29 Resonator

Dr. Ayushi Paliwal and Dr. Monika Tomar

1. Introduction:

Microwave resonators have been widely used in various applications such as filters, oscillatos, amplifiers and so on. Since the operation of microwave resonators is very similar to that of the lumped-element resonators of circuit theory, we will begin by reviewing the basic characteristics of series and parallel RLC resonant circuits. For communications through microwave resonators, waveguides play a crucial role. In the initial stages of the microwave development, the Rectagular waveguide become the dominant waveguide structure largely because high quality components could be designed using it. One of the main issues was its narrow bandwidth due to its cut off frequency characteristics. In the present module, waveguide and types of waveguides will be discussed in detail.

2. Waveguide:

Waveguide is a metallic structure that guides the EM wave in a particular direction. They exist in form of wires, coaxial cables, or parallel plates. The optical fibers are also used as waveguide for EM waves. In general, the transmission of EM radiations through free space is subjected to diffraction, which results in spreading out of energy. This problem has been resolved by waveguides at the expense of material structure. Waveguide enables a signal to propagate with minimal loss of energy by restricting expansion to one or more dimensions. A waveguide is a special form of transmission line consisting of a hollow, metallic structure (Figure 1). The walls of the waveguide represent the distributed inductance, while the empty space between the walls provides distributed capacitance.

Waveguides are utilized practically only for the signals of extremely high frequency, where the wavelength of operation approaches the cross sectional dimensions of the waveguide. For lower frequency, waveguides are of no use.

3. Modes of wave propagation:

The electromagnetic wave is consists of electric and magnetic field perpendicular to each other but propagate in the direction of propagation. The electromagnetic wave propagates through waveguide in three ways which are as follows:

• Transverse electric (TE) mode
• Transverse Magnetic (TM) mode
• Transverse electric and magnetic (TEM) mode

When an electromagnetic wave propagates down a hollow tube, only one of the fields i.e. either magnetic or electric will be transverse to the direction of wave propagation. However, the other field will loop longitudinally to the direction of wave propagation. But the two fields i.e. electric and magnetic fields will remain perpendicular to each other. So, when the electric field is transverse, then it is called transverse electric (TE) mode and when magnetic field is transverse to the wave propagation, it is called transverse magnetic (TM) mode (Figure 2). In TEM mode, both electric field and magnetic field pare perpendicular to the direction of travelling wave along the length of a normal transmission line as seen in figure 3. TEM mode exist only where there are two conductors present and it is dominant mode where the cross sectional dimensions of the transmission line are small as compared to the wavelength of the signal. TEM mode is also termed as differential mode, because the signal flowing on the inner conductor is directed opposite to the ground signal flowing on the outer conductor. In TEM mode, atleast two unconnected conductors and a single insulating material are required for it to exist. Its cut off frequency is 0Hz.

Figure 2: TE and TM waveguides modes

Figure 3: TEM mode of waveguide

There exists one more mode i.e. Quasi TEM mode which is utilized in transmission of wave. As discussed earlier, TEM mode is transverse electric and magnetic mode which exists in structures having two conductors surrounded by dielectric material. Quasi TEM mode is very much similar to the already discussed TEM mode. But, it is present in microstrip resonators and coplanar waveguide resonators. Due to the presence of two different medium having different resistivities, wave propagation occurs where wave propagate with different speeds in both the directions. Quasi TEM mode is also known as hybrid mode which has non zero electric and magnetic fields in the direction of propagation. Hybrid modes are higher order modes with cut off frequencies different from direct current (0 Hz) and are undiserable.

4. Types of Waveguides:

As discussed earlier, waveguide can be developed in three ways: 1) Stripline, 2) Microstrip, and 3) Coplanar waveguide (CPW).

Stripline:

Stripline is the type of waveguide which requires three layers of conductors where the internal conductor is commonly called hot conductor or stub while the other two are always connected at signal ground which is known as cold conductor or simply ground. The stub is embedded in a homogeneous and isotropic dielectric, of dielectric constant εr. So, unlike the case of microstrip, the substrate is not appropriate since the dielectric completely surrounds the signal conductor.

Because the region between the outer plates of stripline contains only a single medium, the phase velocity and the characteristic impedance of the dominant TEM mode do not vary with frequency. In the fundamental mode, the signal conductor is equipotential (every point in it is at the same potential).

Stripline is often required for multilayere circuit boards because it can be placed between the layers, but grounding the stripline requires some care. Id the top and bottom ground planes are not at the same potential, therefore, the a parallel plate mode can propagate between them.

If excited, this mode will not remain confined to the region near the strip, but will be able to propagate wherever the two ground planes exist. Stripline is more sensitive than microstrip to lateral ground planes of a metallic enclosure, since the electromagnetic field is strongly contained near the centre conductor and the top bottom conductor ground planes.

