RELATED APPLICATIONS 
     This application claims the benefit of provisional patent application Ser. No. 60/942,265, filed Jun. 6, 2007, the disclosure of which is hereby incorporated by reference in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to Micro-Electro-Mechanical Systems (MEMS) resonators and more specifically relates to anchor, or support, structures for MEMS resonators. 
     BACKGROUND OF THE INVENTION 
     Micro-Electrical-Mechanical Systems (MEMS) resonators have great potential for on-chip selective radio frequency (RF) applications such as oscillators and filters. One of the most important factors in MEMS resonator design is a quality factor (Q) of the MEMS resonator. The quality factor (Q) generally compares a resonant frequency of the MEMS resonator to a rate at which it dissipates energy. The frequency selectivity of the MEMS resonator is dependant on the quality factor (Q). The higher the quality factor (Q), the greater the frequency selectivity. Thus, a high quality factor (Q) is needed for applications in which frequency selectivity is required. 
       FIGS. 1 and 2  illustrate a conventional MEMS resonator  10  including a resonator body  12  connected to anchors  14  and  16  by support structures  18  and  20 , respectively. The support structures  18  and  20  are more specifically support beams  18  and  20 . The anchors  14  and  16  are either connected to or are part of a substrate  22 . As illustrated in  FIG. 2 , the resonator body  12  and support beams  18  and  20  are separated from the substrate  22  by a gap  24  having some height (h). In the conventional MEMS resonator  10 , the support beams  18  and  20  are each designed such that their lengths are exactly quarter-wavelength (λ/4), which is defined by the equation: 
                 λ   4     =       1     4   ·     f   o         ⁢         E   eff       ρ   eff             ,         
where f O  is a resonant frequency in Hertz of the resonator body  12 , E eff  is the Young&#39;s Modulus of the support beams  18  and  20 , and ρ eff  is a density of the support beams  18  and  20 .
 
