Abstract:
A composite resonator ( 10 ) consisting of a conducting metal ( 14 ) and a dielectric material ( 12 ) is used to provide resonant frequencies lower than can be obtained using the same volume of dielectric alone and with higher unloaded Q than can be obtained using the same volume of metal imbedded into a cavity and used as a resonator. This significantly reduces the cost and size of the resonator ( 10 ) without degrading its performance. An inexpensive metal ( 14 ), such as aluminum, can be substituted for more than half of the dielectric ( 12 ) and stille form a resonator ( 10 ) with substantially equivalent resonant properties. The operative embodiments of the resonator invention ( 1 ) cover composites with doughnut-shaped, i.e., cylindrical, configurations, with the “doughnut” either metal ( 14 ) or dielectric ( 12 ), and the “hole” either dielectric ( 314 ) or metal ( 312 ), respectively.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a resonator composed of a conducting metal ring surrounding a cylindrical dielectric core material which can be incorporated into multi-cavity filters for frequency separation. 
     2. Description of Related Art 
     Dielectric resonator filters are a class of stable microwave filters that are frequently used in radar and communications systems. Dielectric resonators are often utilized in filter circuits because of an intrinsically high Q value. These characteristics allow a filter employing a dielectric resonator to have excellent frequency stability with only a small amount of frequency drift over a wide range of temperatures and environmental conditions. The Q value of a dielectric resonator is defined as the ratio between the energy stored per cycle to the energy dissipated per cycle. 
     Dielectric resonators are typically made of a ceramic type material having a high dielectric constant (∈ r =20 to 90) and a low dissipative loss. These characteristics allow the dielectric resonator to store energy with relatively little internal energy dissipation. This corresponds to a high Q value. 
     One significant limitation of the practical use of dielectric resonator filters is the cost of the dielectric itself. The cost of a typical prior art 6″ ceramic dielectric cylindrical resonator can cost three hundred dollars or more. In addition, the size of the resonator substantially increases the size of any multi-cavity filter in which it might be employed. 
     The following patents are generally representative of typical prior art dielectric resonators: U.S. Pat. Nos. 4,757,289; 5,140,285; and, 5,356,844. 
     Resonators are typically employed in filters for the wireless communication industry. Such filters typically include a plurality of resonators located in adjacent cavities and coupled to each other through a variety of different means. One coupling mechanism known in the prior art is the use of a tunable iris as described in U.S. Pat. No. 5,220,300 entitled “RESONATOR FILTERS WITH WIDE STOPBANDS” and issued on Jun. 15, 1993 and assigned by Richard V. Snyder to RS Microwave Company, Inc., the entire contents and substance of which is incorporated herein by reference. Other cutoff means are also known, but few are known that would be suitable for composite resonators such as described in this disclosure. 
     What is clearly missing in the prior art, therefore, is a relatively inexpensive resonator, of reasonably small size, that can be used in a multi-cavity filter structure without appreciable loss in performance. 
     SUMMARY OF THE INVENTION 
     Briefly described, the invention comprises a composite resonator preferably including a cylindrical ceramic core and an exterior metal layer that surrounds most of the exterior circumference of the core and wherein the resonator resonates in substantially bound modes. This composite configuration is used to provide resonant frequencies lower than can be obtained using the same volume of dielectric alone and with higher unloaded Q than can be obtained using the same volume of metal imbedded into a cavity and used as a resonator. An inexpensive metal, such as aluminum, can be substituted for more than half of the dielectric and still form a resonator with substantially equivalent resonance properties. 
     According to alternative embodiments of the invention, the resonators are incorporated into spectrum filters for separation of frequencies. As contrasted to .conventional prior art implementations, the new technique achieves similar, or better, electrical performance; similar, or reduced, size; and significantly reduced cost for applications in the frequency range below 2.5 Ghz, thus including PC, wireless, AMPS and GSM applications, as well as a myriad of other applications in this frequency range. With spectrum currently selling for up to $45.00 per Hz, filters are very valuable for providing users the opportunity to utilize all spectrum available. Yet, the cost of the filters must ultimately be borne by the users, so reductions in cost are important to commercial applications. The present invention in its various embodiments contributes to such a reduction in cost. 
