Patent Publication Number: US-8111114-B2

Title: MEMS filter with voltage tunable center frequency and bandwidth

Description:
RELATED APPLICATIONS 
     This application claims priority to international PCT patent application PCT/US2007/068018 filed on May 2, 2007 which claims priority to U.S. provisional patent application 60/746,210 filed May 2, 2006 and entitled, “MEMS Filter with Voltage Tunable Center Frequency and Bandwidth.” Therefore, this application also claims priority to the 60/746,210 provisional U.S. patent application as well. The 60/746,210 provisional patent application and the CT/US2007/068018 PCT patent application are hereby incorporated by reference in their entirety. 
    
    
     FIELD 
     The present invention relates to MEMS filters, and, more particularly, to voltage tunable MEMS filters. 
     BACKGROUND 
     High-Q microelectromechanical (MEMS) resonators are ideal replacements for conventional lumped LC components in radio frequency applications. Ladder and lattice filters built from MEMS resonators have better shape factor due to their inherent high mechanical quality factors (Q˜1000-10,000) compared to quality factors of electrical LC components (Q˜200). However, a major disadvantage of current MEMS filters is the lack of frequency and bandwidth tunability. 
     Therefore, what is needed is a MEMS filter with a tunable center frequency and bandwidth. 
     SUMMARY 
     A tunable MEMS filter is disclosed. The tunable filter has a substrate having a first isolated substrate area and a second isolated substrate area. The tunable filter also has first and second anchor points coupled to the substrate. The tunable filter further has a base coupled to the first and second anchor points by first and second coupling beams, respectively. The tunable filter has a dielectric layer coupled to the base. The tunable filter further has an input conductor coupled to the dielectric layer. The tunable light filter also has an output conductor coupled to the dielectric layer, wherein the first isolated substrate area is configured to receive a first substrate voltage with respect to the base; and the second isolated substrate area is configured to receive a second substrate voltage with respect to the base. 
     A method of tuning a center frequency and a bandwidth of a MEMS resonator filter is also disclosed. A first bias voltage is adjusted between a base layer and input and output conductor layers. A second bias voltage is adjusted between the base layer and isolated substrate areas below at least a portion of the base layer. The center frequency and the bandwidth of the MEMS resonator filter are determined until the adjustments to the first bias voltage and the second bias voltage provide a desired center frequency and a desired bandwidth, wherein adjusting the first bias voltage and the second bias voltage comprises: while holding the first bias voltage fixed, adjusting the second bias voltage such that the desired center frequency is obtained, noting the difference between the first bias voltage and the second bias voltage for the desired center frequency, adjusting the first bias voltage and the second bias voltage while maintaining the noted difference between the first bias voltage and the second bias voltage to obtain the desired bandwidth, making the first bias voltage and the second bias voltage the same, while keeping the second bias voltage the same as the first bias voltage, adjusting the first bias voltage to obtain the desired bandwidth, and while maintaining the first bias voltage, adjusting the second bias voltage to obtain the desired center frequency. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a perspective view of one embodiment of a MEMS resonator filter; 
         FIG. 2  is an equivalent circuit of the resonator shown in  FIG. 1 ; 
         FIG. 3  is a plot of a simulation of the variation of the resonator transfer function with the applied structural bias voltage for two resonators with different series resonant frequencies; 
         FIG. 4  is a depiction of the deformation of the resonator shown in  FIG. 1  when tuned by orthogonal frequency tuning; 
         FIG. 5  is a plot of a simulation of the variation of the output transfer function with the voltage difference between the structural bias voltage and the substrate tuning voltage; 
         FIG. 6A  is a plot of the transmission characteristics of a MEMS resonator according an embodiment the present invention for a first DC polarization voltage; 
         FIG. 6B  is a plot of the transmission characteristics of a MEMS resonator according an embodiment the present invention for a second DC polarization voltage; 
         FIG. 6C  is a plot of the transmission characteristics of a MEMS resonator according an embodiment the present invention for a third DC polarization voltage; 
         FIG. 6D  is a plot of the pole-zero separation shown in  FIGS. 6A-6C  as a function of the DC polarization voltage; 
         FIG. 7A  is a perspective view of three of the resonators shown in  FIG. 1  arranged in one embodiment of a ladder filter configuration; 
         FIG. 7B  is a cross-sectional view of two of the resonators shown in  FIG. 7A ; 
         FIG. 7C  is a top view from a Scanning Electron Microscope of the ladder filter shown in  FIG. 7A ; 
         FIG. 8  is a plot of the calculated transfer function of a first example of a MEMS voltage tunable filter according to the present invention; 
         FIG. 9  is a plot of the calculated transfer function of a second example of a MEMS voltage tunable filter according to the present invention; 
         FIG. 10A  is a plot of the transfer function of the filter shown in  FIG. 7A  with the structural bias voltage and the substrate tuning voltage of all of the resonators at 5 volts; 
         FIG. 10B  is a plot showing the transfer function of  FIG. 10A  and the transfer function for a first set of structural bias voltages and substrate tuning voltages for the filter shown in  FIG. 7A ; 
         FIG. 10C  is a plot showing the transfer function of  FIG. 10A  and the transfer function for a second set of structural bias voltages and substrate tuning voltages for the filter shown in  FIG. 7A ; and 
         FIG. 10D  is a plot showing the transfer function of  FIG. 10A  and the transfer function for a third set of structural bias voltages and substrate tuning voltages for the filter shown in  FIG. 7A . 
         FIGS. 11A-11C  schematically illustrate embodiments of ladder filters using MEMS resonators. 
         FIG. 12  schematically illustrates an embodiment of a lattice filter which uses MEMS resonators. 
         FIGS. 13-15  illustrate embodiments of methods for tuning the center frequency and the bandwidth of a MEMS resonator filter. 
     
