Patent Publication Number: US-7589605-B2

Title: Method and apparatus to provide compensation for parasitic inductance of multiple capacitors

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     Not Applicable. 
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
     Not Applicable. 
     BACKGROUND OF THE INVENTION 
     As is known in the art, electromagnetic interference (EMI) filters are used in many types of electrical equipment. Such filters play a role in meeting federal regulations for device compatibility, for example. The size and performance of these filters are often limited by filter capacitor parasitics, such as the equivalent series inductance (ESL) associated with magnetic energy storage in the capacitor and interconnects. Coupled magnetic windings can be used to compensate for the effects of parasitic inductance of capacitors, reducing the filter volume and increasing its attenuation performance. 
       FIG. 1A  shows a prior art LC low pass filter LPF having a capacitor C and an inductor L.  FIG. 1B  shows the filter of  FIG. 1   a  with a high-frequency model  20  of the capacitor C. The high frequency capacitor model  20  includes the equivalent series inductance ESL and resistance R of the capacitor. These parasitics cause non-ideal filter behavior at high-frequencies as the capacitor impedance is no longer dominated by its capacitance. The filter attenuation performance is determined by the impedance mismatch between the series and shunt paths. At frequencies beyond the resonant frequency of the capacitor C, the shunt path impedance increases, and the filter LPF performance degrades as compared to the ideal LC filter. 
     It is known to use coupled magnetic windings to compensate for the effect of the parasitic inductance of the filter capacitor, greatly improving high-frequency filter performance. U.S. Pat. No. 6,937,115 to Perreault et al., for example, which is incorporated herein by reference, discloses a coupled inductor structure used to induce a voltage that counteracts the voltage due to the capacitor equivalent series inductance. This effect arises from the mutual coupling of the magnetic windings and appears as a negative branch inductance in the equivalent “T” model of the coupled magnetic windings. The negative branch inductance is used to compensate for the device parasitic inductance, creating a three terminal component with drastically reduced equivalent shunt path inductance. 
       FIGS. 2A and 2B  show a prior art compensated LC low-pass filter having inductance cancellation with the coupled magnetic winding represented by its equivalent “T” model. An implementation of this inductance cancellation method has been found to be effective for improving the performance of high frequency filters, such as those used in limiting electromagnetic interference (EMI).  FIG. 2A  shows an unsimplified equivalent circuit for the filter and  FIG. 2B  shows an equivalent simplified version. 
     While the inductive component of a single capacitor can be cancelled as described above, providing coupled windings for each capacitor in a circuit can require significant real estate. 
     SUMMARY OF THE INVENTION 
     In one aspect of the invention, a filter comprises first and second capacitors, and a circuit coupled to the first and second capacitors, the circuit including discrete magnetically-coupled windings to compensate for equivalent series inductance effects of a first capacitor electrical path including the first capacitor and a second capacitor electrical path including the second capacitor. Induction of the magnetically-coupled windings generates voltages that counteract voltages due to equivalent series inductance of the first and second capacitor electrical paths and do not counteract voltages due to capacitances of the first and second capacitors. 
     In one embodiment, the coupled windings nullify the equivalent series inductance of the first capacitor electrical path. It is understood that for purposes of this document nullify is a subset of compensate where nullify refers to counteracting completely or substantially completely, equivalent series inductance. 
     In another aspect of the invention, a method of suppressing electrical signals comprises coupling a first capacitor having at least first and second ends to a circuit including discrete magnetically-coupled windings, and coupling a second capacitor having at least first and second ends to the coupled magnetic winding circuit. The coupled windings compensate for an effect of an equivalent series inductance of paths through the first and second capacitors, wherein an effect of the capacitances of the first and second capacitors is not compensated for. 
