Patent Publication Number: US-5293086-A

Title: Polar leapfrog filter

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a polar leapfrog filter which can be constructed in the form of an active filter, and more particularly it pertains to an even-order polar leapfrog low-pass filter. 
     2. Description of the Prior Art 
     Heretofore, such a passive filter as shown in FIG. 1 has been widely used. However, it is the recent trend that so-called active filters are employed in lieu of such passive filters, as the result of peripheral circuits associated therewith being constructed in the form of semiconductor integrated circuit. Generally, an active filter is made up of components, each of which comprises a resistor, a capacitor, and an operational amplifier, and constructed in the form of a sallen-key circuit, a biquad circuit or an FDNR (frequency-dependent negative resistance) circuit by combining such operational amplifiers. Alternatively, a desired filter is constructed by using such circuits as units. In an attempt to change the filter characteristics, with the biquad circuit or the like, it is required that the constants for the resistors and capacitors be changed. With the FDNR filter, on the other hand, difficulties are experienced in an attempt to adjust the filter characteristics thereof since it is the usual practice that several such filters are interconnected with each other and it is required that constants for the elements of each such filter be changed. The other types of filter use variable resistors to make variable the filter characteristics thereof; thus, such filters are constructed inevitably in the form of a hybrid integrated circuit. Alternatively, it is required that chip components of a predetermined resistance value be pre-selected and mounted onto a printed circuit board, which disadvantageously leads to a increase in the size of the filter. In either case, such filters are disadvantageous in that they cannot be constructed in the form of a monolithic integrated circuit since the variable resistors or the pre-selected chip components should be mounted onto the printed circuit board as mentioned above. 
     In an attempt to make a Cauer filter or the like or achieve desired filter characteristics, it is required that the filter be constructed in the form of polar type having a damping pole, i.e., transmission zero point (pole zero) at a definite frequency. With the above-mentioned conventional arrangements, however, when it is attempted to achieve this, problems arise in that a number of parts are required so that the circuit arrangement turns out to be complicated and difficulties are encountered in an attempt to achieve an an odd-order arrangement, though an even-order arrangement is readily achievable, as is the case with a biquad circuit. The problems with a filter constructed in the form of a hybrid integrated circuit having a number of parts mounted on a printed circuit board, are such that most such filters are bulky and difficult to adjust the filter characteristics thereof. 
     SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a polar leapfrog filter constructed in accordance with the leapfrog simulation procedure (refer to M. E. Van Valkenburg: &#34;Analog Filter Design&#34;, 1982, CBS College Publishing), thereby eliminating the above-described drawbacks of the prior art. 
     The present invention employs the leapfrog simulation procedure as mentioned just above, and it is also based on the invention disclosed in U.S. patent application Ser. No. 798,215 filed Nov. 26, 1991, now U.S. Pat. No. 5,177,382, corresponding to Japanese Patent Application No. 333136/1990 filed Nov. 29, 1990. 
     An important advantage of the present invention is such that Cauer active filter can be very easily achieved according to the present invention. Another important advantage is such that a higher order filter can be easily constructed. Still another important advantage is such that the filter according to the present invention can readily be constructed in the form of semiconductor integrated circuit since the main elements of the filter are integrators, so that the number of parts thereof as well as the size thereof can be reduced. 
     Furthermore, the polar leapfrog filter according to the present invention is advantageous in that since the main constitutional elements thereof are integrators each comprising a variable transconductance amplifier, the pass bandwidth thereof can be easily adjusted by adjusting current supplied to the differential transistor pair of such variable transconductance amplifier to cause the internal resistance of such transistor pair to be changed. 
     Other objects, features and advantages of the present invention will become apparent from the ensuing description taken in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 illustrates an example of conventional even-order passive low-pass filter. 
     FIG. 2 is a circuit diagram showing an example of polar network which is used as a constitutional element of the polar leapfrog filter according to the present invention. 
     FIGS. 3A to 3D are block diagrams useful for explaining about the polar network of FIG. 2. 
     FIG. 4 illustrates the frequency characteristics of the polar network shown in FIG. 2. 
     FIG. 5 illustrates the arrangement of FIG. 1 as represented by using admittance and impedance. 
     FIG. 6A illustrates signal flow developed in accordance with the leapfrog simulation procedure. 
     FIG. 6B illustrates an arrangement transformed from that of FIG. 6A. 
     FIG. 6C illustrates an arrangement transformed from that of FIG. 6B. 
     FIG. 7 is a circuit diagram showing the polar leapfrog filter according to an embodiment of the present invention which is constructed by transforming the passive filter shown in FIG. 1. 
     FIG. 8 illustrates the frequency characteristics of the embodiment shown in FIG. 7. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Description will now be made of embodiments of the present invention with reference to FIGS. 2 to 8. 
     Referring to FIG. 2, an input terminal 1 is connected to a non-inverting input terminal of an operational amplifier A 1  of a substantially infinite gain, and an output terminal 2 is led out of the output terminal of the operational amplifier A 1 . The output terminal of the operational amplifier A 1  is connected to a non-inverting input terminal of a variable transconductance amplifier A 2 , the output terminal of which is coupled to an inverting input terminal of the operational amplifier A 1  and grounded through a capacitor C 1 . The variable transconductance amplifier A 2  and the capacitance C 1  constitute an integrator 11 which is arranged to provide negative feedback to the operational amplifier A 1 . The operational amplifier A 1  and the integrator 11 constitute a differentiator 13. Further, the output terminal of the variable transconductance amplifier A 2  is connected to a non-inverting input terminal of another variable transconductance amplifier A 3 , the output terminal of which is grounded through a second capacitor C 3  and connected to an inverting input terminal of the variable transconductance amplifier A 2 . A second integrator 12 is constituted by the variable transconductance amplifier A 3  and the capacitor C 2 . An inverting input terminal of the variable transconductance amplifier A 3  is grounded. 
     As will be appreciated from the above explanation, the circuit of FIG. 2 is a negative feedback circuit comprising in combination the integrator 12 and differentiator 13 and which represents a damping pole (pole zero) at a predetermined frequency as will be seen from FIG. 4. Thus, the circuit of FIG. 2 comprises a polar type circuit having a damping pole, which will be referred to as &#34;polar network&#34; hereinafter. 
     The polar network of FIG. 2 has a damping pole which occurs at a point where the filter characteristics (a) and (b) of the integrator 12 and differentiator 13 as combined cross each other, as shown in FIG. 4. It is possible to shift the damping pole by adjusting the operating currents of the variable transconductance amplifiers A 2  and A 3  to set up the transconductances gm 1  and gm 2  of variable transconductance amplifiers A 2  and A 3 . By adjusting the operating current of the variable transconductance amplifier A 3  constituting the integrator 12, for example, it is possible to cause the damping pole to be shifted from P 1  at a frequency f 0  to P 2  at a frequency f 1 . Further, by adjusting the operating currents of the variable transconductance amplifiers A 2  and A 3  constituting the integrators 11 and 12 respectively at the same time and in the same direction, it is possible to change the damping quantity alone, while keeping the damping pole at the predetermined frequency f 0 . 
     The block diagram of FIG. 3D corresponds to the polar network of FIG. 2; thus, it will be explained that the network of FIG. 2 can be converted directly from the block diagram of FIG. 3D. As mentioned above, the polar network of FIG. 2 is a negative feedback circuit comprising the integrator 12 and differentiator 13 in combination; thus, the transfer function of the polar network shown in FIG. 2 is equivalent to a sum of the transfer functions of the integrator 12 and differentiator 13. In this way, assuming that the transfer functions of the differentiator and integrator are represented by sC 1  /gm 1  and gm 2  /sC 2  respectively, the transfer function of the polar network shown in FIG. 2 is given as follows: 
     