    
    
     It will be appreciated that for purposes of clarity and where deemed appropriate, reference numerals have been repeated in the figures to indicate corresponding features, and that the various elements in the drawings have not necessarily been drawn to scale in order to better show the features of the invention. 
     DETAILED DESCRIPTION 
       FIG. 1  schematically illustrates a perspective view of an embodiment of a MEMS resonator filter  10  using dielectric transduction. The filter  10  has a base  12 . The base  12  can be made, for example, from doped silicon, but in other embodiments, other conductive materials may be used. A dielectric layer  14  is coupled to the base  12 . In the illustrated embodiment, the dielectric layer is divided into two portions, but in other embodiments, the dielectric layer  14  can be one continuous layer. A variety of materials can be used for the dielectric layer  14 , such as, but not limited to, hafnium dioxide. The dielectric layer  14  may be deposited on the base  12 . Coupled to the dielectric layer  14  are an input conductor  16  and an output conductor  18 . Suitable material for the input and output conductors  16 ,  18  can include polysilicon. The base  12  is separated from a substrate  13  except at two anchor points  20  and  22  which are attached to the substrate  13 . The substrate  13  is shown with dashed lines, since a variety of substrate shapes can be used while not changing the nature of the claimed invention. The main rectangular section  24  of the resonator  10  is supported by two tether points  26 , one of which is visible in  FIG. 4 . Returning to  FIG. 1 , below the main rectangular section  24  are two isolated substrate areas  28  and  30  which are electrically isolated from the substrate  13 . This electrical isolation can be from physical separation of the isolation substrate areas  28 ,  30  from the substrate  13 , or it can be the result of doping the isolation substrate areas  28 ,  30  so that they are conductive in an otherwise non-conductive substrate  13 , or the isolation substrate areas  28 ,  30  may be formed by deposition of conductive material on a non-conductive and/or insulated substrate  13 . 
     In operation an input signal is applied to the input conductor  16  at the extension of the input conductor  16  over the input anchor point  22 . The output signal is taken from the output conductor  18  at the extension of the output conductor  18  over the output anchor point  20 . DC polarization voltages, V p ,  32  and  34  are applied between the base  12  and each of the input and output conductors  16  and  18 , respectively. DC substrate bias voltages, V s ,  36  and  39  are applied between the base  12  and each of the two isolated substrate areas  28  and  30 , respectively. 
       FIG. 2  is an equivalent circuit of the resonator  10  consisting of a series RLC circuit of R x , C x , and L x  in parallel with a feedthrough capacitance C ft . For a given transduction efficiency, η, 
     
       
         
           
             
               
                 
                   η 
                   ≡ 
                   
                     
                       V 
                       p 
                     
                     · 
                     
                       
                         ∂ 
                         C 
                       
                       
                         ∂ 
                         x 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     R 
                     x 
                   