     These and other features of the invention will be more fully understood by reference to the following drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a perspective view of a prior art ceramic resonator core. 
     FIG. 2 is a perspective view of a prior art multi-cavity filter including four prior art ceramic resonators. 
     FIG. 3 a  is a perspective view of a composite resonator that comprises a dielectric core surrounded by an exterior metal layer according to the preferred embodiment of the invention. 
     FIG. 3 b  is a vertical cross-sectional view of the preferred embodiment of the resonator shown in FIG. 3 a.    
     FIG. 3 c  is a top cross-sectional view of the resonator according to the preferred embodiment a illustrated in FIGS. 3 a  and  3   b.    
     FIG. 4 is a perspective view of one embodiment of the invention showing the metal dielectric composite resonator according to the preferred embodiment utilized in a basic filter resonator having a ferrite or garnet disk, magnetically tunable in frequency. 
     FIG. 5 is a perspective view of an alternate embodiment illustrating two metal dielectric composite resonators according to the present invention, each with a window facing one another in their metallized circumference, thereby permitting coupling of energy between the two resonators. 
     FIG. 6 is a perspective view of another alternate embodiment of the invention illustrating a coupled filter in which two metal dielectric composite resonators according to the present in vention are coupled by means of a tunable iris. 
     FIGS. 7,  8  and  9  are perspective views of other alternative, hybrid embodiments illustrating cross coupled array filters in which four metal dielectric composite resonators according to the present invention are coupled by various means. 
     FIG. 10 perspective view of another alternative embodiment illustrating a metal dielectric composite resonator according to the present invention employed in a dual mode filter configure. 
     FIG. 11 a  is a perspective view of an alternative embodiment of a resonator with a shape similar to the preferred embodiment shown in FIGS. 3 a - 3   c  but having a metallic core and an external layer of dielectric material surrounding most of the surface of the metallic core. 
     FIG. 11 b  is a vertical cross-sectional view of the alternative embodiment of the resonator shown in FIG. 11 a.    
     FIG. 11 c  is a top cross-sectional view of the alternative embodiment of the resonator shown in FIGS. 11 a  and  11   b.    
     FIG. 12 is a resonator characteristic graph providing an example of how a typical resonator, according to the preferred embodiment of the invention, is structured and designed. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     During the course of this description like numbers will be used to identify like elements according to the different figures that illustrate the invention. 
     FIG. 1 illustrates a typical prior art dielectric resonator. Such prior art resonators are typically relatively large and made of a single material, such as ceramic. Because of their size they can be relatively expensive to manufacture. In addition, their larger size in turn dictates that any filter in which they are used will also be relatively large, and thus display undesirable spurious responses in close proximity to the resonant frequency of the resonator. See FIG. 2 for a typical prior art multi-cavity four-stage filter including four prior art dielectric resonators. 
     A composite resonator  10 , according to the preferred embodiment of the invention, is illustrated in FIGS. 3 a - 3   c . The preferred resonator  10  includes a ceramic core  12  surrounded by a metal layer  14  to form a “doughnut” or “hockey puck” shape. The core  12  includes a top surface or face  22 , a bottom surface or face  24 , and an interior sidewall surface or face  26 . As best seen in FIG. 3 b , the circumference  26  of the core  12  is surrounded by a sidewall metallic band or layer  18 . The metallic layer  14  is, of course, the side layer  18 . The metallic layer  14 , i.e., ring  18 , is preferably at least 2-3 skin depths thick or deep. Ring  18  can be much thicker but must be at least 2-3 skin depths to operate properly. The term skin depth is well known in the prior art and defined as {fraction (1/e)}. 
     FIG. 4 illustrates a relatively simple, basic embodiment  30  in which the resonator  10  is employed as a filter. The resonator  10 , which includes the core  12  and the surrounding metal layer  14 , as described with regard to FIGS. 3 a - 3   b , is located in a structure, or housing,  32  and fed by a conventional probe  36  supported at anchor point  34 . The ∈ r , and μ can be chosen to vary the characteristics of the resonator. The resonator  10  comprises a ferrite or garnet disk, magnetically tunable in frequency. 