     In a further aspect of the invention, a filter comprises a circuit including a first set of discrete magnetically-coupled windings having at least first and second capacitor tap points, a first capacitor having a first end connected to the first capacitor tap point, and a second capacitor having a first end connected to the second capacitor tap point. Induction of the first set of magnetically-coupled windings provides cancellation of an effect of an equivalent series inductance of a first path through the first capacitor and an equivalent series inductance of a second path though the second capacitor path, wherein an effect of capacitances of the first and second capacitors is not cancelled. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention will be more fully understood from the following detailed description taken in conjunction with the accompanying drawings, in which: 
         FIG. 1A  is a circuit diagram for a prior art low-pass LC filter; 
         FIG. 1B  is an equivalent circuit diagram of the prior art filter of  FIG. 1A  with a high frequency model of the capacitor; 
         FIG. 2A  is an equivalent circuit diagram of a prior art LC low-pass filter having inductance cancellation; 
         FIG. 2B  is a simplified equivalent circuit diagram for the prior art circuit of  FIG. 2A ; 
         FIG. 3  is an equivalent circuit diagram of a prior art EMI filter circuit; 
         FIG. 4A  is a circuit diagram of a common mode model of the circuit of  FIG. 3 ; 
         FIG. 4B  is a circuit diagram of a differential mode model of the circuit of  FIG. 3 ; 
         FIG. 5  is an equivalent circuit diagram of an EMI filter circuit having inductance cancellation; 
         FIG. 6  is a pictorial representation of a prior art circuit implementing balanced inductance cancellation; 
         FIG. 7A  is a circuit diagram of a common mode model for the circuit of  FIG. 5 ; 
         FIG. 7B  is a circuit diagram of a differential mode model for the circuit of  FIG. 5 ; 
         FIG. 8A  is a circuit diagram of a circuit having first and second capacitors with inductance cancellation provided by a single coupled winding; 
         FIG. 8   b  is a pictorial representation of a test fixture for the circuit of  FIG. 8A ; 
         FIG. 9  is a schematic depiction of the coupled winding of  FIG. 8A ; 
         FIG. 10  is a graphical representation of insertion loss versus frequency from the test fixture of  FIG. 8B ; 
         FIG. 11  is a pictorial representation of a test box to test coupled inductance cancellation; 
         FIG. 12A  is a schematic representation of a circuit having coupled inductance cancellation coils in the same direction; 
         FIG. 12B  is a schematic representation of a circuit having coupled inductance cancellation coils in the opposite direction; 
         FIG. 13A  is a schematic representation of flux patterns for coupled inductance cancellation coils in the same direction; 
         FIG. 13B  is a schematic representation of flux patterns for coupled inductance cancellation coils in the opposite direction; 
         FIG. 14A  is a graphical depiction of insertion loss of coupled inductance cancellation windings in the same direction; 
         FIG. 14B  is a graphical depiction of insertion loss of coupled inductance cancellation windings in the opposite direction; 
         FIG. 15A  is a schematic representation of a commercially available EMI filter; 
         FIG. 15B  is a pictorial representation of the EMI filter of  FIG. 15A ; 
         FIG. 16A  is a schematic representation of an EMI filter having inductance cancellation of multiple capacitors in accordance with the present invention; 
         FIG. 16B  is a pictorial representation of an EMI filter having inductance cancellation of multiple capacitors in accordance with the present invention; 
         FIG. 17A  is a graphical representation of common mode insertion loss versus frequency with and without inductance cancellation; 
         FIG. 17B  is a graphical representation of differential mode insertion loss versus frequency with and without inductance cancellation; 
         FIG. 18  is a schematic representation of a three-port extended cantilever model of a coupled magnetic circuit; 
         FIG. 19  is a schematic representation of a three-port extended cantilever model in tapped inductor configuration; and 
         FIG. 20  is a schematic representation of a shared terminal three-port circuit for use with the extended cantilever model tapped inductor configuration. 
     