Microsrtip:

Microstrip technology consists of a thin conducting strip placed on to of a dielectric substrate. The bottom side of the substrate is covered entirely by metal and serves as the ground plane which is shown in figure.

The substrate is characterized by its dielectric permittivity and thickness. It serves as a mechanical support and as amedium for electromagnetic field propagation. In 1975, Hammerstad proposed the first equation to compute the characteristic impedence of a microstrip line. It is proportional to the ratio of the strip width W over the substrate thickness h. Since then, additional equations have been proposed to account for the strong dispersion of microstrip lines at millimeter wavelengths. In simple terms, microstrip is the printed circuit version of a wire over a ground plane, and thus it tends to radiate as the Spacing between the ground plane and the strip increases. The microstrip line is dispersive in nature. With increasing frequency, the effective dielectric constant gradually increases towards that of the substrate, so that the phase velocity decreases. However, it is valid for non dispersive medium also.

Coplanar waveguide:

Among the three, CPW is the most successful geometry for transmission of EM wave in the microwave region. In general, Coplanar waveguide consists of a center conductor and two ground planes printed on the same surface of a dielectric slab (Figure 7.1). The coplanar line has basically two dominant modes of propagation. One is quasi-TEM (Transverse EM) mode, often known as odd mode or coplanar mode, where the fields in the two slots are 180o out of phase as shown in figure .. (a). The other is non-TEM mode, called the even mode or the coupled slotline mode, where the fields are in phase as shown in figure(b). In CPW microwave circuits, the coplanar mode is the most desired due to its low radiation property. The excitation of unwanted slotline modes can be avoided by either maintaining the symmetry of the CPW structure or using air-bridges (or bond wires) to connect the ground planes of CPW.

Figure: (a) Coplanar mode and (b) Coupled slotline mode field configuration

5. Applications of microwave resonators

Satellite communication plays an imperative role in the navigation of moving aircrafts, detection of water pollution, oil field monitoring, weather conditions, war-fighters, virtual connectivity on earth, population detection, television communication etc. at RF/Microwave frequencies. The first initiative towards the fabrication of Microwave filters was carried out during second world war by Radio Research Laboratory where the broad band, low band, bandpass, high pass coaxial filters and narrow band tunable co-axial resonator were developed by Radiation Laboratory [Levy and Cohn (1984)].

Wireless communication systems are demanding tighter requirements in terms of electrical specifications as well as drastic reduction in manufacturing costs and development [Boria and Gimeno (2007)]. The area of microwave filters, especially for mobile and space applications, has experienced significant improvement with respect to the theoretical and technological aspects [Zang et al. (2006)]. One of the significant improvements in filter design theory was utilizing cross-coupling between nonadjacent resonators. Early works by Kurzok described 3rd order and 4th order filters with cross-coupling [Kurzok (1966)]. Cross-coupling gives a number of alternative paths in which a signal may be carried between the input and output ports. The multipath effect causes zero transmission to appear in the transfer function, which is dependent on the phasing of the signals, causing zero transmission (or attenuation poles) at finite frequencies or group-delay flattening, or even both [Levy and Cohn (1984)].

Along with the improvement in electrical performance of the filters, many efforts have been made for the improvement in its miniaturization [Tang et al. (2008)]. Especially, reducing the size and volume of filter has been the main topic for filter designs. Since Richtmyer (1939) showed that dielectric materials can function much like metallic cavities, dielectric resonators have been a good candidate for this purpose. Dielectric resonator filters have the advantages of not only small size but also good temperature stability [Reaney and Iddles (2006)]. Another major step in reducing the volume and mass of microwave filters was utilizing dual-mode technique. Williams (1970) describes the fundamental theory for dual mode waveguide filter realization. Dual-mode filters use two orthogonal degenerate modes of each resonator, where the number of resonators can be reduced by a factor of two. Hence, obtaining small volume and mass become possible by using dual-mode filters.

6. Tunable microwave filters:

Another microwave filter design technique having great potential is the tunable microwave filter design technique. For example, frequency agile system for naval target control uses 19 channels from 421.5 MHz to 448.5 MHz. This frequency agile system can obtain improved resistance to interference and increased data link reliability [Thesis: Lee et al. (2009)]. It can sense interference and automatically change the frequencies to avoid any interference by switching from one filter to the other, but the system should have as many filters as the number of the channels. Hence, compared to a bank of switched fixed-frequency filters, tunable filter can make frequency-agile systems much simpler.

