Bandpass Filters With Chebyshev Responses," Microwave Theory and Techniques, IEEE Transactions on, vol. 56, pp. 1516-1523, 2008.

[24] T. G. Bryant and J. A. Weiss, "Parameters of Microstrip Transmission Lines and of Coupled Pairs of Microstrip Lines," Microwave Theory and Techniques, IEEE Transactions on, vol. 16, pp. 1021-1027, 1968.

[25] H. Oraizi, et al., "Optimum design of parallel coupled-line filters," in Communications Systems, 2004. ICCS 2004. The Ninth International Conference on, 2004, pp. 340-344.

[26] S. I. Kim, et al., "Multilayer meander parallel coupled-line microstrip bandpass filter," in Microwave Conference Proceedings, 2005. APMC 2005. Asia-Pacific Conference Proceedings, 2005, p. 4 pp.

[27] S. Amari, et al., "New parallel λ/2-microstrip line filters with transmission zeros at finite frequencies," in Microwave Symposium Digest, 2003 IEEE MTT-S International, 2003, pp. 543-546 vol.1.

[28] W. Chao-Huang, et al., "Compact microstrip bandpass filter with two transmission zeros based on λ/8 coupled-line sections," in Microwave Conference, 2005 European, 2005, p. 4 pp.

[29] R. J. Wenzel, "Exact Theory of Interdigital Band-Pass Filters and Related Coupled Band-Pass Structures," Microwave Theory and Techniques, IEEE Transactions on, vol. 13, pp. 559-575, 1965.

[30] R. J. Wenzel, "Synthesis of Combline and Capacitively Loaded Interdigital Bandpass Filters of Arbitrary Bandwidth," Microwave Theory and Techniques, IEEE Transactions on, vol. 19, pp. 678-686, 1971.

[31] C. D. Nordquist, et al., "X-band RF MEMS tuned combline filter," Electronics Letters, vol. 41, pp. 76-77, 2005.

[32] M. El Sabbagh, et al., "Full-wave analysis of coupling between combline resonators and its application to combline filters with canonical configurations," Microwave Theory and Techniques, IEEE Transactions on, vol. 49, pp. 2384-2393, 2001.

[33] M. Hoft, et al., "Compact combline filter with improved cross coupling assembly and temperature compensation," in Microwave Conference, 2006. APMC 2006. Asia-Pacific, 2006, pp. 781-784.

[34] J. M. Chuma, et al., "An Elliptic Function Combline Filter using Electric Probe to Realise Capacitive Coupling," in Microwave Conference, 1998. 28th European, 1998, pp. 415-417.

[35] Z. Y. Xiao, et al., "Design of compact ultra-wideband filter using stepped impedance dual-mode resonator," in Signals Systems and Electronics (ISSSE), 2010 International Symposium on, 2010, pp. 1-4.

[36] G. L. Matthaei, "Interdigital Band-Pass Filters," Microwave Theory and Techniques, IRE Transactions on, vol. 10, pp. 479-491, 1962.

[37] J. R. Pyle, "Design Curves for Interdigital Band-Pass Filters (Correspondence)," Microwave Theory and Techniques, IEEE Transactions on, vol. 12, pp. 559-567, 1964.

[38] G. L. Matthaei, "Interdigital, Band-Pass Filters," in PGMTT National Symposium Digest, 1962, pp. 41-45.

[39] M. K. Mohd Salleh, et al., "Quarter-Wavelength Side-Coupled Ring Resonator for Bandpass Filters," Microwave Theory and Techniques, IEEE Transactions on, vol. 56, pp. 156-162, 2008.

[40] Z. Lei and W. Ke, "A joint field/circuit model of line-to-ring coupling structures and its application to the design of microstrip dual-mode filters and ring resonator circuits," Microwave Theory and Techniques, IEEE Transactions on, vol. 47, pp. 1938-1948, 1999.

[41] M. K. M. Salleh, et al., "Series-cascaded ring resonators for compact microwave filter with high rejection," in Industrial Electronics & Applications (ISIEA), 2010 IEEE Symposium on, 2010, pp. 518-521.

[42] M. Zhewang, et al., "Design of a wideband bandpass filter using microstrip parallel-coupled dual-mode ring resonator," in Microwave Conference Proceedings (APMC), 2010 Asia-Pacific, 2010, pp. 21-24.

[43] D. Ahn, et al., "Design of 2-pole band pass filters using closed loop resonator and coupled lines," in Microwave Symposium Digest, 1996., IEEE MTT-S International, 1996, pp. 1643-1646 vol.3.

[44] S. Lin and A. M. Ferendeci, "Coupling Parameters for a Side-Coupled Ring Resonator and a Microstrip Line," in Microwave Theory and Techniques, IEEE Transactions on, vol. 47, no 6, June 1996.

