Abstract:
A thin film resonator which combines a microstrip resonator structure and a coplanar resonator structure to form an integrated resonator structure. The resonant frequency of this resonator structure is independent of the substrate thickness within a certain thickness range. This resonator structure also has a very economical size, as compared to other existing resonator designs. Different coupling configurations between the resonators are shown with the resulting coupling coefficients. Also a two-pole, four-pole and an eight-pole filter are designed using the thin film resonator and the insertion loss and return loss characteristics for various filters are shown.

Description:
TECHNICAL FIELD 
   The present disclosure relates generally to electromagnetic resonators, and more particularly, to microstrip electromagnetic resonators. 
   BACKGROUND ART 
   Conventional resonant cavity filters consist of an outer housing made of an electrically conductive material and one or more resonant elements, or resonators, are mounted inside the housing. The resonators may be mounted within the cavity using, for example, a dielectric material. Electromagnetic energy is coupled through a first coupling mechanism in the housing to a first resonator and then to any additional resonators in the housing. A second coupling mechanism is used to output the electromagnetic energy from the housing. 
   Resonators are often used in filters to pass or reject certain signal frequencies. The particular design, shape, materials and spacing of the housing, the resonant elements, and the apertures between resonant elements determine the signal frequencies passed through the filter, as well as the insertion loss of the filter and quality factor (“Q”) of each resonator. Ideally, resonators should have minimum signal loss in their passbands. 
   Resonators generally consist of conductive structures, and are typically of either a two-dimensional type, or a three-dimensional type. Two-dimensional resonators, also known as microstrip resonators, are formed by depositing a conductive layer onto a substrate and removing some of the conductive material from the substrate to leave a length of conductive material behind. The length of conductive material remaining on the substrate forms one or more resonators. Two-dimensional resonators are commonly referred to as thin film resonators. 
   Thin film resonator technology has been used to produce high performance military and commercial wireless devices. One type of two-dimensional resonators uses a thin film of high temperature superconductive (HTS) material disposed onto a dielectric substrate. One major problem associated with the fabrication of thin film resonators is the variation in the thickness of the dielectric substrate. Thickness of the dielectric substrate influences not only the coupling coefficient between adjacent resonators, but also affects the resonant frequency of the resonator. Accordingly, variations in the thickness of the dielectric substrate also results in the variations in the resonant frequency of the thin film resonator. 
   The velocity of an electromagnetic wave in a microstrip is given by Equation 1. 
               v   p     =     c       ɛ   e                 Equation   ⁢           ⁢   1             
 
Where c is the velocity of light in free space and ε e  is the effective dielectric constant of the microstrip. The effective dielectric constant of the microstrip can be approximated by Equation 2. 
               ɛ   e     ≈         1   +     ɛ   r       2     +         [         ɛ   r     -   1     2     ]     ⁡     [     1   +     10   ⁢     h   w         ]         -     1   2                   Equation   ⁢           ⁢   2             
 
Where ∈ r  is the dielectric constant of the substrate, h is the thickness of the substrate, and w is the width of the microstrip. As can be seen from Equations 1 and 2, when h increases, ∈ e  decreases and, therefore, υ p  increases. As a result, the resonant frequency of the microstrip resonator increases as well. In practice, it is not uncommon for even the most precisely fabricated substrates to vary in thickness by as much as ±1%.
 
