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Principle and synthesis of bandpass filter
Filters are often used in electronic systems to emphasize signals in certain frequency ranges and reject signals in other frequency ranges . Filters can be separated in five categories: low-pass, high-pass, band-pass, band-stop, and all-pass or phase shift filter.
A low-pass filter allows signal frequencies below the low-cut off frequency (this is the frequency that define the limits of the filter range, it is the desirable cut-off frequency of the filter) to pass and attenuates frequencies above the cut-off frequency. It is commonly used to reduce environmental noise and provide a smoother signal. The opposite of the low-pass filter is the high-pass filter, which attenuates signals below cutoff frequency; this filter can be used as blocking DC from circuitry sensitive to non-zero voltage or RF devices. It can also be used in conjunction with a low-pass filter, to obtain a band-pass filter that allows signals in a specific band of frequencies to pass, while signals at all other frequencies are attenuated. The band-stop filter is the opposite of the band-pass filter. The all-pass or phase-shift filter has no effect on the amplitude of the signal in the whole frequency range; its function is to change the phase of the signal without affecting its amplitude. In this thesis, only bandpass filter will be considered.
Since filters are defined by their frequency domain effects on signals, the most useful analytical and graphical descriptions of filters also fall into the frequency domain. Thus, curves of gain vs frequency and phase vs frequency are commonly used to illustrate filter characteristics, and the most widely used mathematical tools are based on the frequency domain.
To understand the bandpass filter theory, it is necessary to briefly study transfer functions.
Microwave filter synthesis began with the image impedance parameter method and was useful for low frequency filters . A more modern procedure for designing filters which allows a high degree of control over the passband and stopband amplitude and phase characteristics is the insertion loss method . Filters can be classified into four categories distinguished from each other by terms of locations of transmission zeros and poles of their transfer functions. Namely, Butterworth, Chebyshev, Elliptic and Bessel filter. The Butterworth filter has no ripple in the passband and because of this it is called a maximally flat filter. The Chebyshev filter has a deeper attenuation beyond the cut-off frequency than the same order Butterworth filter, at the expense of ripple in its passband. The Elliptic function response allows obtaining transmission zeros close to the passband in order to increase the rejection slope; thus this filter has a steeper attenuation beyond the cut-off frequency than the Chebyshev filter, such filter is complex in practical realization. The linear phase response concerns the Bessel filter whose transfer function is derived from a Bessel polynomial and Gaussian filter, whose transfer function is derived from a Gaussian function, have a poor selectivity.
In this thesis, only bandpass filter based on Chebyshev prototype filter are considered.
Chebyshev bandpass filter
The Chebychev filtering function can be synthesized by a low pass network composed of capacitive and inductive elements, whose elements values are normalized to make the source resistance or conductance equal to one (as seen in Figure I-1).
Knowing the elements values of the lowpass prototype filter shown in Figure I-1, the next step is to look for the corresponding bandpass filter design, which can be obtained directly from the prototype by a lowpass to bandpass transformation (Figure I-2) .
J and K inverters
The filter structure in Figure I-2 consists of series resonators alternating with shunt resonators, an arrangement which is difficult to achieve in practical microwave structure when implementing filter with a particular type of transmission line. In microwave filter it is much more practical to use a structure which approximates the circuit containing only resonators of the same type based on “impedance and admittance inverters” as shown in Figure I-3.
A load impedance (or admittance) connected at one end is seen as an impedance (or admittance) that has been inverted with respect to the characteristic impedance (or admittance) squared at the input , . Thus, a series LC resonator (or shunt LC resonator) with an inverter on each side looks like a shunt LC resonator (a series LC resonator) from its external terminals.
Modern systems require high performed filters with very low losses, small size, sharp cut-off, and high rejection at the stopband. Thus, different resonators and techniques have been introduced for RF and microwave filters to achieve different filters with these performances using planar transmission lines.
Planar transmission lines are composed of a solid dielectric substrate having one or two layers of metallization, with the signal and ground currents flowing on separate conductors.