                   = 
                   
                     b 
                     
                       η 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     x 
                   
                   = 
                   
                     
                       η 
                       2 
                     
                     K 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     L 
                     x 
                   
                   = 
                   
                     M 
                     
                       η 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     where b, K and M denote damping constant, effective spring constant, and effective mass of the resonator, respectively. The feedthrough capacitance originates from electric field coupling from the input conductor  16  to the output conductor  18  in a two-port resonator and therefore is a function of the structure layout. 
     The series resonance frequency is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ω 
                           series 
                         
                         = 
                         
                           1 
                           
                             
                               
                                 L 
                                 x 
                               
                               ⁢ 
                               
                                 C 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             η 
                             
                               M 
                             
                           
                           · 
                           
                             
                               K 
                             
                             η 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             K 
                             M 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     A convenient expression for the parallel resonance frequency can be obtained through application of Taylor&#39;s expansion: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ω 
                           parallel 
                         
                         = 
                         
                           1 
                           
                             
                               Lx 
                               ⁢ 
                               
                                 CxCft 
                                 
                                   Cx 
                                   + 
                                   Cft 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ω 
                             series 
                           
                           ⁢ 
                           
                             
                               1 
                               + 
                               
                                 Cx 
                                 Cft 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ω 
                             series 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 Cx 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Cft 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               K 
                               M 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   η 
                                   2 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     C 
                                     ft 
                                   
                                   ⁢ 
                                   K 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             
                               K 
                               M 
                             
                           
                           + 
                           
                             
                               η 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 C 
                                 f 
                               
                               ⁢ 
                               t 
                               ⁢ 
                               
                                 KM 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             series 
                           
                           + 
                           
                             
                               η 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 C 
                                 ft 
                               
                               ⁢ 
                               
                                 KM 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Substituting for η for parallel plate actuation: 
                   η   =         V   p     ⁢   ɛ   ⁢           ⁢   A       d   2               (   7   )               
where ∈=dielectric permittivity, A=electrode area, and d=parallel plate gap size.
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ω 
                           parallel 
                         
                         = 
                         
                           
                             ω 
                             series 
                           
                           + 
                           
                             
                               η 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 C 
                                 ft 
                               
                               ⁢ 
                               
                                 KM 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ω 
                             series 
                           
                           + 
                           
                             
                               
                                 V 
                                 p 
                                 2 
                               
                               ⁢ 
                               
                                 ɛ 
                                 2 
                               
                               ⁢ 
                               
                                 A 
                                 2 
                               
                             
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 d 
                                 4 
                               
                               ⁢ 
                               
                                 C 
                                 ft 
                               
                               ⁢ 
                               
                                 KM 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ω 
                             series 
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   
                                     ɛ 
                                     2 
                                   
                                   ⁢ 
                                   
                                     A 
                                     2 
                                   
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     d 
                                     4 
                                   
                                   ⁢ 
                                   
                                     C 
                                     ft 
                                   
                                   ⁢ 
                                   
                                     M 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 V 
                                 p 
                                 2 
                               
                               
                                 K 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             ω 
                             series 
                           
                           + 
                           
                             β 
                             ⁢ 
                             
                               
                                 V 
                                 p 
                                 2 
                               
                               
                                 K 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Differentiating the above equation with respect to K, the following equations can be obtained: 
     
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ω 
                       parallel 
                     
                   
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ω 
                         series 
                       
                     
                     - 
                     
                       β 
                       ⁢ 
                       
                         
                           
                             V 
                             p 
                             2 
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           K 
                         
                         
                           2 
                           ⁢ 
                           
                             K 
                             
                               3 
                               / 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     ⁡ 
                     
                       ( 
                       
                         
                           ω 
                           parallel 
                         
                         - 
                         
                           ω 
                           series 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           η 
                           2 
                         
                         ⁢ 
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         K 
                       
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           ft 
                         
                         ⁢ 
                         K 
                         ⁢ 
                         
                           KM 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ω 
                       series 
                     
                   
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       K 
                     
                     
                       2 
                       ⁢ 
                       
                         KM 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             ω 
                             parallel 
                           
                           - 
                           
                             ω 
                             series 
                           
                         
                         ) 
                       
                     
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ω 
                         series 
                       
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           η 
                           2 
                         
                       
                       
                         2 
                         ⁢ 
                         
                           KC 
                           ft 
                         
                       
                     