   Due to such dependence of the resonant frequency on the thickness of the substrate, the measured frequency response of such a microstrip resonator usually deviates from the frequency response for which the resonator is designed. Tuning of filters designed using such resonators is a very tedious task even for experienced filter engineers, because one has to tune not only the coupling coefficient between the resonators but also the resonant frequency of the individual resonators. 
   Another issue pertinent to thin film filters is the miniaturization of the resonator structure used to design such filters. As the resonant frequency of a microstrip resonator decreases, and, therefore, the resonant wavelength increases, it is necessary to use larger size microstrip resonators, which necessitates the use of bulky resonators to achieve lower resonant frequencies. Substantial effort has been devoted to the miniaturization of the resonator structures.  FIG. 1  shows some exemplary thin film resonator structures that have been used in filters. In  FIG. 1 , reference numeral  12  refers to a standard microstrip resonator, reference numeral  14  refers to a loop resonator formed by removing the central portion from the standard microstrip resonator  12  and reference numeral  16  refers to a capacitively loaded loop resonator. Further, reference numeral  18  refers to an open loop resonator, reference numeral  20  refers to a meander shaped open loop resonator, and reference numeral  22  refers to a folded open loop resonator. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present patent is illustrated by way of example and not limitations in the accompanying figures, in which like references indicate similar elements, and in which: 
       FIG. 1  shows various exemplary thin film resonator structures used in filters; 
       FIG. 2  is an exemplary illustration of a resonator comprising two open loops and a filled microstrip; 
       FIG. 3  is an exemplary plot illustrating of the resonant frequencies of the resonator of  FIG. 2  for various shunting arrangements; 
       FIG. 4  is an exemplary illustration of the resonator of  FIG. 2  further comprising an input coupling microstrip; 
       FIGS. 5A and 5B  illustrate two alternate exemplary coupling configurations used in designing multi-pole filters using the resonator of  FIG. 2 ; 
       FIG. 6  is an exemplary plot illustrating the coupling coefficients as a function of the distance between the resonators for the two coupling configurations illustrated in  FIGS. 5A and 5B ; 
       FIG. 7  is an exemplary plot illustrating the coupling coefficients as a function of the shunting position within the resonators for the coupling configuration illustrated in  FIG. 5A ; 
       FIG. 8  illustrates an exemplary layout of a two-pole filter using the resonator of  FIG. 2 ; 
       FIG. 8A  illustrates an exemplary implementation of the two-pole filter of  FIG. 8  on a substrate; 
       FIG. 8B  illustrates a three dimensional implementation of the two-pole filter of  FIG. 8  in a metallic housing; 
       FIG. 9  is an exemplary plot illustrating a frequency response of the exemplary two-pole filter of  FIG. 8 ; 
       FIG. 10  illustrates an exemplary layout of a four-pole filter using the resonator of  FIG. 2 ; 
       FIG. 11  is an exemplary plot illustrating a frequency response of the exemplary four-pole filter of  FIG. 10 ; 
       FIG. 12  illustrates an exemplary layout of an eight-pole filter using the resonator of  FIG. 2 ; 
       FIG. 13  is an exemplary plot illustrating a frequency response of the exemplary eight-pole filter of  FIG. 12 ; and 
       FIG. 14  is an exemplary plot illustrating another frequency response of the exemplary eight-pole filter of FIG.  12 . 
   