         V.sub.2 /V.sub.1 =sC.sub.1 /gm.sub.1 +gm.sub.2 /sC.sub.2   (1) 
    
     where V 1  is an input voltage, V 2  is an output voltage, and gm 1  and gm 2  are the transconductances of the variable transconductance amplifiers A 2  and A 3  respectively. 
     FIG. 3A is a block diagram illustrating a common negative feedback circuit comprising an adder 3, and blocks 4 and 5. FIG. 3B is a block diagram wherein an integrator having a predetermined transfer function is provided in each of blocks 7 and 8 which correspond to the blocks 4 and 5 of FIG. 3A respectively. FIG. 3C illustrates an arrangement in which blocks 10 and 14 corresponding to the blocks 4 and 5 of FIG. 3A are constituted by an operational amplifier having an infinite gain and a block having a predetermined transfer function, respectively. FIG. 3D is a block diagram showing the above-mentioned polar network which is constituted by the blocks of FIGS. 3B and 3C. The polar network of FIG. 2 can be realized through direct transformation of the blocks of FIG. 3D. 
     The transfer function given by the equation (1) will now be sought. The transfer function of the arrangement shown in FIG. 3A, which comprises the blocks of a negative feedback circuit called basic feedback system, is given by equation (2) as follows: 
     
         (V.sub.1 -αV.sub.2)β=V.sub.2                    (2) 
    
     where α is the variable of the block 4, and β is the variable of the block 5. 
     Thus, from the equation (2), the transfer function of the block diagram shown in FIG. 3A is given as follows: 
     