                     = 
                     
                       
                         C 
                         x 
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           ft 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     The ratio of C x  to C ft  is very small (10 −4 -10 −2 ) for electrostatic actuation. This ratio is also sometimes expressed as the electromechanical coupling factor k e   2 . The pole-zero distance is effectively independent of series resonance frequency shifts. 
     Therefore, β can be redefined to absorb the K dependence in the parallel resonance frequency equation
 
ω paralel =ω series   +βV   p   2   (13)
 
     Therefore, the parallel resonance frequency is an offset from the series resonance frequency; the offset being directly proportional to the square of structural bias voltage. 
     An intuitive explanation of the voltage tunable parallel resonance frequency but voltage independent series resonance frequency is as follows. At series resonance, the feedthrough capacitance is negligible. C x  is proportional to the square of V p  but L x  is inversely proportional to square of V p . The effect of bias voltage cancels perfectly in the expression for series resonance frequency. At parallel resonance, however, the feedthrough capacitance C ft  is in series with C x , and it is no longer negligible. Since C ft  is independent of V p , the effect of bias voltage on the total capacitance and inductance do not cancel perfectly in the parallel resonance frequency expression. Hence the parallel resonance frequency is tunable through structural bias voltage. A simulation of the variation of resonator transfer function with V p  shown as curves  50 - 62  and  70 - 82  for two resonators with different series resonance frequency is shown in  FIG. 3 . Note that the series resonance frequency does not change with V p . 
     Tuning of the series resonance frequency can be done by varying the spring constant of the resonator. For high frequency RF applications, the resonator spring constant must be very high. Hence a large force in the direction of vibration is required to change the spring constant appreciably. One possible method to tune the series resonance frequency of a resonator is through Orthogonal Frequency Tuning. In Orthogonal Frequency Tuning, the resonator is bent by the electrostatic field  86  produced by V s  in a direction orthogonal to the direction of vibration, as shown in  FIG. 4 , where the spring constant is much smaller. A much smaller force is required to time the spring constant and hence the series resonance frequency. 
     The precise operation of Orthogonal Frequency Tuning depends on the device geometry and mode of vibration. For example, consider a released thickness shear mode resonator suspended by quarter-wave tethers. A voltage V p  is applied to the vibrating structure and a voltage V s  is applied to the isolated substrate. The voltage difference V p −V s  causes an electrostatic force that deflects the structure towards the isolated substrate. Bending the structure changes its stiffness and hence its resonance frequency. 
       FIG. 5  shows a simulation of the output transfer function as V p -V s  varies, V p =5V in this simulation as shown by curves  90 - 100  and  110 - 120  for a series resonator in a ladder filter  128  shown in  FIG. 7A  and a shunt resonator in the ladder filter  128 , respectively. The series resonance frequency has a tuning range of ˜5 MHz. The shunt resonator is longer than the series resonator in one embodiment of the invention with a corresponding lower stiffness (therefore lower frequency). The transfer function for different V p −V s  can be determined experimentally.  FIGS. 6A-6D  are the results of one such experiment. 
       FIG. 6A  is a plot of the transmission characteristics of a MEMS resonator  10  with a DC polarization voltage of 5 volts.  FIG. 6B  is shows the transmission characteristics with a 7 volt DC polarization voltage.  FIG. 6C  is shows the transmission characteristics with a 10 volt DC polarization voltage.  FIG. 6D  is a plot of the pole-zero separation shown in  FIGS. 6A-6C  as a function of the DC polarization voltage. 
       FIG. 7A  schematically illustrates a perspective view of an embodiment of a multi-stage MEMS filter  128 . In this embodiment, the multi-stage MEMS filter  128  is an embodiment of a ladder filter having two series resonators  130  and  132 , and a shunt resonator  134 . In a typical ladder filter configuration, ω parallel  of the shunt resonator  134  is matched with the ω series  of series resonators  130  and  132  and defines the filter center frequency (f c ). Filter bandwidth is determined by notches on either side of the passband and is 2× the pole-zero separation of the series and shunt resonators. As a result, it has been discovered that the key to tunable ladder filters is the ability to change the center frequency, f c  and dynamically tune the pole-zero separation of the resonators. 
     Following are two embodiments of methods for tuning a center frequency and a bandwidth for a MEMS resonator filter. 
     Method 1 
     With V p  fixed, change V s  for both the series and shunt resonators such that the desired series and shunt center frequencies are obtained (Orthogonal Frequency Tuning). 
     Next, to keep the center frequencies, tune (V p −V s ) separately for each resonator to obtain the desired V p  for the required bandwidths (Parallel Resonance Frequency Tuning). Since (V p −V s ) remains constant, the bending of the structure remains the same, and hence the center frequencies of the resonators do not change in this second step. 
     Method 2 
     Short V s  and V p  so that there is no orthogonal frequency tuning. Change the value of V p  (and hence V s ) to obtain the desired bandwidth (Parallel Resonance Frequency Tuning). 
     Next, to obtain the center frequencies, tune V s  separately for each resonator (Orthogonal Frequency Tuning). 
     Method 2 is relatively more straightforward compared to Method 1, since V p  and V s  are tuned independently. However, Method 1 is superior to Method 2 in terms of accuracy. In Method 2, the pole-zero distance actually changes a little when V s  is applied (i.e. when center frequency shifts), although the errors introduced are small (on the order of k e   2 Δf pole  from the analysis in Section 1). There are no such issues with Method 1. 
       FIG. 7B  schematically illustrates a cross-sectional view of two of the resonators shown in  FIG. 7A  taken along cross-section line  7 B- 7 B and looking in the direction indicated by the arrows on the end of line  7 B- 7 B. Spacing or insulating layers  122  can be seen in the view of  FIG. 7B . Such spacing or insulating layers  122  can be used to space and/or electrically isolate the base coupled to the anchor points  20 ,  22  from the substrate  13 . Suitable material for the spacing or insulating layers  122  can be silicon-dioxide, which is easily formed on a silicon base. Other embodiments may use other materials or combinations of materials to space and/or insulate the anchor points from the substrate. 
       FIG. 7C  is a top view (from a Scanning Electron Microscope) of an embodiment of a ladder filter similar to the embodiment of the ladder filter  128  shown in  FIG. 7A . A wire bond connection  142  is shown between the shunt resonator  134  and the two series resonators  130  and  132 . Other embodiments may use different techniques to connect the resonators in the multi-stage filter structure. Additionally, one of the tether points  26  shown in  FIG. 2  is identified in  FIG. 7C . 
     Example 1 
     The following section illustrates one embodiment of the filter tuning methods through an example. Let
 