   DETAILED DESCRIPTION 
   As disclosed in detail hereinafter, a resonator is provided which integrates a microstrip resonator structure and a coplanar resonator structure.  FIG. 2  illustrates an exemplary resonator  100  including a first outer loop  102 , a first open slot  104 , a first inner loop  106  and a second open slot  108 . The first open slot  104  is located within the first outer loop  102 . Similarly, the second open slot  108  is located within the first inner loop  106 . The resonator  100  further includes a first rectangular strip  110  located within the second open slot  108 . 
   The first outer loop  102  of the resonator  100  includes a first opening  112 , while the first inner loop  106  of the resonator  100  includes a second opening  114 . The first outer loop  102  and the first inner loop  106  of the resonator  100  illustrated in  FIG. 2  may be fabricated from high temperature superconductive materials, such as YBa2Cu3O7-δ. However, in an alternate embodiment of the resonator  100 , the first outer loop  102  and the first inner loop  106  may be made of any other conductive material used in building microstrip resonators. In the embodiment of the resonator  100  shown in  FIG. 2 , the first outer loop  102  and the first inner loop  106  are of rectangular shape. However, in an alternate embodiment of the resonator  100 , the first outer loop  102  and the first inner loop  106  may be made in any other shapes desired, such as, triangular, circular, etc. 
   The first outer loop  102  of the resonator  100  illustrated in  FIG. 2  includes a first longer side  122 , a second longer side  124 , a first shorter side  126  and a second shorter side  128 . The first inner loop  106  of the resonator  100  illustrated in  FIG. 2  includes a third longer side  132 , a fourth longer side  134 , a third shorter side  136  and a fourth shorter side  138 . In the exemplary embodiment of the resonator  100  illustrated in  FIG. 2 , the first opening  112  is located on the first shorter side  126 , however, in an alternate arrangement, the first opening  112  may be located on any other side of the first outer loop  102 . Similarly, in the exemplary embodiment of the resonator  100  illustrated in  FIG. 2 , the second opening  114  is located on the fourth shorter side  138 . However, in an alternate arrangement, the second opening  114  may be located on any other side of the inner loop  106 . 
   In the exemplary resonator  100  of  FIG. 2 , the first rectangular strip  110  is connected to the inner loop  106  on the fourth shorter side  138 . The resonator  100  further includes a shunting microstrip  140  that connects the first outer loop  102  to the first inner loop  106 . In the exemplary embodiment of the resonator  100 , the shunting microstrip is located between the first longer side  122  and the third longer side  132 . However, in an alternate arrangement, the shunting microstrip may be located in any alternate location between the first outer loop  102  and the first inner loop  106 . The separation of the first outer loop  102  from the first inner loop  106  by the first open slot  104  and the separation of the first inner loop  106  from the first rectangular strip  110  by the second open slot  108  gives the resonator  100  a coplanar structure. 
   In the exemplary implementation of the resonator  100 , the width of the first outer loop  102  and the first inner loop  106  is 200 micrometers (μm), while the width of the first open slot  104  and the second open slot  108  is 100 μm. However, alternate width for the first outer loop  102 , the first inner loop  106 , the first open slot  104  and the second open slot  108  may be provided. In the exemplary implementation, the outer dimensions of the resonator  100  are 1.7 mm by 7 mm, accordingly, in this implementation of the resonator  100 , the length of the first longer side  122  is 7 mm and the length of the first shorter side  126  is 1.7 mm. Also in the embodiment of the resonator  100  illustrated in  FIG. 2 , the width of the first rectangular strip  110  is 500 μm. 
   The exemplary embodiment of the resonator  100  of  FIG. 2  is located on a substrate of Magnesium Oxide (MgO) having the permittivity of 9.6 and a thickness varying between 0.2 mm and 2 mm. However, in an alternate arrangement, the resonator  100  of  FIG. 2  may be located on any of the alternate dielectric substrate material commonly used in the industry. 