    
    
     DETAILED DESCRIPTION 
     In general, the present invention provides inductive impedance cancellation for multiple capacitors with a coupled magnetic winding. In an exemplary embodiment, an EMI filter includes capacitors to attenuate both common-mode and differential-mode EMI. The coupled winding nullifies the parasitic effects of the capacitors in a space-efficient and elegant manner. 
     Before describing the invention in detail, some introductory information is provided.  FIG. 3  shows a known EMI filter  100  with a source circuit  102  to energize the filter  100 , which can be coupled to a load circuit  104 . The source circuit  102  is driven by both common mode and differential mode signals, as shown. 
     In an exemplary embodiment, the filter  100  includes a common-mode choke modeled with inductive elements L 11 , L 12 , L 2 ,  2 Lu 1 ,  2 Lu 2  and capacitive elements, which include a differential capacitor Cx and first and second common mode capacitors Cy 1 , Cy 2 . In one particular embodiment, the circuit  100  is analyzed by separating its common mode and differential mode responses and treating these equivalent circuits as independent circuits. 
     The common and differential mode equivalent circuits  150 ,  160  are shown in  FIGS. 4A and 4B , respectively. The common mode equivalent circuit  150  includes an inductor Lu+L 1 / 2  and a capacitor  2 Cy. The differential mode equivalent circuit  160  includes an inductor  2 L 1  and first and second capacitors Cx/ 2  and Cy/ 2  coupled in parallel. 
       FIG. 5  shows the circuit of  FIG. 3  with inductance cancellation coils for each capacitor. The differential capacitor Cx is fitted with first and second inductance cancellation coils  200 ,  202 . These cancellation coils  200 ,  202  are believed to be as effective as a single coil.  FIG. 6  shows a prior-art circuit in which first and second inductance cancellation coils are used to cancel the capacitance of a single differential capacitor in a balanced fashion. Returning to  FIG. 5 , first ends of the common mode capacitors Cy 1 , Cy 2  are coupled to a reference node and third and forth cancellation coils  204 ,  206  are coupled at the other ends of the capacitors Cy 1 , Cy 2 . 
     It is understood that the first and second cancellation coils  200 ,  202  provide a balanced implementation to avoid inserting an unbalanced circuit element within the otherwise balanced system. Without balancing the series inductances on both sides of the differential-mode capacitor Cx, a cross coupling between the differential and common mode signal sources would result. By avoiding this coupling, the common and differential mode circuit equivalents remain straightforward, as illustrated in  FIGS. 7A and 7B . The common mode equivalent circuit of  FIG. 7A  includes an inductor Lu+L 1 / 2 , a capacitor  2 Cy, and a cancellation winding CW. The differential mode equivalent circuit of  FIG. 7B  includes an inductor  2 L 1 , first and second capacitors Cx, Cy/ 2  and respective cancellation windings CW 1 , CW 2 . It should be noted that a consideration when physically implementing two balanced cancellation windings for a single capacitor is linked magnetic flux between the two sets of windings, as described below. 
     Referring again to  FIG. 5 , the construction of an EMI filter with full, balanced inductance cancellation would require four magnetically coupled windings  200 ,  202 ,  204 ,  206 . However, these windings may take up significant additional space within the filter, or if placed in close proximity, may exhibit significant magnetic coupling, complicating the design. Given these limitations, it is desirable to use a single winding to provide appropriate inductance compensation for two capacitors. It is also desirable to use only two windings to provide inductance compensation for three capacitors. 
     EXAMPLE 
     A test box was created with first and second capacitors C 1 , C 2  and a planar coupled winding CW inside as shown in  FIGS. 8A  (circuit schematic) and  8 B (illustrative fixture implementation). In the test fixture, two Panasonic ECKATS472ME6 4700pF Y2 class ceramic capacitors were used. This test box does not directly examine common and differential mode testing, however it does provide a straightforward example how a single coupled winding can support the compensation of inductance for two capacitors. 
       FIG. 9  shows an illustratively dimensioned drawing of the coil CW, which was cut using a water jet cutter from a single piece of 1 mm thick copper sheet. In the exemplary embodiment, the outer coil dimensions are 24 by 24 millimeters, a conductor width of about 1.2 millimeter, and a spacing of about 0.6 millimeter. It should be noted that this coil was intentionally far oversized for the amount of compensation required; the coil was designed in order to provide maximum design flexibility. It is understood that the coil CW can have a wide range of geometries and fabrication techniques to meet the needs of a particular application without departing from the invention. 
     An exemplary procedure for tuning the filter response of the two capacitors is now discussed. Initially, the first capacitor C 1  is tuned to optimally compensate for its parasitic inductance by adjusting the connection point of the capacitor while observing the filter attenuation (e.g., with a network analyzer). Once the optimal position is found, the first capacitor C 1  is attached to the cancellation winding CW. In one embodiment, the connection point of the first capacitor to the coupled winding circuit is selected by finding the connection point that minimizes the magnitude of an output signal when an input signal is applied. The second capacitor C 2  is then tuned (with the first capacitor C 1  in place) to find an optimal filter response. This gives one combination of capacitor locations on the coupled winding that results in a high performance filter characteristic. 