[45] L. H. Hsieh, K. Chang, "Compact, Low Insertion Loss, Sharp Rejection Wideband Bandpass Filters using Dual-mode Ring Resonators with Tuning Stubs," Electronics Letters, vol. 32, No 22, pp. 1355-1347, 2001.

[46] Kwok-Keung M. Cheng, " Design of Dual-Mode Ring Resonators with Transmission Zeros Electronics Letters, vol. 33, No 16, pp. 1392-1393, 1997.

[48] Z. I. Khan, et al., "Achievable bandwidth of a quarter wavelength side-coupled ring resonator," in Industrial Electronics & Applications, 2009. ISIEA 2009. IEEE Symposium on, 2009, pp. 358-361.

[49] Edwards, T.C., Steer, M.B., "Foundations of Interconnect and Microstrip Design (3rd edition)," New York: John Wiley & Sons., (2000)

[50] Lee, T.H., "Planar Microwave Engineering," New York: Cambridge University Press, 2004.

## Chapter 3: Second Order Dual-Path Coupled Lines Filter and Its Synthesis

## Introduction

Microwave components based on the coupled line structure have been used for over half a century. The coupled line section was first introduced by S. B. Cohn [1] where the idea was used to design bandpass characteristic in numerous applications such as filter, baluns and couplers.

Various topologies with more stringent requirements such as higher performance, smaller size and lower cost have been presented to support the microwave application such as wireless system. The advantages of microstrip parallel coupled lines that are the structure has simple circuit design, thus it leads in ease of synthesis either using Chebyshev or maximally flat approximation[2]-[7]. Generally, parallel coupled line produced narrow bandwidth and quite difficult to design for large due to the gaps limitation in parallel coupled line structure, thus led to the difficulties to design a filter of wide bandwidth. However, a large bandwidth response can be achieved by choosing the line dimensions appropriately [12].

Basically, the capacity of gaps of the coupled lines controls the characteristics of the filter response. The dependency of gaps between the coupled lines leads to the difficulties of workmanship of the design. Therefore, level of the coupling in the resonator can be control by playing with type of coupling such as end coupling.

One of the important things in this job of thesis is the development of a new concept of resonator which is used quarter wavelength of coupled lines. The advantage of this concept is that the presence of two transmission zeros at both sides of the passband help to ensure the selectivity of the response. The bandwidth, in-band matching level and order of the filter can be controlled by varying the elements of the coupled lines.

In this chapter, a new topology of coupled line microwave filter is proposed where two identical coupled lines are combined in series are interconnected to form a dual-path coupled lines interconnection which operates as a bandpass filter. For this topology, we develop a global synthesis for the case of second order to determine the characteristic impedances of the essential elements of the coupled lines in order to satisfy the specifications in central frequency, in frequency of the transmission zeros and wave in the band. Two types of second order filter will be introduced for two types of configurations; symmetrical and asymmetrical. These two types of configurations are in term of dimension elements of the conductors. A total of two filters are designed which centered at 2 GHz on the microstrip using FR4 epoxy-glass substrates for the realization of the filters.

## Topology Of The Filter

In this section, a procedure for designing a novel topology of coupled line microwave filter is introduced where a number of identical coupled lines in series are duplicated to form a dual-path coupled lines interconnections. In the basic topology of this filter, the total length of the coupled line is equal to the length of wave guided in frequency, f0. The topology is made by the complete lines of the quarter wave coupled lines. There are two parameters involved in this topology which act to control the frequency responses of the dual-path coupled line filter which are the characteristic impedances of even modes (Z0e) and odd modes (Z0o) of the coupled lines.

Figure 3.1: Second order of a dual-path coupled lines filter topology.

The use of series lines coupled in the topology of dual-path filter allows simply simulations by using the models of ideal lines which simulated using EM simulator software as ADS. Fig. 3.2 illustrates an example of the electrical response of the filter simulated using ADS for value of Z0e = 40 Î© and Z0o = 20 Î©.

Figure 3.2: Ideal response of second order bandpass filter.

The frequency response of the topology dual-path coupled lines filter is centred in central frequency f0 = 1 GHz where the topology results in bandpass response of xth order which is equal to . The transmission zeros are found at both side of the passband help to ensure the selectivity of the response. The first harmonic is found at 3f0 which at 3 GHz. In this filter, the rejection level of the passband, the level of attenuation in the band, the positions of the transmission zeros and the breadth of band are strongly controlled by the level of characteristic impedance even and odd modes, (Z0o and Z0e).

Figure 3.3: Variations of the filter responses in functions of parameters (a) Z0e and (b) Z0o.

Based on the ideal response of the filter, this topology exhibits sufficient bandwidth that covers wideband application. Bandwidth level of this filter is around 40 to 70 percent. The breadth of the band can be determined by varying the value of even mode characteristic impedances. The more value of impedances Z0e increases the more breadth of the band is increased. The face in 3.3 illustrates the variations of the characteristics impedances controlled the electrical response of the filter for certain variations of parameters Z0o and Z0e.