     However, the conventional MEMS resonator  10  having the quarter-wavelength (λ/4) support beams  18  and  20  has several issues that limit its quality factor (Q) or result in a reduction in its quality factor (Q). One issue with the conventional MEMS resonator  10  that limits or reduces the quality factor (Q) is that there is a significant amount of vibrational energy at the anchor points of the support beams  18  and  20  that is dissipated through the anchors  14  and  16  into the substrate  22  when the conventional MEMS resonator  10  is in vibration mode. Note that the points at which the support beams  18  and  20  are connected to the anchors  14  and  16  are referred to herein as anchor points.  FIG. 3  presents a modal analysis of the conventional MEMS resonator  10 . In general, the quarter-wavelength (λ/4) support beams  18  and  20  do not vibrate with mechanically symmetric mode. For the quarter-wavelength (λ/4) support beams  18  and  20 , a boundary condition is created (i.e., the anchor points are fixed with no vibration) in order to provide vibration at the desired frequency. If all boundary conditions are removed, the quarter-wavelength (λ/4) support beams  18  and  20  will not vibrate at the desired frequency of the conventional MEMS resonator  10 . 
     The quarter-wavelength (λ/4) support beams  18  and  20  have maximum stress at the anchor points and, therefore, the anchors  14  and  16  must provide counter action in order for there to be no movement at the anchor points. In reality, the anchors  14  and  16  close to anchoring points are floating together with the resonator body  12  and the quarter-wavelength (λ/4) support beams  18  and  20  because of manufacturing issues. As a result, near the anchor points, the anchors  14  and  16  vibrate with some displacement in order to counter act with quarter-wavelength (λ/4) movement on the quarter-wavelength (λ/4) support beams  18  and  20 . As a result, a significant amount of vibrational energy is dissipated through the anchors  14  and  16  into the substrate  22 . The dissipation of energy through the anchors  14  and  16  into the substrate  22  is another anchor loss that limits or reduces the quality factor (Q) of the conventional MEMS resonator  10 . 
     Another issue with the conventional MEMS resonator  10  is that the quarter-wavelength (λ/4) support beams  18  and  20  are relatively long. For example, for a desired resonant frequency (f O ), the length of the support beams  18  and  20  may be on the order of tens of microns. The relatively long support beams  18  and  20  raise fabrication issues and may further result in the resonator body  12  bending and touching the substrate  22 , which would of course degrade the quality factor (Q) of the conventional MEMS resonator  10 . 
     Thus, there is a need for an improved MEMS resonator that eliminates or reduces anchor losses, thereby improving a quality factor (Q) of the MEMS resonator. 
     SUMMARY OF THE INVENTION 
     The present invention provides Micro-Electro-Mechanical Systems (MEMS) resonators having support structures that minimize or substantially reduce anchor losses, thereby improving a quality factor (Q) of the MEMS resonators. In general, a MEMS resonator includes a resonator body connected to anchors via support structures. The anchors are connected to or are part of a substrate on which the MEMS resonator is formed. The support structures operate to support the resonator body in free space to enable vibration. The support structures are designed to minimize or substantially reduce energy loss through the anchors into the substrate. 
     In one embodiment, the support structures are mechanically symmetric support structures. More specifically, each of the mechanically symmetric support structures includes a mechanically symmetric support component and a coupling beam. The mechanically symmetric support component is coupled to the anchor, and the coupling beam has a first end coupled to the resonator body and a second end coupled to the mechanically symmetric support component. In addition, anchor points for the mechanically symmetric support component may be selected as nodal points on the mechanically symmetric support component at which there is essentially no vibration. As such, the mechanically symmetric support component vibrates as if does not experience any forced boundary condition. As a result, anchor losses are reduced and, therefore, the quality factor of the MEMS resonator is increased. 
     In another embodiment, the support structures include coupling beams having first ends connected to the resonator body and second ends connected directly or indirectly to the anchors. Unlike the support beams used in conventional MEMS resonators, a length of each of the coupling beams may be of any desired length and is not limited to being quarter-wavelength (λ/4). Each of the coupling beams may have a minimum length obtainable with the fabrication process used to fabricate the MEMS resonator. In another embodiment, each of the coupling beams may have a length that is substantially less than quarter-wavelength (λ/4). The support structures may additionally include mechanically symmetric support components. Each mechanically symmetric support component is coupled to the anchor, and the coupling beam has a first end coupled to the resonator body and a second end coupled to the mechanically symmetric support component. Further, anchor points for the mechanically symmetric support component may be selected as nodal points on the mechanically symmetric support component about which there is symmetric vibration when the MEMS resonator is in vibration mode and at which there is minimal or essentially no vibration when the MEMS resonator is in vibration mode. As a result of the mechanical symmetry of the support structure and the selection of the anchor points as the nodal points having minimal or essentially no vibration, anchor losses are reduced and, therefore, the quality factor of the MEMS resonator is increased. 