   The thickness of the substrate on which the resonator  100  is located influences the resonant frequency of the resonator  100 . As explained above with respect to Equations 1 and 2, the resonant frequency of the resonator  100  increases as the thickness of the substrate increases due to increase in the effective dielectric constant ∈ e  of the substrate. The coplanar structure of the resonator  100  gives rise to stray capacitance between various microstrips. For example, there is stray capacitance between the first outer loop  102  and the first inner loop  106 . Similarly, there is stray capacitance between the first between the microstrips increases when the thickness of the substrate increases. The increase in the stray capacitance between the microstrips of the resonator  100  results in a decrease in the resonant frequency of the resonator  100 . This effect of decrease in the resonant frequency of the resonator  100  due to increase in the thickness of the substrate due to the stray capacitance of the resonator  100  is opposite to the effect of increase in the resonant frequency of the resonator  100  upon an increase in the thickness of the substrate due to the change in effective dielectric constant ∈ e  of the substrate. Accordingly, by properly trading off the increasing and decreasing capacitances that occur as substrate thickness varies, the resonant frequency of the resonator may be made relatively immune to substrate thickness variations. 
   The amount of stray capacitance between various microstrips of the resonator  100  depends on the width of the first open slot  104  and the width of the second open slot  108 , as well as on the location of the shunting microstrip  140 . In the exemplary illustration of the resonator  100 , where the thickness of the substrate may vary between 0.5 mm and 0.51 mm, the shunting microstrip  140  may be located at a distance of 1.4 mm from the outer edge of the second shorter side  128 . However, for different thickness of the substrate, the shunting microstrip  140  may be located at a different location in the resonator  100 . 
     FIG. 3  is an exemplary plot illustrating of the resonant frequencies of the resonator  100  of  FIG. 2  as a function of the location of the shunting microstrip  140  from the outer edge of the second shorter side  128 . The resonant frequencies of the resonator  100  illustrated in  FIG. 3  are measured for the thickness of the substrate on which the resonator  100  is located being equal to 0.5 mm and 0.51 mm. In  FIG. 3 , the horizontal axis indicates the distance of the shunting microstrip  140  from the outer edge of the second shorter side  128 . The vertical axis on the left-hand side indicates the resonant frequency of the resonator  100 . The line  302  in  FIG. 3  shows the resonant frequency of the resonator  100  for various distances of the shunting microstrip  140  from the outer edge of the second shorter side  128  when the thickness of the substrate is equal to 0.5 mm, while the line  304  shows the resonant frequency of the resonator  100  at various distances of the shunting microstrip  140  from the outer edge of the second shorter side  128  when the thickness of the substrate is equal to 0.51 mm. In  FIG. 3  the vertical axis on the right-hand side indicates the percent change in the resonant frequency between the 0.5 mm and the 0.51 mm substrate thicknesses. The line  306  in  FIG. 3  shows the percentage change in the resonant frequency of the resonator  100  when the substrate thickness changes from 0.5 mm to 0.51 mm for various distances of the shunting microstrip  140  from the outer edge of the second shorter side  128 . 
   As can be seen from the  FIG. 3 , when the distance of the shunting microstrip  140  from the outer edge of the second shorter side  128  is equal to 1.4 mm, the same resonant frequency is obtained for the resonator  100  at the substrate thickness of 0.5 mm and 0.51 mm. This indicates that when the shunting microstrip  140  is located at distance of 1.4 mm from the outer edge of the second shorter side  128  in the resonator  100 , the increase on the resonant frequency of the resonator  100  due to the increase in the thickness of the substrate from 0.5 mm to 0.51 mm is offset by the decrease in the resonant frequency of the resonator  100  due to the stray capacitance between various microstrips of the resonator  100 . 
   Another advantage of the resonator  100 , is that, due to the stray capacitance between various microstrips, for a given size, the resonator  100  may be used at much lower resonant frequencies than the conventional resonators illustrated in FIG.  1 . In other words, to achieve a given resonant frequency, the resonator  100  may be designed to have a much smaller size than the conventional resonators described in FIG.  1 . 
   The compact nature of the resonator  100  is illustrated in Table 1, which shows the resonant frequencies for the various resonator types described in FIG.  1  and FIG.  2 . For this illustration, each of these resonators is constructed to have the dimension of 1.4 mm by 7 mm and they are deposited on an MgO substrate of the thickness of 0.5 mm. Column B in the Table 1 indicates the resonant frequency for the specific resonator listed in Column A. While Column C indicates the resonant frequency listed in Column B as a percentage of the resonant frequency of the microstrip resonator  12  described in FIG.  1 . 
   