Δ f= ( V   p   −V   s )×10 5   (14)
 
so that a 50V difference is required to tune the center frequency by 5 MHz. For purpose of this example, consider the following values for the equivalent RLC model of the series resonator.
 
     
       
         
           
             
               
                 
                   
                     C 
                     x 
                   
                   = 
                   
                     6.6087 
                     × 
                     
                       10 
                       
                         - 
                         17 
                       
                     
                     ⁢ 
                     
                       V 
                       p 
                       2 
                     
                     ⁢ 
                     F 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     L 
                     x 
                   
                   = 
                   
                     
                       
                         4.6799 
                         × 
                         
                           10 
                           
                             - 
                             4 
                           
                         
                       
                       
                         V 
                         p 
                         2 
                       
                     
                     ⁢ 
                     H 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     R 
                     x 
                   
                   = 
                   
                     
                       332.6365 
                       
                         V 
                         p 
                         2 
                       
                     
                     ⁢ 
                     Ω 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     C 
                     0 
                   
                   = 
                   
                     9.9563 
                     × 
                     
                       10 
                       
                         - 
                         13 
                       
                     
                     ⁢ 
                     F 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     The shunt resonator is modeled as a 0.5% mass loaded series resonator to obtain the inherent frequency separation, so the only change is in the motional inductance. 
     
       
         
           
             
               
                 
                   
                     L 
                     x 
                   
                   = 
                   
                     
                       
                         4.7033 
                         × 
                         
                           10 
                           
                             - 
                             4 
                           
                         
                       