The resonators can be categorized into planar resonators and non-planar resonators. Most common planar resonators are implemented using the microstrip structure. Microstrip patch resonators and microstrip line resonators are the representative of planar resonators. Regarding the reduction in size of the resonator, the microstrip resonator line has great advantages over cavity resonators [Thomson and Hong (2007)]. Most successful method to reduce the size of the microstrip line resonator is meandering the microstrip line of the resonator. The meandered microstrip line open loop resonator is smaller than λg0/8 by λg0/4, where λg0 is the guided wavelength at the center frequency [Hong and Lancaster (1996)]. For further reduction of the resonator size using meandering the microstrip line, the line width of the microstrip line could be reduced. However in general, the quality factor of the resonator becomes worse as the line becomes narrower [Hong and Lancaster (1996)]. This disadvantage of the meandered microstrip line resonator has been overcomed by using high temperature superconducting materials. On the other hand, non-planar resonators have relatively higher quality factor than planar resonators. Although the non-planar resonators are bulky, these filters are employed by the wireless systems where the electrical performances are much more important than the size and volume of the filter. For example, the waveguide cavity filters have applications in base stations of cellular systems and satellite transponders, although these systems, compared with handsets, allocate large spaces for the filters [Martin et al. (2003)]. As delineated in the above, two major methods, the dielectric resonators and dual-mode techniques, have been utilized for designing small non-planar microwave filters. Another approach for producing non-planar filters with reduced size is utilizing the evanescent mode of the waveguide [Craven and Mok (1971)]. The waveguides of the cavity filters operate above their cutoff frequencies, while the evanescent mode filters operate below the cutoff frequencies [Craven and Mok (1971)].

7. Summary:

• Waveguide is a metallic structure that guides the EM wave in a particular direction. They exist in form of wires, coaxial cables, or parallel plates. Waveguide enables a signal to propagate with minimal loss of energy by restricting expansion to one or more dimensions. The walls of the waveguide represent the distributed inductance, while the empty space between the walls provides distributed capacitance.
• The electromagnetic wave is consists of electric and magnetic field perpendicular to each other but propagate in the direction of propagation. The possible mode of propagation is TE, TM and TEM modes.
• waveguide can be developed in three ways: 1) Stripline, 2) Microstrip, and 3) Coplanar waveguide (CPW). Among the three, CPW is the most successful geometry for transmission of EM wave in the microwave region. In general, Coplanar waveguide consists of a centre conductor and two ground planes printed on the same surface of a dielectric slab

1. Introduction

Major challenge of the twenty first century is to develop self powered systems, especially those which can be operated using energy harvester as electrical power sources [Jones et al. (2001)]. Self powered devices are used in a wide range of wireless applications ranging from encapsulated implants to industrial process monitoring. Furthermore, the increasing demands of wireless sensor networks in mobile devices and the recent advent of extremely low power operated electrical and mechanical devices make such energy harvester sources very attractive. Traditional power sources such as batteries have various limitations in current wireless remote sensor systems which include their large volume, limited lifetime, contribution to environmental pollution and huge maintenance requirements etc. Exploring the possibilities of renewable, sustainable and green energy sources, to replace fossil fuels, is one of the most significant and challenging issues in the energy research. A number of ambient sources such as sunlight, heat, magnetic energy and mechanical vibrations have been studied for generating useful electrical voltage. Among them, mechanical vibrations are independent of weather conditions and offer great potential in various applications as energy harvesters. So, for harvesting mechanical energy, piezoelectric cantilevers are the promising candidates. Let us discuss about the piezoelectric cantilevers in detail in this module.

2. Cantilever

A structural element anchored at one end to a support and subjected to load transverse to its axis at the other end is known as a cantilever. A cantilever is classified under a broader category of beam. In general, a beam can be either free from any axial force or the effect of this force may be negligible. Usually, a beam is considered in horizontal direction and load in vertical direction. The load can be of two types, 1) Concentrated load and 2) Distributed load. The concentrated load is assumed to act at a particular point, though in practice it may be distributed over a small area. On the other hand, distributed load is the one which is spread over the length of the cantilever. However, the rate of loading may be uniform or may vary from one point to another. There are different types of supports for beam which are as follows:

1. Roller support: In case of roller support, a beam rests on a sliding surface like a roller or a flat surface (Figure 1). The roller support can sustain a force normal to its surface as the possible movement on the supporting surface does not allow any resistance in that direction. Therefore, the reaction (R) along the rolling surface is zero and it is present only normal to the surface.

Figure 1: Schematic of roller support

2. Hinged Support: In case of hinged support, the possibility of translational displacement of the beam is zero, however, rotation is possible. In this case, there can be reactions in vertical (R) as well as in horizontal direction (H) (Figure 2).

Figure 2: Schematic of Hinged support

3. Fixed or encastre or built-in support: A built-in rigid support which does not allow any type of movement or rotation is known as fixed or encastre or built-in support. A fixed support exerts a fixed moment (M) and a reaction (R) on the beam (Figure 3).