       FIG. 10  shows results for insertion loss versus frequency for the circuit and test fixture of  FIGS. 8A and 8B . Measurements were made using an Agilent 4395A network analyzer, which provides 50Ω source and load impedances. Results are shown for no capacitors  250 , C 1  not canceled  252 , C 1  canceled  254 , C 1  and C 2  not canceled  256 , C 1  canceled C 2 , not canceled  258 , and C 1  and C 2  canceled  260 . 
     To characterize the filter attenuation performance with different combinations of capacitor connections, measurements  250  were first taken without either capacitor connected to show the baseline filter response with only the coupled winding CW. When tuning the response with only C 1 , two measurements were taken for comparison: one with the capacitor connected directly at the input (source-side) terminal providing no cancellation, and one where the capacitor was connected to the cancellation coil CW at a location where the output response was optimal. The same approach was taken when tuning the response for the combination of C 1  and C 2 ; C 2  was connected either directly at the filter output (response-side) terminal or at a position optimizing the filter response with both capacitors. 
     The characterization results of the filter attenuation performance  260  clearly show a dramatic improvement (as much as 35 dB at high frequency) from the case where no compensation is provided (Both Not Canceled)  256  to the case where inductance compensation is provided for both capacitors (Both Canceled)  260 . 
     These results demonstrate that a single coupled magnetic winding CW can be used to provide inductance compensation for two capacitors C 1 , C 2 , with dramatic performance improvement at high frequencies. 
     When physically implementing multiple cancellation windings in a single filter, linked magnetic flux between the windings will affect the inductance cancellation and filter performance. As discussed above,  FIG. 6  shows a test filter with balanced cancellation windings for a single capacitor. In a more compact design, or those with larger coils, flux linkage effects may be increased substantially. 
     Based on the test box in  FIG. 8B , two additional boxes (using the same winding structure as the first box) were created to test two general coupling cases.  FIG. 11  shows the test box and its internal layout. 
       FIGS. 12A-B  and  13 A-B illustrate the difference between the two coupling cases. When considered from a common mode perspective, windings CW 1 , CW 2  placed in the same direction ( FIG. 12A ) throw flux in a way that opposes the flux of the other winding. For coils CW 1 , CW 2  wrapped in the opposite direction ( FIG. 12B ), the flux from one winding is reinforced by the other acting in a manner similar to a common-mode choke. 
     The illustrative tuning procedure described here is similar to the two-capacitor case, except that there are now two common-mode capacitors Cy 1 , Cy 2  that are tuned in the same step. Initially, capacitors Cy 1  and Cy 2  are tuned simultaneously in the common-mode case to compensate for their parasitic inductances. Once the optimal positions are found, the common-mode capacitors Cy 1 , Cy 2  are attached to their respective windings CW 1 , CW 2 . Following this, the differential mode capacitor Cx 1  is tuned in the differential mode case by moving its connections on both coils (symmetrically or asymmetrically) to find an optimal output response. 
       FIGS. 14A and 14B  show the results for the same and opposite direction cases, respectively.  FIG. 14A  shows the insertion loss for same direction coils for common mode no cancellation  300 , common mode optimal cancellation  302 , and differential mode optimal cancellation  304 .  FIG. 14B  shows the insertion loss for opposite direction coils for common mode no cancellation  310 , common mode optimal cancellation  312 , and differential mode optimal cancellation  314 . It can be seen that in this case, the relative winding orientations do not have a significant impact on the achievable filtration performance. It is understood, however, that the impact of the relative winding orientations may vary with the geometry and fabrication of the coupled windings without departing from the invention. 
     Having discussed above that a single inductance cancellation winding can be used with two capacitors to improve filter performance, its use in the context of common and differential mode EMI filtering is examined below 
     EXAMPLE 
     A commercially available filter was used as a starting point as shown in  FIG. 15A , which shows a filter schematic, and  FIG. 15B , which shows a pictorial representation of the filter. As can be seen in  FIG. 15B , the relatively large (15 μH) series inductors L 11  and L 12  are particularly bulky, heavy, and expensive components of the commercial filter. Thus, it would be desirable to eliminate them while preserving filter performance. In the illustrated filter, exemplary component values are set forth below in Table 1: 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Impedance Values 
               