Planar transmission lines used in microwave frequencies can be broadly divided into two categories: those that can support a TEM (or Quasi-TEM) mode of propagation, and those that cannot. For TEM (or Quasi-TEM) modes the determination of characteristic impedance and phase velocity of single and coupled lines reduces to finding the capacitances associated with the structure.
The most widely used transmission lines are microstrip line, coplanar waveguide (CPW), and stripline.
a) Microstrip line
Microstrip line (Figure I-4) is a transmission line which consists of a conducting strip separated from a ground plane by a dielectric layer. The dielectric constant varies between 2 to 10 times that of free space, with the penalty that the existence of two different dielectric constants (below and above the strip) introduces a variability of propagation velocity with frequency that causes some complications in microstrip analysis and design , . Thus a concept of effective dielectric constant was introduced, which is expected to be greater than the dielectric constant of air and less than that of the dielectric substrate. In addition, the substrate discontinuity causes its dominant mode to be hybrid (Quasi TEM). Figure I-4 shows the general structure of a microstrip line. The main advantages of microstrip line are that it is a suitable technique for MICs (Microwave Integrated Circuit), and the radiation that it provides can be used for antenna design. The main disadvantage of this type of transmission lines is that via holes are necessary in the case of shunt connection, inducing a complex modelling of these vias. The characteristic impedance of the microstrip line is depending on the width of the guided wave line, substrate thickness, and the effective dielectric constant of the substrate .
b) Coplanar waveguide
Coplanar waveguide (CPW) is basically a single strip located between two ground planes on the same side of the substrate, see Figure I-5. This type of transmission line can be easily integrated with the MIC’s. Furthermore shunt connections can be easily realized without the need of via holes. In addition it is less sensitive to the substrate thickness that in the case of microstrip. Both ground planes must be at the same potential to avoid the asymmetric propagation mode.
Stripline is essentially a flat metal transmission line between two ground planes, with the ground planes separated by a dielectric substrate material. The metal transmission line is embedded in a homogeneous and isotropic dielectric, thus the phase velocity and the characteristic impedance remain constant with frequency. The width of the transmission line, the thickness of the substrate, and the relative dielectric constant of the substrate material determine the characteristic impedance of the transmission line. Difficulties with stripline include ground planes that must be shorted together, requiring electrical via connections between the two metal ground planes. Stripline second ground plane also results in a narrower transmission line widths, for a given substrate thickness and characteristic impedance, than for microstrip.
As previously mentioned, transmission lines in stripline technology are embedded in a homogeneous medium. Thus, the phase velocity for coupled transmission lines of both even- and odd- modes are the same, on the contrary for a microstrip coupled lines. Thus, the stripline technology is more convenient to validate synthesis formulas for a new filter topology. Hence, the stripline will be considered herein to validate the Parallel-Coupled Stub-Loaded-Resonator (PC-SLR) bandpass filter (as will detailed in Chapter II). However, the design procedure of a stripline topology is more difficult and expensive to fabricate than microstrip. Consequently, microstrip and CPW topologies remain more used than stripline one.
Different planar resonators have been introduced for RF and microwave bandpass filters to achieve filters with high performance (low losses, small size, sharp cut-off, and high out-of-band rejection). The main used resonators are described below.
a) Half wavelength resonator
It basically consists of transmission line section having a length of half wavelength at the corresponding center frequency. Early filters based on these resonators are end-coupled and parallel-coupled bandpass filters  (as illustrated in Figure I-7). Their design equations have been well documented -. The parallel arrangement (Figure I-7(b)) gives compact size (the length of the filter is reduced approximately by half), compared to the end-coupled bandpass. Also it gives relatively large coupling for a given spacing between resonators, thus this filter structure is particularly more convenient for constructing wider bandwidth than the end-coupled filter.