     Those skilled in the art will appreciate the scope of the present invention and realize additional aspects thereof after reading the following detailed description of the preferred embodiments in association with the accompanying drawing figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING FIGURES 
       The accompanying drawing figures incorporated in and forming a part of this specification illustrate several aspects of the invention, and together with the description serve to explain the principles of the invention. 
         FIG. 1  illustrates a conventional Micro-Electro-Mechanical Systems (MEMS) resonator having quarter-wavelength (λ/4) support structures; 
         FIG. 2  provides a cross-sectional view of the conventional MEMS resonator of  FIG. 1 ; 
         FIG. 3  is a graphical illustration of a modal analysis of the conventional MEMS resonator of  FIGS. 1 and 2 ; 
         FIG. 4  is a block diagram of a MEMS resonator according to one embodiment of the present invention; 
         FIG. 5  illustrates the effects of a mismatch between a resonant frequency of a clamped-clamped supporting structure of the MEMS resonator and a resonant frequency of a resonator body of the MEMS resonator on a composite resonant frequency of the MEMS resonator; 
         FIG. 6  is a graphical illustration of a quality factor (Q) degradation ratio with respect to the resonant frequency of the supporting structure; 
         FIG. 7  is a block diagram of a MEMS resonator having support structures including coupling beams of a length substantially shorter than quarter-wavelength (λ/4) according to the present invention; 
         FIG. 8  illustrates an exemplary MEMS resonator having coupling beams of a length substantially shorter than quarter-wavelength (λ/4) according to a first embodiment of the present invention; 
         FIG. 9  provides a perspective view of the MEMS resonator of  FIG. 8  according to one embodiment of the present invention; 
         FIG. 10  is a graphical illustration of a modal analysis of the MEMS resonator of  FIG. 8 ; 
         FIG. 11  illustrates an exemplary MEMS resonator having modified clamped-clamped support structures including coupling beams of a length substantially shorter than quarter-wavelength (λ/4) according to a second embodiment of the present invention; 
         FIG. 12  is a graphical illustration of a modal analysis of the MEMS resonator of  FIG. 11 ; 
         FIG. 13  illustrates an exemplary MEMS resonator having mechanically symmetric support structures and anchor points located at points on the mechanically symmetric support structures having minimal or essentially no vibration according to a third embodiment of the present invention; 
         FIG. 14  is a graphical illustration of a modal analysis of the MEMS resonator of  FIG. 13 ; 
         FIG. 15  illustrates an exemplary MEMS resonator having mechanically symmetric support structures designed to provide multimode symmetric vibration and having anchor points located at points on the mechanically symmetric support structures having minimal or essentially no vibration according to a fourth embodiment of the present invention; and 
         FIG. 16  is a graphical illustration of a modal analysis of the MEMS resonator of  FIG. 15 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The embodiments set forth below represent the necessary information to enable those skilled in the art to practice the invention and illustrate the best mode of practicing the invention. Upon reading the following description in light of the accompanying drawing figures, those skilled in the art will understand the concepts of the invention and will recognize applications of these concepts not particularly addressed herein. It should be understood that these concepts and applications fall within the scope of the disclosure and the accompanying claims. 
     The present invention provides support, or anchor, structures for MEMS resonators that reduce or eliminate one or more of the anchor losses occurring in the conventional MEMS resonator  10  of  FIGS. 1 and 2 . Before discussing specific embodiments of the present invention, a mathematical understanding of a MEMS resonator may be beneficial.  FIG. 4  is a block diagram of a MEMS resonator  26 . The MEMS resonator  26  includes a resonator body  28  connected to anchors  30  and  32  by support structures  34  and  36 , respectively. While not illustrated, like the anchors  14  and  16  of  FIGS. 1 and 2 , the anchors  30  and  32  are connected to or are part of a substrate. Note that the anchors  30  and  32  may be part of a single anchor structure surrounding the resonator body  28  or may be two separate anchor structures. The support structures  34  and  36  and the resonator body  28  are separated from the substrate by a gap of free space enabling vibration. 
     The MEMS resonator  26  can be modeled as a second order linear system expressed as: 
                   x   =       force       k   r     ⁡     (     1   +     j   ⁡     (     ω       ω   r     ·     Q   r         )       -       (     ω     ω   r       )     2       )         ⁢           ⁢   and             (     Eqn   .           ⁢   1     )                 x   =     force       k   s     ⁡     (     1   +     j   ⁡     (     ω       ω   s     ·     Q   s         )       -       (     ω     ω   s       )     2       )           ,                           
where x represents the displacement of both the resonator body  28  and the support structures  34  and  36 , k r , ω r , and Q r  are a spring stiffness, resonant frequency in radians, and quality factor for the resonator body  28 , and k s , ω s , and Q s  are a spring stiffness, resonant frequency in radians, and quality factor for the support structures  34  and  36 . The composite system can be modeled as a parallel connection of two spring systems and expressed as:
 