            
           
           
               
               
               
            
               
                   
                 Component 
                 Value 
               
               
                   
                   
               
               
                   
                 L11, L12 
                  15 μH 
               
               
                   
                 CY11, CY12 
                  47 nF film capacitor 
               
               
                   
                 CX1, CX2 
                 2.2 μF film capacitor 
               
               
                   
                 CY21, CY22 
                  15 nF ceramic capacitor 
               
               
                   
                   
               
            
           
         
       
     
     As shown in  FIGS. 16A  (schematic) and  16 B (pictorial representation), the series inductors (L 11 , L 12  of  FIG. 15B ) were thus removed to provide working space for installing inductance cancellation windings CW 1 , CW 2 . Removal of the series inductors is offset through use of the significantly smaller cancellation windings CW 1 , CW 2 , which each have a total inductance on the order of 300 nH. 
     Common and differential mode measurements were taken of the unmodified filter of  FIG. 15B , as well as an intermediate step before the inductance cancellation winding were installed. In this intermediate step, the large inductors L 11  and L 22  were removed and straight solid 14-gauge wire was installed in their place. This configuration, referred to here as ‘Without Series Inductor’ was used as a baseline comparison for improvements based on inductance cancellation. 
     The tuning procedure outlined here is similar as that described above for the circuit of  FIG. 12A  and directed to tuning the filter response of common and differential mode capacitors. Initially, common-mode capacitors Cy 11 , Cy 12  are tuned simultaneously to compensate (in a symmetric fashion) for their parasitic inductances. Once the optimal positions are found, the capacitors Cy 11 , Cy 12  are permanently attached to their respective windings. Following this, the differential-mode capacitor Cx 1  is tuned by moving its connections on both coils (symmetrically or asymmetrically) to find an optimal output response. This order of tuning makes sense: in the case of an EMI filter, the common and differential mode capacitors do not impact system performance in the same way. In  FIG. 7A  described above it is shown that the common mode equivalent circuit is not influenced by the differential mode capacitance (or the inductance cancellation, other than the fixed series inductance introduced by the cancellation winding); the common-mode filtration operates as if the differential-mode capacitor were an open circuit. However, the differential mode filtration is dependent on the common mode capacitance and inductance cancellation. Thus, if the inductance cancellation for the common-mode capacitance is optimized first, the inductance compensation for the differential mode capacitor can be tuned subsequently without influencing the common mode performance. 
     The results of the completed tuning are shown in  FIGS. 17A  (common mode) and  17 B (differential mode) along with baseline filter and stock filter configurations. It should be noted that tuning both the common and differential modes is based on compromises between high and low frequency performance. This particular optimal output response chosen here may not be the highest achievable performance for another particular range of frequencies This is also influenced by the symmetry of connections Cx 1  has with the two coils in the process of tuning. 
     The result of incorporating the inductance cancellation coils show a dramatic improvement in the filtration performance for both common and differential modes with the large differential-mode inductors removed. The common mode shows improvement across its full range, and the differential mode shows improvement over its full range except for the small resonance around 2 MHz. The performance with the inductance cancellation windings exceeded the required performance specification of the stock filter, without requiring the large, expensive series inductors of the stock filter. One interesting aspect of the differential mode performance is that the inductance-compensated version has equal or better performance than the stock filter (with the large series inductors) except for the small resonance. 
     The results from this commercial EMI filter, as well as those from the test box in  FIG. 8 , show clearly that a single magnetically coupled winding can provide effective inductance compensation for two capacitors, and that two coupled windings can provide effective inductance cancellation for three capacitors. 
     Analysis of inductance cancellation windings with a single capacitor is relatively straightforward since one can use a two port transformer for the windings, which has only three independent terms. The number of independent terms needed to completely describe coupled magnetics with n terminals is given by n(n+1)/2, which grows as the square of n. 
     Adding to the complexity is the fact that many models for multiple winding transformers either do not adequately model real-world behavior or have poor correlation and numeric conditioning to attempted measurements or model parameters. One model that is effective, and well conditioned for experimental parameter extraction, is the Extended Cantilever Model, which is well known to one of ordinary skill in the art. 
     The Extended Cantilever Model of a coupled system yields an equivalent circuit with directly measurable parameters and provides a direct mapping between circuit parameters and the inductance matrix parameters. It is also well conditioned numerically when dealing with small leakage fluxes or high coupling factors. The extended cantilever circuit model for a three-port system is shown in  FIG. 18 , with circuit parameters related to impedance matrix parameters as follows: 
     
       
         
           
             
               
                 
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       FIG. 19  shows the application of the extended cantilever model to a center-tapped cancellation winding circuit with two tap points.  FIG. 20  shows the circuit model including tap capacitors and source and load networks. As set forth in the below equations, the transfer function from input voltage to load voltage for the circuit in  FIG. 20  was found using direct circuit analysis, with the source network consisting of an input voltage source V in  with series impedance z s  and a load network comprised of an impedance z 1 . T 2  and T 3  are arbitrary impedances representing the two capacitors. 
     