     According to the preferred embodiment of the present invention, composite resonators are used in a resonator apparatus that operates in a substantially bound mode. In a substantially bound mode, the signal is essentially contained within the high dielectric material and is essentially non-radiating. This is due to the almost perfect reflecting boundary conditions resulting from both the selective use of conductive metallization on the periphery and the critical angle of reflection at the non-metallized boundaries of high dielectric constant material (∈ r ≧10, and typically ∈ r =24 or greater) with the low dielectric (∈ r =1) air filling the enclosures. What is important is the ratio of dielectric constant filling the resonator to that filling the cavity, external to the resonator. To ensure almost perfect reflection and thus resonance of substantially bound modes, the ratio should be at least 15:1. Examples of substantially bound modes function in this application are the TE oin  modes which exist substantially without leakage in the structure described herein. In the example mode, the subscripts refer to the number of circumferential, radial and longitudinal magnetic field variations (for the case of a cylinder). 
     The invention  10  is not limited to round doughnut shapes, as the principle also applies to planar configurations or parallelepiped resonator configurations. The invention also applies to planar configurations in which metal dielectric composites are used to form artificial dielectric screens for application to antennas and similar devices. 
     Substantially bound modes become unbound only at specific interfaces wherein coupling mechanisms such as irises, tuning screws, or other perturbations are present, and then only for purposes of enhancing coupling of a portion of the substantially bound mode to another structure such as another resonator or port. FIGS. 5-10 depict such coupling mechanisms in various combinations. 
     The foregoing invention is described primarily in the context of a cylindrical example. It should be understood, however, that it can operate in any of the recognized nine “separable geometries”. “Separable geometries” is a term known in the prior art and is described, for example, in “Methods of Theoretical Physics”, by Morse and Feshbach, McGraw Hill, 1953. The geometries, which are included in the nine separable modes, are believed to be the only ones which can support more than one orthogonal mode simultaneously. 
     FIG. 5 illustrates a filter embodiment  40  housed in a structure  42  having a cavity  49  and a standard energy feed port  44 . A first and a second composite resonator  46   a  and  46   b  are attached at opposite ends of the cavity  49 . The first resonator  46   a  includes a small window  48   a  located in the metallic sidewall sufficient to expose the underlying dielectric core  12 . Similarly, the second composite resonator  46   b  includes a window  48   b  in its metallic sidewall that faces window  48   a  of the first resonator  46   a . Energy from the first resonator  46   a  is coupled through window  48   a  to window  48   b  of the second composite resonator  46   b.    
     Another coupling embodiment  50  is illustrated in FIG.  6 . Filter, or coupling, embodiment  50  comprises a housing structure  52  that includes a pair of cavities  60   a  and  60   b . Energy is coupled into the cavity by a standard fitting  54 . Cavity  60   a  includes a composite resonator  56   a  which sits atop a pedestal support  58 . Similarly, a second composite resonator  56   b  sits atop a pedestal  58  in cavity  60   b . In real life, all resonators  10  et seq. shown in FIGS. 4-10 sit on pedestals like  58  but are not shown because they are well known in the prior art. Such pedestals, sometimes referred to as “toadstools”, typically have a low ∈ (in range of 2-6) and are made of foam or B e O. Partition, or wall,  62  separates cavity  60   a  from  60   b . A window  64  is located in wall  62  and includes a tunable iris  66  for selectively coupling energy from composite resonator  56   b  to composite resonator  56   a . A tunable resonator having an acceptable iris structure is described in U.S. Pat. No. 5,220,300 issued on Jun. 15, 1993 and assigned by Richard V. Snyder to RS Microwave, Inc., Butler, N.J. 
     A cross-coupled array filter  70  embodiment is illustrated in FIG.  7 . The housing structure  72  includes a standard energy port  74  and defines a pair of interior cavities  78   a  and  78   b . A first and a second composite resonator  76   a  and  76   b , respectively, are located within cavity  78   b . Similarly, a third and fourth composite resonator  76   c  and  76   d  are located within cavity  78   a . A partition, or wall,  82  separates cavities  78   a  and  78   b . A pair of windows  82   a  and  82   b  is located in partition  82 . Window  82   a  includes a tunable iris  84   a . Likewise, window  82   b  includes a tunable iris  84   b . Tunable irises  84   a  and  84   b  can be identical to those described in U.S. Pat. No. 5,220,330. Energy from the first composite resonator  76   a  can be selectively coupled through iris  84   a  to the third composite resonator  76   c . Likewise, energy from the second composite resonator  76   b  can be coupled through iris  84   b  to the fourth composite resonator  76   d.    