                       
                         V 
                         p 
                         2 
                       
                     
                     ⁢ 
                     H 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     The resonance frequencies for the series and shunt resonator are 905 MHz and 902.74 MHz, the difference being 2.2582 MHz. The filter pass-band can start anywhere from 897.74 MHz to 902.74 MHz since orthogonal frequency tuning can only tune the frequency downwards (by 5 MHz in this example). With the additional requirement that the parallel resonance frequency of the shunt resonator coincide with the series resonance frequency of the series resonator, and a symmetric filter is desired, then the maximum (notch-to-notch) bandwidth is 2(905−897.74) MHz=14.52 MHz. 
     The simplest instance of a ladder filter is a T-network, with a shunt resonator sandwiched in between two series resonators. In the first example, a filter with first notch at 900 MHz and notch-to-notch bandwidth of 5 MHz is desired. 
     Using Method 1: 
     First, fix V p  at 5V. To shift the center frequency of the shunt resonator to 900 MHz, a substrate bias=(5−27.4) V=−22.4V is applied to the shunt resonator. To shift the center frequency of the series resonator to 902.5 MHz, a substrate bias=(5−25) V=−20V is applied to the series resonator. 
     Next, the pole-zero separation is given by 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       pole 
                     
                     ⁢ 
                     
                       
                         C 
                         x 
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           ft 
                         
                       
                     
                   
                   = 
                   
                     2.5 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     MHz 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     The required V p  for the shunt resonator is 9.1486V. For the series resonator, the required V p  is 9.1359V due to the slightly higher resonance frequency. To keep V p −V s  a constant, the shunt substrate bias becomes −22.4V+9.1486V=−13.2514V. The series substrate bias becomes −20V+9.1359V=−10.8641V. 
     Using these values and a termination resistance of 400Ω, the output transfer function of the ladder filter as shown in  FIG. 8  can be obtained through Kirchoff&#39;s Law. Note that the synthesis method gives the exact notch frequencies and bandwidth. In  FIG. 8  the curve  150  is the calculated transfer function of the shunt resonator  134 , curve  152  is the calculated transfer function of the series resonators  130  and  132 , and curve  154  is the calculated transfer function of the ladder filter  128 . 
     Example 2 
     Obtain a filter with first notch at 900 MHz and notch-to-notch bandwidth of 10 MHz. 
     Using Method 2 
     The required pole-zero separation is 5 MHz for both the series and shunt resonators. Using the equation 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       pole 
                     
                     ⁢ 
                     
                       
                         C 
                         x 
                       
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           C 
                           ft 
                         
                       
                     