       
         
           
             
               
                 
                   
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                                       Z 
                                       33 
                                     
                                   
                                   - 
                                   
                                     Z 
                                     23 
                                     2 
                                   
                                   - 
                                   
                                     2 
                                     ⁢ 
                                     
                                       Z 
                                       13 
                                     
                                     ⁢ 
                                     
                                       Z 
                                       23 
                                     
                                   
                                   - 
                                   
                                     Z 
                                     13 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 T 
                                 2 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       Z 
                                       11 
                                     
                                     ⁢ 
                                     
                                       Z 
                                       22 
                                     
                                   
                                   - 
                                   
                                     Z 
                                     12 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 Z 
                                 33 
                               
                             
                             - 
                             
                               
                                 Z 
                                 11 
                               
                               ⁢ 
                               
                                 Z 
                                 23 
                                 2 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 Z 
                                 12 
                               
                               ⁢ 
                               
                                 Z 
                                 13 
                               
                               ⁢ 
                               
                                 Z 
                                 23 
                               
                             
                             - 
                             
                               
                                 Z 
                                 13 
                                 2 
                               
                               ⁢ 
                               
                                 Z 
                                 22 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     By analogy to the case for cancellation with a single capacitor, it is desired to find conditions that drive this transfer function to zero (or close to zero) when the impedances T 2  and T 3  are the equivalent series impedances of the two capacitor paths. Finding where the transfer function goes to zero, the numerator of the transfer function is the point of interest. The goal of making the transfer function zero can be accomplished by finding conditions where the numerator is zero, and where the denominator remains non-zero. Starting from the numerator of Equation 8, one finds:
 
0= z   l (( T   2   −Z   13   −Z   12 ) T   3 +(− Z   23   −Z   13 ) T   2   +Z   12   Z   23   −Z   13   Z   22 )  (9)
 
     refactoring and dividing by the (non-zero) value of z l :
 
0= Z   13 ( Z   12   +Z   23   +Z   13   +Z   22 )−( T   2   −Z   12   −Z   13 )( T   3   −Z   13   −Z   23 )  (10)
 
     Equation 10 gives two degrees of freedom in which to generate a zero. An optimal relationship can be found for any two impedances T 2  and T 3 . However in the case of common mode and differential mode filtering there are additional constraints to be considered. 
     As described above, in the case of an EMI filter, the common and differential mode capacitors do not impact system performance in the same way. In  FIG. 7A  it is shown that the common mode equivalent circuit is not influenced by the differential mode capacitance (or the inductance cancellation, other than the fixed series inductance introduced by the cancellation winding); the common-mode filtration operates as if the differential-mode capacitor were an open circuit. However, the differential mode filtration is dependent upon the common mode capacitance and inductance cancellation. This means that if the inductance compensation for the common-mode capacitance is optimized first, the inductance compensation for the differential mode capacitor can be tuned subsequently without influencing the common mode performance. 
     To find the optimal cancellation for the common mode capacitors, the transfer function in Equation 10 is considered when T 3 =∞ (since the differential-mode capacitance is a virtual open circuit for common-mode signals). 
     
       
         
           
             
               
                 
                   
                     H 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             T 
                             2 
                           
                           - 
                           
                             Z 
                             13 
                           
                           - 
                           
                             Z 
                             12 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         z 
                         l 
                       
                     
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     z 
                                     l 
                                   
                                   + 
                                   
                                     T 
                                     2 
                                   
                                   + 
                                   
                                     Z 
                                     33 
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                       Z 
                                       23 
                                     
                                   
                                   + 
                                   
                                     Z 
                                     22 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 z 
                                 s 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     T 
                                     2 
                                   
                                   + 
                                   
                                     Z 
                                     11 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 z 
                                 l 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     Z 
                                     33 
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                       Z 
                                       23 
                                     