               x   =     force         k   r     ⁡     (     1   +     j   ⁡     (     ω       ω   r     ·     Q   r         )       -       (     ω     ω   r       )     2       )       +       k   s     ⁡     (     1   +     j   ⁡     (     ω       ω   s     ·     Q   s         )       -       (     ω     ω   s       )     2       )             ,         
which may be further reduced to:
 
                     x   =     force       k   o     ⁡     (     1   +     j   ⁡     (     ω       ω   o     ·     Q   o         )       -       (     ω     ω   o       )     2       )           ,           (     Eqn   .           ⁢   2     )               
where k o , ω o , and Q o  are a composite spring stiffness, composite resonant frequency in radians, and composite quality factor for the MEMS resonator  26 . The composite spring stiffness k o  may be expressed as:
 
 k   o   =k   r   +k   s ,  (Eqn. 3)
 
the composite resonant frequency ω o  may be expressed as:
 
                       1     ω   o   2       =           k   s       k   o       ·     1     ω   s   2         +         k   r       k   o       ·     1     ω   r   2             ,   and           (     Eqn   .           ⁢   4     )               
the composite quality factor Q o  may be expressed as:
 
     
       
         
           
             
               
                 
                   
                     
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     From equations 1 through 5 it can be seen that the resonant frequency (ω s ) of the support structures  34  and  36  does not need to match the resonant frequency (ω r ) of the resonator body  28 . This is further illustrated in  FIG. 5 . Additionally, as illustrated in  FIG. 6 , a quality factor (Q) degradation ratio, Q r /Q o , is not substantially affected if the resonant frequency (f s ), which in this figure is shown in MHz, of the support structures  34  and  36  does not match the resonant frequency (f r ) of the resonator body  28 . This is especially true as the quality factor (Q s ) of the support structures  34  and  36  becomes large. Also note that equation 5 illustrates that the composite quality factor (Q o ) of the MEMS resonator  26  may be increased by increasing the quality factor (Q s ) of the support structures  34  and  36  and/or decreasing the spring stiffness (k s ) of the support structures  34  and  36 . 
       FIG. 7  is a block diagram of a MEMS resonator  26  according to the present invention. Again, the MEMS resonator  26  includes the resonator body  28  connected to the anchors  30  and  32  by the support structures  34  and  36 , respectively. Note that while two support structures  34  and  36  are illustrated, the MEMS resonator  26  may include any number of one or more support structures depending on the implementation. As illustrated, the support structure  34  more specifically includes a support component  38  and a coupling beam  40 . Likewise, the support structure  36  includes a support component  42  and a coupling beam  44 . The coupling beams  40  and  44  interconnect the support components  38  and  42 , respectively, to the resonator body  28 . Since the resonant frequency of the support structures  34  and  36  does not need to match that of the resonator body  28 , the coupling beams  40  and  44  are other than quarter-wavelength (λ/4) if desired. In one embodiment, the coupling beams  40  and  44  have the minimum lengths possible with the fabrication process utilized to fabricate the MEMS resonator  26 . 
     Preferably, the coupling beams  40  and  44  are as short as allowed by the manufacturing techniques used to fabricate the MEMS resonator  26 . For example, in one embodiment, a fabrication process may be used that enables the coupling beams  40  and  44  to have lengths of 1 or 2 microns whereas corresponding quarter-wavelength (λ/4) support beams would have lengths of approximately 20 microns. In another embodiment, the coupling beams  40  and  44  may have lengths substantially less than quarter-wavelength (λ/4). As an example, the coupling beams  40  and  44  may have lengths less than a fraction of the quarter-wavelength (λ/4) such as, for example, ¾, ½, or ¼ of the quarter-wavelength (λ/4) of the resonant frequency of the resonator body  28  and/or the resonant frequency of the MEMS resonator  26 . Note that for higher frequencies, such as for example 1-2 gigahertz (GHz), the fabrication process may not enable the coupling beams  40  and  44  to have lengths less than or equal to quarter-wavelength (λ/4). While this may be an issue with the conventional MEMS resonator  10  of  FIG. 1 , the support structures  34  and  36  of the MEMS resonator  26  of the present invention may have resonant frequencies that are different than that of the resonator body  28  without substantially affecting the composite quality factor (Q o ). As such, the coupling beams  40  and  44  may also have lengths greater than quarter-wavelength (λ/4) if desired. 
     In one embodiment, ω r ≈ω s ≈ω o , k s &lt;&lt;k r  such that k o ≈k r , and composite quality factor (Q o ) of at least Q r /2 is desired. Thus, starting with equation 5 from above: 
                 k   o         ω   o     ·     Q   o         =             k   r         ω   r     ·     Q   r         +       k   s         ω   s     ·     Q   s           →     Q   o       =       1         (         ω   o       ω   r       ·       k   r       k   o         )     ·     1     Q   r         +       (         ω   o       ω   s       ·       k   s       k   o         )     ·     1     Q   s             .             
Since ω r ≈ω o , k o ≈K r , ω s ≈ω o , Equation (5) yields:
 