     FIG. 8 illustrates another alternative embodiment  100  which is a combination, or hybrid, of the window and iris coupling mechanisms. As illustrated in FIG. 8, the combination embodiment  100  is housed in a structure  102  and includes a standard energy coupling  104 . Housing  102  includes interior cavities  110   a  and  110   b . Cavity  110   b  houses a first, second, and fourth composite resonator  106   a ,  106   b  and  106   d , respectively. Cavity  110   a  houses third composite resonator  106   c . The second and fourth resonators  106   b  and  106   d  each include a window  108   b  and  108   d , respectively, which face each other and which couple energy from the second composite resonator  106   b  to the fourth composite resonator  106   d  in the manner previously described with reference to FIG.  5 . The interior cavity  110   a  is defined by right angle panels or partitions  112  and  114 , respectively. Panel  112  includes a window  116  and a tunable iris  118 , similar to that described in U.S. Pat. No. 5,220,300 for coupling energy from the first composite resonator  106   a  to the third composite resonator  106   c . Similarly, panel  114  includes a window  120 , and a tunable iris  122 , for selectively coupling energy from the fourth composite resonator  106   d  to the third composite resonator  106   c.    
     Another combination or hybrid coupling embodiment  150  is illustrated in FIG.  9 . The resonators are located within a housing structure  152  which includes the standard energy port  154 . Housing  152  defines a single interior cavity  160  which houses a first, second, third and fourth composite resonator  156   a ,  156   b ,  156   c  and  156   d , respectively. The second and fourth composite resonators  156   b  and  156   d  each include windows, or apertures,  158   b  and  158   d , respectively, which couple energy from the second composite resonator  156   b  to the fourth composite resonator  156   d . A partition, or wall,  162  separates the first composite resonator  156   a  from the third composite resonator  156   c . A window  164  is located in the wall  162  and includes a tunable iris  166 , similar to that described in U.S. Pat. No. 5,220,300 for coupling energy from the first composite resonator  156   a  to the third composite resonator  156   c.    
     FIG. 10 illustrates a single resonator  204 , dual mode, configuration  200  that takes advantage of the fact that substantially bound modes become unbound where perturbations are present. The resonator  204  has a distinctly rectangular shape and is located within housing  202 . It is feed energy by a conventional probe  206  and includes a rectangular dielectric core  208  partially, but not entirely, surrounded by a metallized peripheral layer  210 . A three-dimensional orthogonal notch  212  is taken out of one comer of the composite resonator  204 . The notch  212  provides for coupling of dual TM 529  modes in the resonator  204 . A high dielectric constant structure can support more than one bound mode simultaneously, either as degenerate (i.e., field orthogonal, but resonant, at the same frequency) or as separate modes separated in the frequency domain. Consequently, multimode filter configurations are attainable, as depicted in FIG.  10 . 
     As depicted in each of FIGS. 4-10, the structure of the apparatus&#39;s enclosure is too small to be resonant at frequencies at or below that of the high dielectric constant structure. Consequently, the enclosure is not a fundamental resonator in itself, but rather is below cutoff (i.e. only propagation of evanescent modes is possible within the enclosure, external to the high dielectric constant structure). 
     A composite resonator element  300 , according to an alternative embodiment of the invention, is illustrated in FIGS. 11 a - 11   c . The size and shape of resonator  300  is essentially the same as the preferred embodiment  10  shown in FIGS. 3 a - 3   c  except with the metallic and dielectric materials reversing roles and positions. Accordingly, the alternative resonator  300  includes a metallic core  312  surrounded by a dielectric layer  314  to form a “doughnut” or “hockey puck” shape. The metallic core includes a top surface or face  322 , a bottom surface or face  324  and an interior sidewall surface or face  326 . As best seen in FIG. 11 b , the circumference  326  of metallic core  312  is surrounded by a sidewall dielectric band or layer  318 . The dielectric layer  314  is, therefore, composed of the sidewall band or layer  318 . 
     EXAMPLE 1 
     For comparison purposes, a calculation was made with the standard Trans-Tech Dielectric Resonator Design package (available from Trans-Tech, 552 Adamstown Road, Adamstown, Md. 21710) for a conventional prior art resonator with an ∈ r =80 to obtain a desired frequency of 0.733 GHz. The ultimate dielectric required a width of 1.940″ by 0.873″. The volume then is πr 2 h=2.58 in 3 . 
     In contrast, using commercial available Mathcad™ 7 program distributed by MathSoft, Inc., 101 Main Street, Cambridge, Mass. 02142, the following calculations were obtained: 
     