                                   
                                   + 
                                   
                                     Z 
                                     22 
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                       Z 
                                       13 
                                     
                                   
                                   + 
                                   
                                     2 
                                     ⁢ 
                                     
                                       Z 
                                       12 
                                     
                                   
                                   + 
                                   
                                     Z 
                                     11 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 T 
                                 2 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 Z 
                                 11 
                               
                               ⁢ 
                               
                                 Z 
                                 33 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 Z 
                                 11 
                               
                               ⁢ 
                               
                                 Z 
                                 23 
                               
                             
                             + 
                             
                               
                                 Z 
                                 11 
                               
                               ⁢ 
                               
                                 Z 
                                 22 
                               
                             
                             - 
                             
                               Z 
                               13 
                               2 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 Z 
                                 12 
                               
                               ⁢ 
                               
                                 Z 
                                 13 
                               
                             
                             - 
                             
                               Z 
                               12 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     From this result, it is shown that if T 2 =Z 12 +Z 13  then full cancellation in the common-mode can be achieved. If this result is then taken along with the original transfer function in Equation 10, one finds: 
     
       
         
           
             
               
                 
                   
                     H 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     - 
                     
                       
                         
                           ( 
                           
                             
                               
                                 Z 
                                 13 
                               
                               ⁢ 
                               
                                 Z 
                                 23 
                               
                             
                             + 
                             
                               
                                 Z 
                                 13 
                               
                               ⁢ 
                               
                                 Z 
                                 22 
                               
                             
                             + 
                             
                               Z 
                               13 
                               2 
                             
                             + 
                             
                               
                                 Z 
                                 12 
                               
                               ⁢ 
                               
                                 Z 
                                 13 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           z 
                           l 
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               T 
                                               3 
                                             
                                             + 
                                             
                                               Z 
                                               22 
                                             
                                             + 
                                             
                                               Z 
                                               13 
                                             
                                             + 
                                             
                                               Z 
                                               12 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           z 
                                           l 
                                         
                                       
                                       + 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             Z 
                                             33 
                                           
                                           + 
                                           
                                             2 
                                             ⁢ 
                                             
                                               Z 
                                               23 
                                             
                                           
                                           + 
                                           
                                             Z 
                                             22 
                                           
                                           + 
                                           
                                             Z 
                                             13 
                                           
                                           + 
                                           
                                             Z 
                                             12 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         T 
                                         3 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   Z 
                                                   22 
                                                 
                                                 + 
                                                 
                                                   Z 
                                                   13 
                                                 
                                                 + 
                                                 
                                                   Z 
                                                   12 
                                                 
                                               
                                               ) 
                                             
                                             ⁢ 
                                             
                                               Z 
                                               33 
                                             
                                           
                                           - 
                                           
                                             Z 
                                             23 
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         z 
                                         s 
                                       
                                     
                                     + 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               Z 
                                               13 
                                             
                                             + 
                                             
                                               Z 
                                               12 
                                             
                                             + 
                                             
                                               Z 
                                               11 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           T 
                                           3 
                                         
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             
                                               Z 
                                               13 
                                             
                                             + 
                                             
                                               Z 
                                               12 
                                             
                                             + 
                                             
                                               Z 
                                               11 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           22 
                                         
                                       
                                       + 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                   ⁢ 
                                                   
                                                     Z 
                                                     12 
                                                   
                                                 
                                                 + 
                                                 
                                                   Z 
                                                   11 
                                                 
                                               
                                               ) 
                                             
                                             ⁢ 
                                             
                                               Z 
                                               13 
                                             
                                           
                                           + 
                                           
                                             Z 
                                             12 
                                             2 
                                           
                                           + 
                                           
                                             
                                               Z 
                                               11 
                                             
                                             ⁢ 
                                             
                                               Z 
                                               12 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         z 
                                         l 
                                       
                                     
                                     + 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               Z 
                                               13 
                                             
                                             + 
                                             
                                               Z 
                                               12 
                                             
                                             + 
                                             
                                               Z 
                                               11 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           33 
                                         
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             
                                               2 
                                               ⁢ 
                                               
                                                 Z 
                                                 13 
                                               
                                             
                                             + 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 Z 
                                                 12 
                                               
                                             
                                             + 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 Z 
                                                 11 
                                               