     
       
             
             
             
           
             
             
             
           
         
             
               TABLE 1 
             
             
                 
             
             
                 
               Resonant 
               Percentage 
             
             
                 
               Frequency 
               Resonant 
             
             
               Resonator Type 
               (MHz) 
               Frequency (%) 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               Standard Microstrip Resonator 12 
               7539 
               100 
             
             
               Loop Resonator 14 
               7330 
               97.2 
             
             
               Capacitively Loaded Loop Resonator 16 
               6107 
               81 
             
             
               Open Loop Resonator 18 
               3810 
               50.5 
             
             
               Meander Open Loop Resonator 20 
               2355 
               31.2 
             
             
               Folded Open Loop Resonator 22 
               1932 
               25.6 
             
             
               Shunted Open Loop Resonator 100 
               1822 
               24.1 
             
             
                 
             
           
        
       
     
   
   As shown in Table 1, the resonator  100  can achieve a resonant frequency which is only 24.1% of the resonant frequency of the microstrip resonator  12 . This property of the resonator  100  allows it to be used in building of smaller and less bulky filters that can operate at lower frequencies. 
     FIG. 4  illustrates the resonator  100  of  FIG. 2  with a coupling microstrip  402  that can be used as an input port. The coupling microstrip  402  is a microstrip of conducting material that can be connected to a signal input port. In the exemplary coupling arrangement illustrated in  FIG. 4 , the distance between the coupling microstrip  402  and the resonator  100  is 0.1 mm, however, in an alternate embodiment the coupling microstrip  402  may be located at a different distance from the resonator  100 . The coupling strength (i.e., the loaded quality factor) of the coupling between the resonator  100  and the coupling microstrip  402  increases when the distance between the coupling microstrip  402  and the resonator  100  decreases. The coupling strength is also a function of the length of the coupling microstrip  402 . For example, in the illustrated embodiment of  FIG. 4 , the loaded quality factor of the coupling arrangement for various lengths of the coupling microstrip  402  is as listed below in Table 2. 
   
     
       
             
             
             
           
             
             
             
           
         
             
                 
               TABLE 2 
             
             
                 
                 
             
             
                 
               Length of the Coupling 
                 
             
             
                 
               Microstrip 
               Loaded Quality Factor 
             
             
                 
                 
             
           
           
             
                 
             
           
        
         
             
                 
               1.0 
               1450 
             
             
                 
               2.0 
               471 
             
             
                 
               3.0 
               229 
             
             
                 
               4.0 
               137 
             
             
                 
               5.0 
               91.5 
             
             
                 
               6.0 
               65.4 
             
             
                 
               7.0 
               49.6 
             
             
                 
                 
             
           
        
       
     