       
         
               
               
               
             
               
               
               
               
             
               
               
               
             
               
             
               
               
             
               
               
             
               
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
               
             
               
               
             
               
               
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
           
               
                   
               
             
             
               
                 Structure Inputs: 
                 radius in inches 
                 a: = .784 
               
               
                   
                 height in inches 
                 d: = .63 
               
               
                   
                 relative permittivity of dielectric 
                 ε r : = 80 
               
             
          
           
               
                   
                 conductivity of metal 
                 met: = 3 metal 
                 = 1 aluminum .3817 
               
               
                   
                   
                   
                 = 2 silver .6173 
               
               
                   
                   
                   
                 = 3 copper .58 
               
             
          
           
               
                   
                 relative permittivity of metal 
                 μmet: = .9999736 
               
               
                   
                 cut plane distance 
                 zd: = 1 d 
               
               
                   
                 (decimal percentage of total height) 
               
             
          
           
               
                 Field Plot Inputs: 
               
             
          
           
               
                 Choose a value [0, 1] for TE: 
                 TE: = 1 
               
               
                 TE = 1 for TE calculations 
               
               
                 TE = 1 for TM calculations 
               
               
                   
               
             
          
           
               
                   
                 
                   
                     
                       
                         
                           
                             
                               
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                 Choose Mode number 
                   
               
               
                 N is the number of circumferential variations in the field 
                 N: = 0 M: = 1 L: = 1 
               
               
                 M is the number of radial variations 
               
               
                 L is the number of axial variations 
               
               
                   
               
             
          
           
               
                   
                 
                   
                     
                       
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                 TE nml  = root NM   
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
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                 TE nml  root NM   
               
               
                   
                   
               
               
                   
                 check = “okay” 
                 TM φrz  = root NM   
               
             
          
           
               
                 Define cutplane for field plots: 
                 Option 1 - φ cut with φ = 90° 
                 option: = 3 
               
               
                   
                 Option 2 - φ cut with φ = 0° 
               
               
                   
                 Option 3 - Z cut with 0 &lt; z &lt; d 
               
             
          
           
               
                 Constants: 
                 ε0: =  8.854187817 · 10 −12   
                 μ0: = 4 · π · 10 −7   
                 j: = {square root over (−1)} 
               
               
                   
               
             
          
           
               
                 
                   
                     
                       
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                 TOL: = 10 −8   
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
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                                   5.520 
                                 
                                 
                                   
                                     
                                         
                                     
                                      
                                     8.652 
                                   
                                 
                               
                               
                                 
                                   3.832 
                                 
                                 
                                   7.016 
                                 
                                 
                                   10.176 
                                 
                               
                               
                                 
                                   5.135 
                                 
                                 
                                   8.417 
                                 
                                 
                                   
                                     11.62 
                                      
                                     
                                         
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
               
                   
                 jroot(n,r): = jn(n,guess(n,r)) 
               
               
                   
                 range variables:  n: = 0 . . . 4  m: = 1 . . . 4 
               
             
          
           
               
                   
                 roots nm : = jroot(n,m) 
                 roots NM  = 2.405 
               
             
          
           
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         roots 
                         = 
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 2.405 
                               
                               
                                 
                                   5.52 
                                    
                                   
                                       
                                   
                                 
                               
                               
                                 
                                   
                                       
                                   
                                    
                                   8.654 
                                 
                               
                               
                                 11.792 
                               
                             
                             
                               
                                 0 
                               
                               
                                 3.832 
                               
                               
                                 7.016 
                               
                               
                                 
                                   10.173 
                                    
                                   
                                       
                                   
                                 
                               
                               
                                 13.324 
                               
                             
                             
                               
                                 0 
                               
                               
                                 5.136 
                               
                               
                                 8.417 
                               
                               
                                 
                                   11.62 
                                    
                                   
                                       
                                   
                                 
                               
                               
                                 14.796 
                               
                             
                             