The Hairpin bandpass filters also known as U-shape , and open-loop - bandpass filters are also based on half wavelength resonators. They are obtained by folding the resonators of the parallel-coupled filter, thus the same design equations can be used. Open-loop resonators have great advantages in reducing filter size. They have different coupling nature depending on the coupling sides  which can be used to obtain transmission zeros in the out-of-band rejection. If the two arms of each hairpin resonator are closely spaced, they function as a pair of coupled line themselves, which can have an effect on the coupling as well.
When designing in microstrip, these half wavelength resonator filters suffer from the spurious response at twice the passband working frequency. This causes response asymmetry in the upper stopband and could greatly limit its applications.
b) Quarter wavelength resonator
Interdigital filters (Figure I-9) consists of parallel-coupled quarter-wavelength resonators which are short-circuited at one end and open-circuited at the other end. Interdigital microstrip filters have the first spurious frequency at three times the working frequency, thus they present a large out-of-band rejection as compared to the parallel-coupled and end-coupled microstrip bandpass filters. Coupling between interdigital lines is strong thus gap between resonators can be large, making interdigital filters simpler to fabricate for wide bandwidth applications, with small filter dimension .
Theory and design procedure for interdigital bandpass filters depends on the configuration of the feeding ports. Thus the design equations for filters with coupled-line Input/Output (I/O) (Figure I-9 (a)) can be found in . Based on , design equations for the type of interdigital bandpass filter with tapped-line I/O (Figure I-9 (b)) are presented.
Miniaturization techniques in planar technology
Filters based on parallel-coupled microstrip line structure have been widely used in the RF front end of microwave and wireless communication systems for decades. One of the most common filter topologies is based on half wavelength resonators with quarter wavelength coupled line sections (as seen in Figure I-7 (b)). Good repetition and easy design are attractive features of this filter topology. However, since the conventional bandpass filters are relatively bulky, large efforts were invested in the miniaturization of these types of filters.
Many methods for the miniaturization of planar filters have been proposed and they are summarized in four categories. The first is based on the use of high dielectric constant substrate. The second proposed a lumped element approach. The third is based on the use of meandered transmission lines. Finally, the fourth category deals with the use of slow-wave transmission lines.
High dielectric constant substrate
Dielectric constant, is an index representing the degree to which substrate can store electric charge. The higher the value is, the more electric charge can be stored. Thus, the dielectric constant of the substrate can determine the size of a circuit. Indeed, this technique produces a reduction of the wave propagation velocity on the transmission line and the consequent reduction of the wavelength for a given frequency (I-3). Thus the slow-wave effect can be evaluated.
This technique has proved its effectiveness in miniaturization of bandpass filters as demonstrated for example in -. In  seven-pole stripline end-coupled bandpass filters were fabricated using a high dielectric constant substrate. This substrate is composed of Zirconium-tin-titanium oxid [(ZnSn)TiO4] (dielectric constant , dielectric loss tangent 4). These bandpass filters were designed with a 140-MHz bandwidth at 3 dB, and center frequencies of 6.04 GHz and 8.28 GHz with an insertion loss of 6.6 dB. In , two miniaturized hairpin 5% bandwidth bandpass filters were fabricated on two very high dielectric constant substrates: the first one being composed of barium and strontium titanates (dielectric constant , dielectric loss tangent 4, substrate thickness 2 ) and the second one composed of niobium-niodinum titanate (dielectric constant , substrate thickness ). A high reduction gain was obtained. Nevertheless, difficulties were highlighted about the realization of 50-Ω transmission-line characteristic impedance on these substrates due to the high dielectric constant value. Indeed, the characteristic impedance of a transmission line (in microstrip, stripline, or CPW technology…) is inversely proportional to the relative permittivity. Thus, a high dielectric constant substrate leads to low line widths. This is the main drawback of using this technique.