                         Q   o     ≈     1       1     Q   r       +       k   so       Q   s             ⁢     →       Q   o     ≥       Q   r     2         ⁢         1     Q   r       ≥       k   so       Q   s         →       k   so     ≈       k   s       k   r       ≤       Q   s       Q   r             ,           (     Eqn   .           ⁢   6     )               
where:
 
     
       
         
           
             
               
                 
                   
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     Thus, the composite quality factor (Q o ) of the MEMS resonator  26  is controlled by k so  and Q s . The parameters k so  and Q s  mitigate the requirement of each other, i.e., a higher Q s  allows a higher k s  and a lower k s  allows a lower Q s . Therefore, high Q s  and low k s  are desirable for the design of support structures in order to maximize the composite quality factor (Q o ). In one exemplary embodiment, in the MEMS resonator  26 , the quality factor (Q s ) of the support structures  34  and  36  is in the approximate range of 1/10 to 1/1000 of the quality factor (Q r ) of the resonator body  28 , and the spring stiffness (k s ) of the support structures  34  and  36  is in the approximate range of 1/10 to 1/1000 of the spring stiffness (k r ) of the resonator body  28 . As a result, the composite quality factor (Q o ) of the MEMS resonator  26  is substantially improved as compared to that of the conventional MEMS resonator  10  ( FIG. 1 ). Note that the ranges for Q s  and k s  given above are exemplary and are not intended to limit the scope the present invention. The ranges may vary depending on the particular implementation, as will be appreciated by one of ordinary skill in the art. 
       FIGS. 8 and 9  provide a more detailed illustration of the MEMS resonator  26  according to a first exemplary embodiment of the present invention. Again, the MEMS resonator  26  includes the resonator body  28  connected to the anchors  30  and  32  by the support structures  34  and  36 , respectively. In this example, the resonator body  28  is disc-shaped. However, the present invention is not limited thereto. The resonator body  28 , anchors  30  and  32 , and support structures  34  and  36  are preferably made of acoustic materials such as, for example, silicon or piezo crystals such as quartz, lithium tantalate, lithium niobate, aluminum nitride, zinc oxide, or the like. As shown in  FIG. 9 , the anchors  30  and  32  are connected to or part of a substrate  46 , and the support structures  34  and  36  and the resonator body  28  are separated from the substrate  46  by a gap of free space having a height (h) in order to enable vibration. The resonator body  28  may be any type of vibrating MEMS resonator such as, for example, a piezo-transduced MEMS resonator, an electrostatically transduced MEMS resonator, a Thin Film Bulk Acoustic Resonator (FBAR), a piezoelectric resonator, or the like. 
     In this embodiment, the support structures  34  and  36  are clamped-clamped support structures. More specifically, in this embodiment, the support component  38  ( FIG. 7 ) of the support structure  34  is formed by a clamped-clamped beam  48 , which is connected to the resonator body  28  by the coupling beam  40 . The clamped-clamped beam  48  has a first end connected to the anchor  30  at an anchor point  50  and a second end connected to the anchor  30  at an anchor point  52 . The coupling beam  40  has a first end connected to the clamped-clamped beam  48  at an intermediate point of the clamped-clamped beam  48  between the first and second ends of the clamped-clamped beam  48 . A second end of the coupling beam  40  is connected to the resonator body  28 . Likewise, the support component  42  ( FIG. 7 ) of the support structure  36  is formed by a clamped-clamped beam  54 , which is connected to the resonator body  28  by the coupling beam  44 . The clamped-clamped beam  54  has a first end connected to the anchor  32  at an anchor point  56  and a second end connected to the anchor  32  at an anchor point  58 . The coupling beam  44  has a first end connected to the clamped-clamped beam  54  at an intermediate point of the clamped-clamped beam  54  between the first and second ends of the clamped-clamped beam  54 . A second end of the coupling beam  44  is connected to the resonator body  28 . Note that while the discussion herein uses the phrase “connected to” with respect to the anchors  30  and  32 , the support structures  34  and  36 , and the resonator body  28 , one of ordinary skill in the art will recognize that the anchors  30  and  32 , the support structures  34  and  36 , and the resonator body  28  may completely or partially be formed from a common epitaxial material by, for example, etching. 