## Equivalent Circuit Of Coupled Line Network

To start the development of the synthesis equation of new topology of the coupled lines filter, the first step is necessary to define an equivalent schematic of an identical coupled lines network. In the design work of coupled line filter, the characteristic impedances and admittance of the coupled line is correspond to the even and odd excitations. It can be expressed using the admittance matrices formula. This concept of admittance matrix is able to use for symmetrical coupled lines. The concepts were proposed by [ref 19, 18, 19] and the formula of admittance matrix is given as follows:

(III.1)

where:

According to Sato [19], the equivalent circuits of the coupled lines network can be represent by using graph transformation method. Fig. 3.4 the two-wire coupled line network which is has port 3 and port 4. The length of lines represents a unit element of length l where the inductor symbols used in the diagram represent a short-circuited stub of length l [20].

Figure 3.4: Graph representation of two wire coupled line network [20].

Some equivalent circuit of the coupled line has shown in table 3.1. This equivalent circuit has been simplified using graph transformation method. In this topology, one of the points of the coupled line is open circuit.

TABLE 3.1: Multiport Distributed Line Network Transformation

Original Circuit

Equivalent Circuit

1

2

3

4

The topology consists of equivalent circuit of the coupled line that has open circuited at their ends. Table 3.2 indicates several derivation of the equivalent for a pair of coupled line which has open circuited at their ends. The concept of coupled line network transformation is used to derive the equivalent circuit model of the coupled line that will illustrate in the next section.

TABLE 3.2: Simplification of Three Port Coupled Line Section

Transformation of coupled line

(a)

Based on figure 1 in table 3.1

(b)

(c)

(d)

## Synthesis Of The Dual-Path Quarter-Wavelength Coupled Line

In this section, the new concept of this topology is synthesized to describe the proposed filter theory which is used as an essential element to control the characteristic of the frequency by using parameters Z0e and Z0o. The development of the synthesis is to find the relationship between this impedance with the electrical characteristics of this filter such as frequency of the zeros transmission, central frequency and breadth of the band in order to reduce the degrees of freedom. The design procedures can be summarized into three steps and the detail description of each step is discussed in the following section.

## Model identical circuit of the filter

The study of this synthesis start with the basic model of the coupled line that proposed by Ozaki in his paper [19]. Based on this simple circuit model, each coupled line can be simplified in three-port section which has been introduced in previous section. Fig. 3.5 illustrates the equivalent-circuit diagram of three-port coupled line network.

Figure 3.5: Equivalent-circuit diagram of a three-port coupled line network

This equivalent-circuit is used to develop the filter synthesis where unit element parameters, Yue and AYue are given in as follows:

In the equivalent-circuit diagram, the parameter Aue and Yue are the characteristic admittances of the unit element and both parameters in term of ABCD matrix. Herein, both the even-mode and odd-mode characteristic impedance of the coupled lines are assumed identical, ie., Z0e=Z0e1 and Z0o=Z0o1. Therefore, the complete resonator filter can be presented as the simplified diagram as shown in Fig. 3.6.

Figure 3.6: Simplified diagram of topology dual-path coupled lines filter.

Another parameter involved in developing the synthesis equation is given as follows:

## Transmission Zeros Frequency

Frequency of zeros transmission is one of the important characteristic in electrical response of the filter. Thus, it is more interesting if we able to place this zero frequency to the place that we wish. The proposed procedure for finding the transmission zeros starts with the determination of Y -matrix equation. The equation consists of the coupling capacitor of admittance YC which is acts as a control parameter for out-of-band response, while the transformer of ratio T is frequency independent which is used to control the out-of-band response magnitude. The coupling capacitor of admittance YC consist in the topology puts into the matrix chain given by:

Knowing that all the parameter involves are changed into ABCD matrix, the schematic diagram of the topology are simplified and illustrated in the Fig. 3.7:

Figure 3.7: Equivalent circuit of dual-path coupled lines filter.

with;

To represent the whole matrix in the central loop, the matrices at both upper and lower part are transformed into admittance matrix. Both matrix chain of ATop and ABot change into admittance matrix, respectively.

and,

Therefore, by adding both part of admittance matrices, we get a quadripole that presenting the whole loop that called Ym. The expressions of Ym are expressed as following and simplified diagram is depicted in Fig. 3.8.

Figure 3.8: Overall simplified diagram in Y - matrix of the topology.

with;

By inserting the expression of and s based on the equation () and , the total admittance Ym can be simplified and presents as follows:

To find the location of the transmission zeros, we assume Ym12 is equal to zero:

So that, the equation can be simplified as follows:

According to the equation (III.36), we plot a graph as compared to the S12 of the ideal response. The simulation response for this equation is i