   
     FIGS. 5A and 5B  illustrate two alternate coupling configurations used in designing multipole filters using the resonator  100  of FIG.  2 .  FIG. 5A  illustrates a coupling arrangement  500  of two resonators  502  and  504  where the first longer side  506  of resonator  502  is adjacent to the first longer side  508  of resonator  504 . In this configuration each of the first longer sides  506  and  508  that are shunted by shunting microstrips  510  and  512  to the inner loops  514  and  516  are adjacent to each other.  FIG. 5B  illustrates a coupling arrangement  550  of two resonators  552  and  554  where the second longer side  556  of resonator  552  is adjacent to the second longer side  558  of resonator  554 . In this configuration each of the first longer sides  560  and  562  which are shunted by microstrips  564  and  566  to the inner loops  572  and  574  are not adjacent to each other. 
     FIG. 6  illustrates the coupling coefficients as a function of the distance between the resonators for various coupling configurations illustrated in  FIGS. 5A and 5B . In  FIG. 6 , the horizontal axis indicates the distance between the resonators  502  and  504  in FIG.  5 A and the distance between the resonators  552  and  554  in FIG.  5 B. The vertical axis in  FIG. 6  indicates the coupling coefficients between the resonators for the coupling configurations illustrated in  FIGS. 5A and 5B . The line  602  illustrates the coupling coefficients between the resonators  502  and  504  of  FIG. 5A  for various distances between the resonators  502  and  504 . The line  604  illustrates the coupling coefficients between the resonators  552  and  554  of  FIG. 5B  for various distances between the resonators  552  and  554 . For the illustration in  FIG. 6 , the distance of the shunting microstrip  510 ,  512 ,  564  and  566  from the second shorter sides  518 ,  520 ,  568  and  570  respectively, is assumed to be 1.4 mm. 
   As can be seen in  FIG. 6 , for the same distance between the resonators, the coupling arrangement depicted by line  604  and illustrated in  FIG. 5B  has a higher coupling coefficient than the coupling arrangement depicted by line  602  and illustrated in FIG.  5 A. 
     FIG. 7  illustrates the coupling coefficients as a function of the shunting position within the resonators  502  and  504  for the coupling configuration illustrated in FIG.  5 A. In  FIG. 7 , the horizontal axis indicates the distance between the shunting microstrips  510  and the second shorter side  518  of the resonator  502 , and between the shunting microstrip  512  and the second shorter side  520  of the resonator  504  of FIG.  5 A. The vertical axis in  FIG. 7  indicates the coupling coefficient between the resonators  502  and  504 . For the illustration in  FIG. 7  it is assumed that the distance between the resonators  502  and  504  is 1 mm. As can be seen from the line  702 , the coupling coefficient between the resonators  502  and  504  increases as the distance of the shunting microstrips  510  and  512  from the second shorter sides  518  and  520  increases. Therefore, the coupling coefficients can be adjusted in a broad range by changing the distance of the shunting microstrips  510  and  512  from the second shorter sides  518  and  520 , which allows for the realization of filters of wide bandwidth, as well as filters of narrow bandwidth where the resonators are nevertheless closely spaced. 
     FIG. 8  illustrates an exemplary layout of a two-pole filter  800  using two resonators similar to the resonator  100  illustrated in FIG.  2 . In  FIG. 8  two resonators  802  and  804  are located adjacent to each other such that the distance between a first longer side  806  of resonator  802  and a first longer side  808  of filter  804  is 0.4 mm. The two-pole filter of  FIG. 8  also includes a first coupling microstrip  810  adjacent to a second longer side  812  of the resonator  802  and a second coupling microstrip  814  adjacent to a second longer side  816  of the resonator  804 . Note that the arrangement of the resonators  802  and  804  adjacent to each other is similar to that illustrated in FIG.  5 A. In the two-pole filter  800  illustrated in  FIG. 8 , the lengths of the first coupling microstrip  810  and the second coupling microstrip  814  are both 6.6 mm. In the two-pole filter illustrated in  FIG. 8 , the distances of the coupling microstrips  810  and  814  from the resonators  802  and  804  are 0.1 mms respectively. 
     FIG. 8A  illustrates an exemplary implementation of the two-pole filter  800  on a substrate. In this exemplary implementation,  820  illustrates the top-view of the two-pole filter  800 ,  822  illustrates the side-view of the two-pole filter  800 , and  824  illustrates the front-view of the two-pole filter  800 . The HTS ground plane  830  may be made of any of the commonly used HTS material such as YBa2Cu3O7-δ or metals such as gold. The substrate  832  may be made of any of the commonly used substrate material such as MgO, sapphire and LaAlO3. 
     FIG. 8B  illustrates a three dimensional implementation  850  of the two-pole filter  800  in a metallic housing. The metallic housing  852  may be made of any of the commonly used metal such as aluminum.  854  and  856  are coaxial cable connectors used to couple energy in and out of the two-pole filter  800 . The bottom layer  858  of the metallic housing is made of any of the carrier material such as titanium alloy. The HTS ground plane is coated by an additional metal layer  862  made of a metal such as gold for improvement of electrical and thermal conductivity. 