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           23 
                                         
                                       
                                       + 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             Z 
                                             13 
                                           
                                           + 
                                           
                                             Z 
                                             12 
                                           
                                           + 
                                           
                                             Z 
                                             11 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         Z 
                                         22 
                                       
                                     
                                     + 
                                     
                                       Z 
                                       13 
                                       2 
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           
                                             2 
                                             ⁢ 
                                             
                                               Z 
                                               12 
                                             
                                           
                                           + 
                                           
                                             Z 
                                             11 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         Z 
                                         13 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             Z 
                                             12 
                                             2 
                                           
                                           + 
                                           
                                             
                                               Z 
                                               11 
                                             
                                             ⁢ 
                                             
                                               Z 
                                               12 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         T 
                                         3 
                                       
                                     
                                     + 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               Z 
                                               13 
                                             
                                             + 
                                             
                                               Z 
                                               12 
                                             
                                             + 
                                             
                                               Z 
                                               11 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           22 
                                         
                                       
                                       + 
                                       
                                         
                                           ( 
                                           
                                             
                                               2 
                                               ⁢ 
                                               
                                                 Z 
                                                 12 
                                               
                                             
                                             + 
                                             
                                               Z 
                                               11 
                                             
                                           
                                           ) 
                                         
                                         ⁢ 
                                         
                                           Z 
                                           13 
                                         
                                       
                                       + 
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             Z 
                                             12 
                                             2 
                                           
                                           + 
                                           
                                             
                                               Z 
                                               11 
                                             
                                             ⁢ 
                                             
                                               Z 
                                               12 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         Z 
                                         33 
                                       
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           
                                             - 
                                             
                                               Z 
                                               13 
                                             
                                           
                                           - 
                                           
                                             Z 
                                             12 
                                           
                                           - 
                                           
                                             Z 
                                             11 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         Z 
                                         23 
                                         2 
                                       
                                     
                                     - 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   Z 
                                   13 
                                   2 
                                 
                                 ⁢ 
                                 
                                   Z 
                                   23 
                                 
                               
                               - 
                               
                                 
                                   Z 
                                   13 
                                   2 
                                 
                                 ⁢ 
                                 
                                   Z 
                                   22 
                                 
                               
                               - 
                               
                                 Z 
                                 13 
                                 3 
                               
                               - 
                               
                                 
                                   Z 
                                   12 
                                 
                                 ⁢ 
                                 
                                   Z 
                                   13 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     which leaves the requirement of
 
0= Z   13 ( Z   12   +Z   23   +Z   13   +Z   22 )  (13)
 
to make the transfer function zero. This result contains no terms of T 3  in which to tune in comparison to the terms of the impedance matrix. Additionally, the terms of the impedance matrix in the center tapped configuration are positive. This seems to give no opportunity for the numerator to become zero, save for the possibility of making Z 13 =O. If Z 13  becomes zero, and thus the transfer function numerator as well, the denominator becomes
 
( T   3   z   l   +Z   22   z   l   +Z   12   z   l   +Z   33   T   3 +2 Z   23   T   3   +Z   22   T   3   +Z   12   T   3   +Z   22   Z   33   +Z   12   Z   33   −Z   23   2 )( z   s   +Z   12   +Z   11 )  (14)
 
     which is finite, and thus Z 1 =0 does represent a zero for the transfer function. 
     It is known that depending on the frequency range of interest, the performance with imperfect cancellation may be significantly better than the uncanceled case. In this case a new term Δ 2  can be defined as the residual shunt path impedance of the capacitor. More specifically, Δ 2 =T 2 −(Z 12 +Z 13 ). If this is substituted into Equation 10, and the resulting equation massaged a bit, Equation 15 results giving a relation where Z 13  is not explicitly required to be zero for the transfer function to be zero.
 
0 =Z   13 ( T   2   +Z   22   +Z   23 )−Δ 2 ( T   3   −Z   23 )  (15)
 
     Hence, one may gain good performance in both common-mode and differential-mode by realizing substantial (but not perfect) cancellation in common-mode to benefit differential-mode performance. 
     One skilled in the art will appreciate further features and advantages of the invention based on the above-described embodiments. Accordingly, the invention is not to be limited by what has been particularly shown and described, except as indicated by the appended claims. All publications and references cited herein are expressly incorporated herein by reference in their entirety.