     The clamped-clamped beam support structures  34  and  36  of  FIG. 8  are designed such that the spring stiffness (k s ) of the support structures  34  and  36  is substantially less than that of the quarter-wavelength (λ/4) support beams  18  and  20 . Thus, according to equation 5 above, by reducing the spring stiffness (k s ) of the support structures  34  and  36 , the composite quality factor (Q o ) is increased. 
     However, the composite quality factor (Q o ) of the MEMS resonator  26  of  FIGS. 8 and 9  is limited or reduced by the fact that the support components  38  and  42 , which are the clamped-clamped beams  48  and  54 , of the support structures  34  and  36  of this embodiment are not mechanically symmetric. As used herein, a mechanically symmetric support component is a support component that vibrates as if it experiences no, or essentially no, forced boundary condition. As discussed below, the composite quality factor (Q o ) can be substantially improved by using a mechanically symmetric support structure and anchoring the mechanically symmetric support structure at nodal points that naturally experience essentially no vibration when the MEMS resonator  26  is in vibration mode. Thus, the clamped-clamped beams  48  and  54  are not mechanically, or vibrationally, symmetric because the clamped-clamped beams  48  and  54  are forced to be anchored at both ends, where the ends of the clamped-clamped beams  48  and  54  would not be nodal points if the clamped-clamped beams  48  and  54  were in vibration without any forced boundary condition. This is illustrated in  FIG. 10 , which provides a modal analysis of the MEMS resonator  26  of  FIGS. 8 and 9 . As shown in  FIG. 10 , the anchor points  50 ,  52 ,  56 , and  58  are at points on the clamped-clamped beams  48  and  54  having significant vibrational energy. As such, a significant amount of stress is induced in order to keep the anchor points  50 ,  52 ,  56 , and  58  fixed, thereby resulting in a significant amount of energy loss through the anchors  30  and  32  into the substrate  46 . 
       FIG. 11  is a more detailed illustration of the MEMS resonator  26  according to a second exemplary embodiment of the present invention. This embodiment is similar to the embodiment of  FIGS. 8 and 9 . However, the support structures  34  and  36  in this embodiment are modified clamped-clamped support structures that further reduce the spring stiffness (k s ) and approach vibration mode close to mechanical symmetry. 
     In this embodiment, the support structures  34  and  36  are modified clamped-clamped support structures. More specifically, the support component  38  ( FIG. 7 ) of the support structure  34  is formed by a modified clamped-clamped beam  48 ′, which is connected to the resonator body  28  by the coupling beam  40 . As illustrated, the modified clamped-clamped beam  48 ′ includes detours  60  and  62 , which in turn modify the anchor points  50  and  52  at which the support structure  34  is attached to the anchor  30 . Likewise, the support component  42  ( FIG. 7 ) of the support structure  36  is formed by a modified clamped-clamped beam  54 ′, which is connected to the resonator body  28  by the coupling beam  44 . As illustrated, the modified clamped-clamped beam  54 ′ includes detours  64  and  66 , which in turn modify the anchor points  56  and  58  at which the support structure  36  is attached to the anchor  32 . 
     By detouring the anchor points  50 ,  52 ,  56 , and  58 , the spring stiffness (k s ) of the modified clamped-clamped support structures  34  and  36  is reduced as compared to that of the clamped-clamped support structures  34  and  36  of  FIGS. 8 and 9 . In addition, as illustrated in  FIG. 12 , by detouring the anchor points  50 ,  52 ,  56 , and  58 , the modified clamped-clamped support structures  34  and  36  approach mechanical symmetry. As a result, the quality factor (Q s ) is substantially improved. Since the spring stiffness (k s ) is reduced and since the modified clamped-clamped support structures  34  and  36  have a vibration mode that is close to mechanical symmetry, the composite quality factor (Q o ) is substantially increased as compared to that of the conventional MEMS resonator  10  of  FIGS. 1 and 2  and the MEMS resonator  26  having the clamped-clamped support structures  34  and  36  of  FIG. 8 . For example, in one embodiment, the clamped-clamped support structures  34  and  36  of  FIG. 8  may be formed of single crystal silicon, have dimensions of 1×6×2 micrometers (μm), have a spring stiffness (k s ) of approximately 8,300 Newtons per meter (N/m), and have a quality factor (Q s ) of 132. In contrast, a similar embodiment of the clamped-clamped support structures  34  and  36  of  FIG. 11  provide a spring stiffness (k s ) of approximately 7,700 N/m and, more importantly, a quality factor (Q s ) of 646. 