Figure 3: Schematic of built-in support

A beam with one end fixed and the other end free is called cantilever (Figure 3). There is a vertical reaction

(R) and a moment (M) at the fixed end and is called fixed moment. In the present chapter, cantilever beam is made which is supported from one end (fixed support) and free from other ends. In this case, cantilever beam transfers the load to the rigid support where it manages the moment of force and shear stress [Duan et al. (2014)].

Shear force

Shear force is one of the most important parameters in case of a cantilever. It is an unbalanced vertical force on one side (other than fixed support) of the cantilever beam and is the sum of all the normal forces [Duan et al. (2014)]. In other words, it represents the tendency of either portion of the cantilever to slide or shear laterally relative to the other. Shear force is considered positive when the resultant of the forces to the left of a section is upwards or to the right downwards.

Bending moment

Bending moment is another parameter of interest while understanding the theory behind cantilever. Bending moment at some section of a beam is defined as the algebraic sum of the moments about the section of all the forces on one side of the section [Duan et al. (2014)].

Natural frequency of cantilever

In this section, the natural frequency of the cantilever having tip mass at the free end has been calculated, followed by the frequency calculation of the cantilever with distributed mass. Subsequently, the frequency of the cantilever with distributed mass and tip mass has also been shown.

Figure 4: (a) Cantilever beam having mass (mt) mounted at its free end (mt >> m), and (b) free body diagram of the cantilever system

Consider a cantilever beam having tip mass (mt) mounted on its free end as shown in figure 4 (a). Assume that the end-mass (mt) is much greater than the mass (m) of the cantilever. Free body diagram of the cantilever system is also shown in figure 4 (b), where, E is the modulus of elasticity of the material of cantilever, I is the moment of inertia of cantilever about the fixed support and normal to its surface, L is length of the cantilever, g is acceleration due to gravity, mt is the tip mass mounted at free end of cantilever, R is the reaction force and, MR is the reaction bending moment. Applying Newton’s law for static equilibrium, algebraic sum of rotational force (ѺF) and translational force (ҐF) must be zero [Duan et al. (2014)]. Therefore, at the free end of the cantilever (Figure 4 (b)), we have

Consider the vibration of cantilever of length L about the fixed end and due to tip mass (mt) present at the free end. Under dynamic equilibrium, the cantilever is expected to vibrate with its natural frequency (fn). To determine the value of fn, consider a small segment of length x of the cantilever, starting from the fixed end as shown in figure 5.

Figure 5: Small section of length x of the cantilever with the deflection in y-direction

Let the tip mass (mt) in the cantilever result in a deflection y at small distance x from the fixed end of the cantilever beam. Let M be the moment due to motion of the cantilever segment. Then the algebraic sum of the moments at a distance x from free end of the segment is

Let us now consider a cantilever beam having r as mass per unit length as shown in figure 6. We assume that the cantilever has a uniform cross section. The natural frequency and effective mass of the cantilever (without tip mass) is determined, where the distributed mass is represented by a discrete end-mass.

Figure 6: Schematic of cantilever beam having ρ as mass per unit length

As discussed in previous case, consider a segment of cantilever of small length x from the fixed end and having a displacement y towards normal to cantilever surface at distance x. The governing differential equation of the segment cantilever is given by [Duan et al. (2014)]

where yo is the displacement of cantilever at free end i.e. x=L. It is important to note that the given solution (equation (13)) meets all the boundary conditions except for the zero shear force at the free end of the cantilever (i.e. x=L). Despite the failure of quarter cosine wave solution (equation (13)) to satisfy zero shear force condition at x=L, it is accepted as an approximate solution in order to describe the deflection of cantilever [Eysden and Sader (2006)]. The Rayleigh method [Turner et al. (2011)] is used to find the natural frequency of the cantilever with distributed mass (without tip mass). The total potential energy P of the cantilever is given by [Turner et al. (2001)]

6. Summary:

Piezoelectric cantilevers are the sole candidates which can convert the mechanical vibrations into electrical energy efficiently. The cantilever can be defined as a structural element anchored at ne end to a support and subjected to a load transverse to its axis at the other end. Usually a beam is considered in horizontal direction and load in a vertical direction. The load can be of two types, 1) Concentrated load and 2) Distributed load. On the other hand, distributed load is the one which is spread over the length of the cantilever. However, the rate of loading may be uniform or may vary from one point to another. There are two major forces which are acting on the cantilever. One is shear force and the other one is bending moment. Shear force is an unbalanced vertical force on one side (other than fixed support) of the cantilever beam and is the sum of all the normal forces. Bending moment is another parameter of interest while understanding the theory behind cantilever. Bending moment at some section of a beam is defined as the algebraic sum of the moments about the section of all the forces on one side of the section.

Then we have derived the expression for the natural frequency of the cantilever having tip mass at the free end. The similar study has been carried for the cantilever without tip mass also.