Lumped element approach
Using lumped elements (fabricated entirely using printed circuit or thin film technologies) is another approach to filter miniaturization . This technique allows high out-of-band rejection. Lumped inductors can be designed as spiral or meander transmission lines and capacitors consist of interdigital or MIM (Metal-Insulator-Metal) structures . These elements have the advantage of small size, and low cost. However, it is very difficult to realize a truly lumped element and thus parasitic effects must be considered inducing spurious appearing in the out-out-band. For example, the shunt parasitic capacitance to the ground may affect the performance of the inductor.
Meandered transmission lines
The distributed approach considering meandered transmission lines can be used -. For example, in , a dual-mode microstrip bandpass filter was fabricated based on a meander loop resonator. This dual-mode meander loop achieves more than 50% size reduction against ring, square patch and disk resonators. In , a CPW bandpass filter was miniaturized to half of its initial size using this technique. However, this technique needs an important work to model parasitic couplings.
Slow-wave transmission lines
From transmission line theory, the propagation constant and phase velocity of a lossless transmission line are given respectively by √ and 1 , and are the inductance and capacitance per unit length along the transmission line.
Thus, slow-wave propagation can be accomplished by effectively increasing the and values. One way to do this consists in introducing periodic variations along the direction of propagation. Slow-wave resonators are generally realized by loading basic transmission lines by inductances and/or capacitances. These elements affect the transmission line parameter and thus reduce the phase velocity of the periodically loaded transmission line. In microstrip, these elements can be added to the conducting strip (series inductance or parallel capacitance) - or by drilling holes in the substrate or by etching patterns in microstrip ground plane. These patterned structures are sometimes referred to as defected ground structures or slotted ground structure , .
Table of contents :
Chapter I. Filter Basics
I.2. Principle and synthesis of bandpass filter
I.2.1. Signal filtering
I.2.2. Transfer function
I.2.3. J and K inverters
I.3. Planar Filter
I.4. Miniaturization techniques in planar technology
I.4.1. High dielectric constant substrate
I.4.2. Lumped element approach
I.4.3. Meandered transmission lines
I.4.4. Slow-wave transmission lines
I.5. Out-of-band rejection techniques
Chapter II. Compact Stub-Loaded Parallel-Coupled Bandpass Filter: Synthesis and Miniaturization Rules
II.2. Parallel-Coupled Stub-Loaded Resonator bandpass filter structure
II.3. Theoretical study of the stub-loaded resonator
II.3.1. Topology of the SLR
Fundamental II.3.2. resonance frequency of the SLR
II.3.3. Miniaturization rules of the SLR
II.4. Synthesis method of the PC-SLR bandpass filter in a homogeneous technology
II.4.1. Design procedure
II.4.2. Validation of the theory
II.5. Synthesis method of bandpass filters in an inhomogeneous medium
II.5.1. Microstrip versus Stripline
II.5.2. Microstrip Filter design problems
II.6. Chapter conclusion
Chapter III. Parallel-Coupled Microstrip Bandpass Filters with Wide Spurious Response Suppression
III.2. Resonance modes of the Stub-Loaded Resonator
III.2.1. Odd-mode resonance frequencies
III.2.2. Even-mode resonance frequencies
III.2.3. Optimization of the resonator topology for improving the out-ofband rejection
III.2.4. Intermediate conclusion
III.2.5. Convenient choice of the substrate
III.2.6. Third-order proof-of-concept filter
III.3. Design of feed lines with”U Corner”
III.3.1. Optimized U Corner for Transmission Zero Placement
III.3.2. Validation of the concept on Parallel-coupled Bandpass Filter
III.3.3. Validation of the concept on PC-SLR bandpass filter
III.4. State-of-the-art comparison
Chapter IV. Tunable Compact Filters Based on Stub-Loaded Resonators
IV.2. Theoretical study of capacitor loaded SLR
IV.2.1. Fundamental resonance frequency of the capacitor loaded SLR
IV.2.2. Resonance modes of the capacitor loaded SLR
IV.3. The varactor diode
IV.4. Experimental results
IV.4.1. Filter 1
IV.4.2. Stub reduction versus capacitance value
IV.4.3. Filter 2 simulation and measurement responses