       FIG. 13  illustrates the MEMS resonator  26  according to a third exemplary embodiment of the present invention. In this embodiment, the support structures  34  and  36  are mechanically symmetric support structures providing fundamental mode symmetric vibration when the resonator body  28  is vibrating or in vibration mode. More specifically, the support component  38  ( FIG. 7 ) of the support structure  34  is formed by a mechanically symmetric support component  68 , which is connected to the resonator body  28  by the coupling beam  40 . In this exemplary embodiment, the mechanically symmetric support component  68  is a rectangular ring structure, which may also be referred to herein as a double-ended tuning fork structure, formed by beams  70  and  72  arranged as shown. Note that, in this embodiment, the length of each of the beams  70  and  72  may be selected to provide fundamental mode vibration when the MEMS resonator  26  is in vibration mode. Note that the resonant frequency of the support structure  34  does not need to match resonant frequency of resonator body  28 . 
     The mechanically symmetric support component  68  vibrates as if it experiences no, or essentially no, forced boundary condition and is anchored at anchoring points  74  and  76  that are nodal points experiencing essentially no vibration. In other words, the mechanically symmetric support component  68  provides symmetric vibration about mirroring points having minimal or essentially no vibration when the MEMS resonator  26  is in vibration mode. The mirroring points are selected as the anchoring points  74  and  76 . When vibrating, the vibration mode on the beam  70  is the same as the vibration mode on the beam  72  with out-of-phase providing momentum cancellation, resulting in mirroring points at which there is minimal or essentially no vibration. The mirroring points are also referred to herein as nodal points having minimal or essentially no vibration. By selecting the mirroring points as the anchoring points  74  and  76 , dissipation of energy through the anchor  30  into the substrate  46  is reduced or minimized. 
     Likewise, the support component  42  ( FIG. 7 ) of the support structure  36  is formed by a mechanically symmetric support component  78 , which is connected to the resonator body  28  by the coupling beam  44 . In this exemplary embodiment, the mechanically symmetric support component  78  is a rectangular ring structure, which may also be referred to herein as a double-ended tuning fork structure, formed by beams  80  and  82  arranged as shown. Again, the mechanically symmetric support component  78  provides symmetric vibration about mirroring points when in vibration mode. The mirroring points are selected as anchoring points  84  and  86 . In other words, when vibrating, the vibration mode on the beam  80  is the same as the vibration mode on the beam  82 , resulting in mirroring points at which there is minimal or essentially no vibration. The mirroring points are also referred to herein as nodal points having minimal or essentially no vibration. By selecting the mirroring points as the anchoring points  84  and  86 , dissipation of energy through the anchor  32  into the substrate  38  is reduced or minimized. 
     The mechanically symmetric support components  68  and  78  provide cancellation of momentum at the anchoring points  74 ,  76  and  84 ,  86 , respectively, causing essentially no vibration of the anchors  30  and  32 . As a result, vibration is confined to the resonator body  28  and the support structures  34  and  36 . As a result of using the mechanically symmetric support components  68  and  78  and anchoring the mechanically symmetric support components  68  and  78  at their respective mirroring points, energy losses through the anchors  30  and  32  into the substrate  46  are substantially reduced and, therefore, the composite quality factor (Q o ) of the MEMS resonator  26  is substantially increased. As an example, by using the mechanically symmetric support components  68  and  78  and by having the short coupling beams  40  and  44 , the composite quality factor (Q o ) of the MEMS resonator  26  may be, for example, at least approximately 10 times that of the conventional MEMS resonator  10  of  FIGS. 1 and 2 . Note, however, that these improvement factors are exemplary and may vary from device to device depending on various factors, as will be apparent to one of ordinary skill in the art upon reading this disclosure. 