                               
                                 0 
                               
                               
                                 
                                   6.38 
                                    
                                   
                                       
                                   
                                 
                               
                               
                                 9.761 
                               
                               
                                 
                                   13.015 
                                    
                                   
                                       
                                   
                                 
                               
                               
                                 16.223 
                               
                             
                             
                               
                                 0 
                               
                               
                                 7.588 
                               
                               
                                 
                                   11.065 
                                    
                                   
                                       
                                   
                                 
                               
                               
                                 
                                   14.373 
                                    
                                   
                                       
                                   
                                 
                               
                               
                                 17.616 
                               
                             
                           
                           ] 
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           guess 
                            
                           
                               
                           
                            
                           
                             ( 
                             
                               n 
                               , 
                               m 
                             
                             ) 
                           
                         
                         := 
                         
                           
                             π 
                             · 
                             
                               ( 
                               
                                 m 
                                 + 
                                 
                                   n 
                                   2 
                                 
                                 - 
                                 
                                   1 
                                   4 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               
                                 4 
                                 · 
                                 
                                   n 
                                   2 
                                 
                               
                               - 
                               1 
                             
                             
                               8 
                               · 
                               
                                 [ 
                                 
                                   π 
                                   · 
                                   
                                     ( 
                                     
                                       m 
                                       + 
                                       
                                         n 
                                         2 
                                       
                                       - 
                                       
                                         1 
                                         4 
                                       
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
             
          
           
               
                 TOL: = 10 −3   
               
             
          
           
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           
                             j 
                             ′ 
                           
                            
                           
                             n 
                              
                             
                               ( 
                               
                                 x 
                                 , 
                                 n 
                               
                               ) 
                             
                           
                         
                         := 
                         
                           
                             root 
                              
                             
                                 
                             
                              
                             
                               ( 
                               
                                 
                                   
                                      
                                     
                                        
                                       x 
                                     
                                   
                                    
                                   
                                     Jn 
                                      
                                     
                                       ( 
                                       
                                         n 
                                         , 
                                         x 
                                       
                                       ) 
                                     
                                   
                                 
                                 , 
                                 x 
                               
                               ) 
                             
                              
                             
                                 
                             
                              
                             
                               j 
                               ′ 
                             
                              
                             0 
                              
                             
                               ( 
                               x 
                               ) 
                             
                           
                           := 
                           
                             root 
                              
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     J 
                                   
                                    
                                   
                                       
                                   
                                    
                                   1 
                                    
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 , 
                                 x 
                               
                               ) 
                             
                           
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
               
                   
                 j′root(n,m) := if(n = 0,j′0(guess(1,m)),j′n(n,guess(n − 1,m))) 
               
             
          
           
               
                 roots′ n,m : = j′root(n,m) 
               
               
                   
               
             
          
           
               
                   
                 
                   
                     
                       
                         
                           roots 
                           ′ 
                         
                         = 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     0 
                                   
                                   
                                     3.832 
                                   
                                   
                                     7.016 
                                   
                                   
                                     10.173 
                                   
                                   
                                     13.324 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     1.841 
                                   
                                   
                                     5.332 
                                   
                                   
                                     
                                       
                                           
                                       
                                        
                                       8.535 
                                     
                                   
                                   
                                     11.705 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     3.056 
                                   
                                   
                                     6.707 
                                   
                                   
                                     9.97 
                                   
                                   
                                     13.168 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     4.199 
                                   
                                   
                                     8.016 
                                   
                                   
                                     11.346 
                                   
                                   
                                     14.581 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     5.318 
                                   
                                   
                                     9.285 
                                   
                                   
                                     12.682 
                                   
                                   
                                     15.964 
                                   
                                 
                               
                               ) 
                             
                              
                             
                                 
                             
                              
                             
                               roots 
                               
                                 N 
                                 , 
                                 M 
                               
                               ′ 
                             
                           
                           = 
                           3.832 
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
               
                   
                 
                   
                     
                       
                         
                           p 
                           
                             n 
                             , 
                             m 
                           
                         
                         := 
                         
                           
                             
                               
                                 roots 
                                 
                                   n 
                                   , 
                                   m 
                                 
                               
                               acm 
                             
                              
                             
                                 
                             
                              
                             