     FIG. 9  illustrates a frequency response of the exemplary two-pole filter  800  illustrated in FIG.  8 . The horizontal axis in  FIG. 9  indicates the frequency in MHz, the left-hand side vertical axis indicates the return loss in decibels (dB) and the right-hand side vertical axis indicates the insertion loss in dBs. The graph depicted by the line  902  shows the return loss characteristics of the two-pole filter illustrated in  FIG. 8 , and the graph depicted by the line  904  shows the insertion loss characteristics of the two-pole filter illustrated in FIG.  8 . As can be seen from the frequency response in  FIG. 9 , the passband center, the bandwidth and the passband ripple of the filter of  FIG. 8  are 1809.2 MHz, 18.8 MHz and 0.026 dB respectively. 
     FIG. 10  illustrates an exemplary layout of a four-pole filter  1000  using four resonators similar to the resonator  100  illustrated in FIG.  2 . In  FIG. 10  four resonators  1002 ,  1004 ,  1006  and  1008  are located adjacent to each other such that the gap between the resonators  1002  and  1004  is 1.5 mm, the gap between the resonators  1004  and  1006  is 1.9 mm, and the gap between the resonators  1006  and  1008  is 1.5 mm. The four-pole filter  1000  of  FIG. 10  also includes a first coupling microstrip  1010  adjacent to the resonator  1002  and a second coupling microstrip  1012  adjacent to the resonator  1008 . The lengths of the coupling microstrips  1010  and  1012  are 2.9 mm. In the four-pole filter  1000  illustrated in  FIG. 10 , the distances of the coupling microstrips  1010  and  1012  from the resonators  1002  and  1008  are 0.1 mm. In the embodiment illustrated in  FIG. 10 , the overall size of the four-pole filter  1000  is 7.4 mm by 14.3 mm. 
     FIG. 11  illustrates the frequency response of the exemplary four-pole filter  1000  illustrated in FIG.  10 . The horizontal axis in  FIG. 11  indicates the frequency in MHz, the left-hand side vertical axis indicates the return loss in dBs and the right-hand side vertical axis indicates the insertion loss in dBs. The graph depicted by  1102  shows the return loss characteristics of the four-pole filter  1000  illustrated in  FIG. 10 , while the graph depicted by  1104  shows the insertion loss characteristics of the four-pole filter  1000  illustrated in FIG.  10 . 
     FIG. 12  illustrates an exemplary layout of an eight-pole filter  1200  using eight resonators similar to the resonator  100  illustrated in FIG.  2 . In  FIG. 12  eight resonators  1202 ,  1204 ,  1206 ,  1208 ,  1210 ,  1212 ,  1214  and  1216  are located adjacent to each other such that the gap between the resonators  1202  and  1204  is 1.6 mm, the gap between the resonators  1204  and  1206  is 2.1 mm, the gap between the resonators  1206  and  1208  is 1.9 mm, the gap between the resonators  1208  and  1210  is 2.2 mm, the gap between the resonators  1210  and  1212  is 1.9 mm, the gap between the resonators  1212  and  1214  is 2.1 mm, and the gap between the resonators  1214  and  1216  is 1.6 mm. The eight-pole filter  1200  of  FIG. 12  also includes a first coupling microstrip  1218  adjacent to the resonator  1202  and a second coupling microstrip  1220  adjacent to the resonator  1216 . The lengths of the coupling microstrips  1218  and  1220  are 2.9 mm. In the eight-pole filter  1200  illustrated in  FIG. 12 , the distances of the coupling microstrips  1218  and  1220  from the resonators  1202  and  1216  are 0.1 mm. In the illustrated embodiment, the overall size of the eight-pole filter  1200  illustrated in  FIG. 12  is 7.5 mm by 29.6 mm. 
     FIG. 13  illustrates the frequency response of the exemplary eight-pole filter  1200  illustrated in  FIG. 12  where the eight-pole filter  1200  is located on a substrate of the thickness of 0.5 mm. The horizontal axis in  FIG. 13  indicates the frequency in MHz, the left-hand side vertical axis indicates the return loss in dBs and the right-hand side vertical axis indicates the insertion loss in dBs. The graph depicted by  1302  shows the return loss characteristics of the eight-pole filter  1200  illustrated in  FIG. 10 , while the graph depicted by  1304  shows the insertion loss characteristics of the eight-pole filter  1200  illustrated in FIG.  12 . 
     FIG. 14  illustrates the frequency response of the exemplary eight-pole filter  1200  illustrated in  FIG. 12  where the eight-pole filter  1200  is located on a substrate of the thickness of 0.51 mm. The horizontal axis in  FIG. 13  indicates the frequency in MHz, the left-hand side vertical axis indicates the return loss in dBs and the right-hand side y-axis indicates the insertion loss in dBs. The graph depicted by  1302  shows the return loss characteristics of the eight-pole filter  1200  illustrated in  FIG. 10 , while the graph depicted by  1004  shows the insertion loss characteristics of the eight-pole filter  1200  illustrated in FIG.  12 . 
   Many modifications and variations may be made in the techniques and structures described and illustrated herein without departing from the spirit and scope of the present invention. Accordingly, it should be understood that the apparatus and systems described herein are illustrative only and are not limiting upon the scope of the present patent.