                   
                   = 
                   
                     5 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     MHz 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     V p  for the series and shunt resonators are 12.9023V and 12.9184V respectively. 
     To move the first notch frequency to 900 MHz, (V p −V s )=27.4V. From the shunt V p  obtained above, the shunt resonator substrate bias is (12.9184-27.4) V=−14.4816V. No orthogonal frequency tuning is needed for the series resonator because it is already at the right frequency at 905 MHz. 
       FIG. 9  shows results from the same ladder filter, with the single modification of structure bias V p  and substrate bias V s  for both series and shunt resonators as calculated above. There is only minor pass-band ripple degradation with larger bandwidth. In  FIG. 9  the curve  160  is the calculated transfer function of the shunt resonator  134 , curve  162  is the calculated transfer function of the series resonators  130  and  132 , and curve  164  is the calculated transfer function of the ladder filter  128 . 
     These two examples demonstrate the feasibility of center frequency tuning of approximately 0.5% and bandwidth tuning of 1% the center frequency through this real time bias voltage tuning scheme. 
     Example 3 
     A ladder filter consisting of one shunt and two series resonators is fabricated in an SOI process and characterized. The resonators are 310 μm (and 300 μm)×100 μm×3.1 μm released bars topped with 20 mm hafnium dioxide as the dielectric transducer layer. With V p =5 V yields a passband  170  with f c =817.2 MHz, 0.6 MHz bandwidth and insertion loss (IL) of 3.2 dB as shown in  FIG. 10A . By applying V sub =15 V to all the resonators in the ladder, we are able to tune the filter center frequency from 817 MHz to 809 MHz, without affecting IL (3.5 dB) and shape factor (1.3) as shown by the passband  172  in  FIG. 10B .  FIG. 10C  shows a passband  174  with bandwidth tuning from 0.6 MHz to 2.8 MHz while maintaining the center frequency at 817.2 MHz. However, the passband ripple increased from 0.4 dB to 1.8 dB. Finally, a combination of bandwidth and center frequency tuning is shown by the passband  176  in  FIG. 10D . A passband with f c =810.8 MHz and 1.4 MHz bandwidth is obtained. 
       FIGS. 11A and 11B  are sections of the ladder filter  128  shown in  FIG. 7A  with  FIG. 11A  having an input shunt resonator, such as resonator  132  in  FIG. 7A , and the series resonator  134  of the ladder filter  128 .  FIG. 11B  has the series resonator  134  and an output resonator such as resonator  130  of the ladder filter  134 .  FIG. 11C  is a ladder filter with two shunt resonators  180  and  182  separated by a series resonator  184 . All three of resonators shown in  FIGS. 11A ,  11 B, and  11 C are tuned in the manner discussed above with respect to the ladder filter  134  shown in  FIG. 7 . 
       FIG. 12  is a schematic diagram of a tunable lattice filter with two series resonators  186  and  188  and two cross resonators  190  and  192 . By fabricating all four resonators  186 - 192  with substantially identical static capacitance, their impedances will match very well off-resonance and thus the out-of-band attenuation for the filter will be very high. 
     Similar to ladder filter synthesis, the zeros of the resonators  186  and  188  are aligned with the poles of the resonators  190  and  192 . The passband edges are defined by the outermost singularities of the lattice arm (i.e., the series resonance frequency of the resonators  190  and  192  and the parallel resonance frequency of the resonators  186  and  188 ). In order to obtain a lattice filter with tunable center frequency and bandwidth, the two tuning methods described above for the ladder filter can be applied, with the series resonators  186  and  188  being tuned similarly as the series resonators  130  and  132  in the ladder filter  128  shown in  FIG. 7A , and the cross resonators  190  and  192  being tuned similarly as the shunt resonator  134  in the ladder filter  128 . 
     Based on the embodiments of tuning methods described above,  FIG. 13  illustrates another, more generic tuning method which can be used with the disclosed system and its equivalents. A first bias voltage between a base layer and input and output conductor layers of a resonator is adjusted  200 . A second bias voltage between the base layer and isolated substrate areas below at least a part of the base is also adjusted  202 . The center frequency of the resonator filter and the bandwidth of the filter are determined  204  until the adjustments to the first bias voltage and the second bias voltage provide a desired center frequency and a desired bandwidth. While this method may not be as efficient as the methods previously described, given the control over the filter&#39;s center frequency and bandwidth provided by the first bias voltage and the second bias voltage, this method is still viable. 
       FIG. 14  illustrates another embodiment of a method of tuning a center frequency and a bandwidth of a MEMS resonator filter. A first bias voltage between a base layer and input and output layers of the filter is provided. A second bias voltage between the base layer and isolated substrate areas below at least a part of the base is provided. While holding the first bias voltage fixed, the second bias voltage is adjusted  206  such that the desired center frequency is obtained. The difference between the first bias voltage and the second bias voltages is noted  208  for the desired center frequency. Both the first bias voltage and the second bias voltage are adjusted  210  while maintaining the noted difference between the first bias voltage and the second bias voltage to obtain the desired bandwidth. 
       FIG. 15  illustrates a further embodiment of a method of tuning a center frequency and a bandwidth of a MEMS resonator filter. A first bias voltage between a base layer and input and output layers of the filter is provided. A second bias voltage between the base layer and isolated substrate areas below at least a part of the base is provided. The first bias voltage and the second bias voltage are made to be the same  212 . While keeping the second bias voltage the same as the first bias voltage, the first bias voltage is adjusted  214  to obtain the desired bandwidth. While maintaining the first bias voltage, the second bias voltage is adjusted  216  to obtain the desired center frequency. 
     Those skilled in the art will understand that the basic filter types described herein can be combined in many different ways and can also combined with other electrical elements in which the structure of various sections of the filter can be fabricated using the resonators and tuning methods described herein. 
     While the invention has been described by reference to various specific embodiments, it should be understood that numerous changes may be made within the spirit and scope of the inventive concepts described. Accordingly, it is intended that the invention not be limited to the described embodiments, but will have full scope defined by the language of the following claims. 
     All features disclosed in the specification, including the claims, abstract, and drawings, and all the steps in any method or process disclosed, may be combined in any combination, except combinations where at least some of such features and/or steps are mutually exclusive. Each feature disclosed in the specification, including the claims, abstract, and drawings, can be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise. Thus, unless expressly stated otherwise, each feature disclosed is one example only of a generic series of equivalent or similar features. 
     Any element in a claim that does not explicitly state a means for performing a specified function or a step for performing a specified function should not be interpreted as a means or a step clause as specified in 35 U.S.C. 112.