       FIG. 14  illustrates a modal analysis of the MEMS resonator  26  of  FIG. 13 . As can be seen, the vibration modes of the beams  70  and  72  of the mechanically symmetric support component  68  are symmetric. Likewise, the vibration modes of the beams  80  and  82  of the mechanically symmetric support component  78  are symmetric. Further, there are naturally occurring nodes having minimal or essentially no vibration at the locations of the anchor points  74 ,  76 ,  84 , and  86 . 
       FIG. 15  illustrates the MEMS resonator  26  according to a fourth exemplary embodiment of the present invention. This embodiment is substantially the same as that of  FIG. 13 . However, the length of the beams  70 ,  72 ,  80 , and  82  are selected to achieve multimode symmetric vibration rather than fundamental mode vibration. More specifically, as the desired resonant frequency (ω s ) for the support structures  34  and  36  increases, the length of the beams  70 ,  72 ,  80 , and  82  of the mechanically symmetric support structures  68  and  78  would have to decrease in order to maintain fundamental mode operation. However, as the length of the beams  70 ,  72 ,  80 , and  82  is decreased to accommodate higher resonance frequencies, the spring stiffness (k s ) increases and the quality factor (Q s ) decreases. Further, at some point, fabrication limitations may prevent further shortening of the beams  70 ,  72 ,  80 , and  82 . Thus, in this embodiment, the lengths of the beams  70 ,  72 ,  80 , and  82  are selected to provide multimode symmetric vibration, thereby increasing the lengths of the beams  70 ,  72 ,  80 , and  82  as compared to those used for fundamental mode symmetric vibration. As an example, by using the mechanically symmetric support components  68  and  78  with multimode symmetric vibration and by having the short coupling beams  40  and  44 , the composite quality factor (Q o ) of the MEMS resonator  26  may be, for example, at least approximately 10 times that of the conventional MEMS resonator  10  of  FIGS. 1 and 2 . With careful design, the composite quality factor (Q o ) may exceed 100,000. Note, however, that the improvement factors given above are exemplary and may vary from device to device depending on various factors, as will be apparent to one of ordinary skill in the art upon reading this disclosure. 
       FIG. 16  illustrates a modal analysis of the MEMS resonator  26  of  FIG. 15 . As shown, there is multimode symmetric vibration of the beams  70 ,  72 ,  80 , and  82 . More specifically, each of the beams  70 ,  72 ,  80 , and  82  has multiple points of high vibration and multiple points of low or essentially zero vibration. Further note that the anchor points  74 ,  76 ,  84 , and  86  remain at mirroring points, which are nodes having minimal or essentially no vibration. 
     It should be noted that while the mechanically symmetric support components  68  and  78  of  FIGS. 13-16  are illustrated as being rectangular rings or double-ended tuning fork shaped, the present invention is not limited thereto. For example, the mechanically symmetric support components  68  and  78  may be rectangular rings, square rings, circular rings, oval rings, hexagons, octagons, or similar structures that are mechanically symmetric and have nodal points where there is minimal or essentially no vibration that can be used for anchoring points. Still further, the mechanically symmetric support components  68  and  78  may be arrays of two or more rectangular rings, square rings, circular rings, oval rings, hexagons, octagons, similar structures, or any combination thereof. 
     Those skilled in the art will recognize improvements and modifications to the preferred embodiments of the present invention. All such improvements and modifications are considered within the scope of the concepts disclosed herein and the claims that follow.
Micro-Electro-Mechanical Systems (MEMS) resonator designs having support structures that minimize or substantially reduce anchor losses, thereby improving a quality factor (Q) of the MEMS resonators, are provided. In general, a MEMS resonator includes a resonator body connected to anchors via support structures. The anchors are connected to or are part of a substrate on which the MEMS resonator is formed. The support structures operate to support the resonator body in free space to enable vibration. The support structures are designed to minimize or substantially reduce energy loss through the anchors into the substrate.