                               p 
                               
                                 n 
                                 , 
                                 m 
                               
                               ′ 
                             
                           
                           := 
                           
                             
                               roots 
                               
                                 n 
                                 , 
                                 m 
                               
                               ′ 
                             
                             acm 
                           
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
             
          
           
               
                 Part 1: Calculate Cutoff Frequency: 
               
               
                   
               
             
          
           
               
                   
                 
                   
                     
                       
                         fcut 
                         := 
                         
                            
                           
                             
                               
                                 
                                   
                                     
                                       
                                         roots 
                                         
                                           N 
                                           , 
                                           M 
                                         
                                         ′ 
                                       
                                       
                                         
                                           2 
                                           · 
                                           π 
                                           · 
                                           acm 
                                         
                                          
                                         
                                             
                                         
                                          
                                         
                                           
                                             μ 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               0 
                                               · 
                                               ɛ 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               0 
                                               · 
                                               ɛ 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             r 
                                           
                                         
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     if 
                                      
                                     
                                         
                                     
                                      
                                     TE 
                                   
                                   = 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         roots 
                                         
                                           N 
                                           , 
                                           M 
                                         
                                         
                                             
                                         
                                       
                                       
                                         2 
                                         · 
                                         π 
                                         · 
                                         
                                             
                                         
                                          
                                         
                                           
                                             μ 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               0 
                                               · 
                                               ɛ 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             
                                               0 
                                               · 
                                               ɛ 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             r 
                                           
                                         
                                       
                                     
                                      
                                     
                                         
                                     
                                      
                                     if 
                                      
                                     
                                         
                                     
                                      
                                     TE 
                                   
                                   ≠ 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
             
          
           
               
                 Part 2: Calculate Resonant Frequency: 
               
               
                   
               
             
          
           
               
                   
                 
                   
                     
                       
                         
                           
                             
                               
                                 fres 
                                 := 
                                 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       
                                         
                                           
                                             
                                               [ 
                                               
                                                 
                                                   c 
                                                   
                                                     4 
                                                     · 
                                                     π 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             roots 
                                                             
                                                             N 
                                                             , 
                                                             M 
                                                             
                                                             ′ 
                                                           
                                                           acm 
                                                         
                                                         ) 
                                                       
                                                       2 
                                                     
                                                     + 
                                                     
                                                       
                                                         ( 
                                                         
                                                           
                                                             L 
                                                             · 
                                                             π 
                                                           
                                                           dcm 
                                                         
                                                         ) 
                                                       
                                                       2 
                                                     
                                                   
                                                 
                                               
                                               ] 
                                             
                                              
                                             
                                                 
                                             
                                              
                                             if 
                                              
                                             
                                                 
                                             
                                              
                                             TE 
                                           
                                           = 
                                           1 
                                         
                                       
                                     
                                     
                                       
                                         
                                           [ 
                                           
                                             
                                               
                                                 c 
                                                 
                                                   4 
                                                   · 
                                                   π 
                                                 
                                               
                                                
                                               
                                                 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           roots 
                                                           
                                                             N 
                                                             , 
                                                             M 
                                                           
                                                         
                                                         acm 
                                                       
                                                       ) 
                                                     
                                                     2 
                                                   
                                                   + 
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           L 
                                                           · 
                                                           π 
                                                         
                                                         dcm 
                                                       
                                                       ) 
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               ] 
                                               
                                                   
                                               
                                                
                                               if 
                                                
                                               
                                                   
                                               
                                                
                                               TE 
                                             
                                             ≠ 
                                             1 
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   fres 
                                   = 
                                   
                                     7.332 
                                     · 
                                     
                                       10 
                                       8 
                                     
                                   
                                 
                                  
                                 
                                     
                                 
                               
                             
                           
                         
                          
                       
                     
                             
                     
                         
                     
                   
                 
               
               
                   
                   
               
             
          
         
       
     
     Here the volume is πr 2 h=1.21 in 3 . Therefore, the metal ring resonator  10  has 1.21/2.58=47% of the volume of a conventional all dielectric resonator as shown in FIG. 1 with the same ∈r=80. 
     While the invention has been described with reference to the preferred embodiment thereof, it will be appreciated by those of ordinary skill in the art that various modifications can be made to the structure and function of the individual parts of the system without departing from the spirit and scope of the invention as a whole.