         V.sub.2 /V.sub.1 =1/(α+1/β)                     (3) 
    
     In the block diagram of FIG. 3A, let it be assumed that the variables α and β are substituted with integrator transfer functions as shown below. ##EQU1## 
     Then, the block diagram of FIG. 3A can be transformed to the block diagram of FIG. 3B comprising blocks 7 and 8. Thus, by substituting the equation (4) for the equation (3), the transfer function can be expressed as follows: 
     
         V.sub.2 /V.sub.1 =1/(sC.sub.1 /gm.sub.1 +gm.sub.2 /sC.sub.2)(5) 
    
     Further, the variables α and β of the blocks 4 and 5 in the block diagram of FIG. 3A are rewritten as follows: ##EQU2## 
     The resulting block diagram turns out to be as shown in FIG. 3=C, the transfer function of which is given as follows, by substituting the equations (6) for the variables α and β in the equation (3): 
     
         V.sub.2 /V.sub.1 =sC.sub.1 /gm.sub.1 +gm.sub.2 /sC.sub.2   (7) 
    
     The equation (7) indicates that the block diagram of FIG. 3C provides a transfer function which is equivalent to one obtained by adding up the characteristics of the differentiator and integrator such as represented by the equation (1). 
     The transfer function of the block 14 shown in FIG. 3C is given as 1/(sC 1  /gm 1  +gm 2  /sC 2 ), which is identical with the transfer function represented by the equation (5). 
     Thus, by combining the block diagrams of FIGS. 3B and 3C, the transfer function of the equation (7) can be illustrated as in the block diagram of FIG. 3D. In this way, it has been found that the block diagram of FIG. 3D can be transformed to the circuit arrangement of FIG. 2. 
     As will be appreciated from the above discussion, according to the present invention, an even-order polar leapfrog filter is constructed on the basis of the matters explained hereinabove with reference to FIGS. 2 to 4. The even-order leapfrog filter is arranged such that in the case where at least one said polar network is incorporated therein, an integrator is provided at each of the input and output portions thereof, and a third integrator is provided immediately preceding the output side integrator; in the case where two or more said polar networks are incorporated therein, a further integrator is provided between said polar networks, the number of all the circuits being selected to be even and equal to the order of the filter; leapfrog type negative feedback is provided between respective adjacent ones of the above-mentioned circuits; and all the integrators except the output side integrator are constructed in the form of self negative feedback type. 
     Referring to FIG. 1, there is shown a conventional even-order passive LPF (low-pass filter), where n is an even number which is equal to or greater than 4. Referring to FIG. 7, there is illustrated the polar leapfrog filter according to an embodiment of the present invention, which is achieved by transforming the passive filter of FIG. 1 to an active filter in accordance with the leapfrog simulation procedure mentioned in the preamble portion of the present specification. 
     Description will now be made of the process of designing the active filter of FIG. 7, which is equivalent to the passive filter of FIG. 1, by using the leapfrog simulation procedure. 
     In FIG. 1, it is assumed that the values for the resistor, capacitors and coils constituting the passive filter shown therein are R&#39;, C&#39; and L&#39;, respectively, and the element values and frequency are subjected to scaling so that the value for the resistor R&#39; and the cut-off frequency become equal to 1 Ω and 1 rad/sec respectively. It is also assumed that the values for the resistor, capacitor and coil, after having been subjected to the scaling, are R&#34;, C&#34; and L&#34; respectively. With the coil being regarded as identical with the capacitor, the element values and frequency are again subjected to scaling so that the value for R&#34; becomes equal to 1 Ω and the cut-off frequency becomes the original value. Assuming that the values for the resistor, capacitor and coil, after having been subjected to the re-scaling, are R&#39;&#34;, C&#39;&#34; and L&#39;&#34;, then these values are given by equation (8) as follows: ##EQU3## 
     Admittance Y and impedance Z in the block diagram of FIG. 5 are given by equation (9) as follows: ##EQU4## 
     In FIG. 5, the admittance and impedance can be regarded as transfer functions which are given by equation (10) as follows: ##EQU5## 
     The arrangement of FIG. 5 can be developed into such a signal flow as shown in FIG. 6A, in accordance with the leapfrog simulation procedure disclosed in the &#34;Analog Filter Design&#34; cited in the preamble portion of the present specification. In FIG. 6A, the characters in the respective blocks represent transfer functions corresponding to the equations (10) respectively, and the circles indicate adders. 
     By putting the equations (10) in the respective blocks of FIG. 6A, the arrangement of FIG. 6A is transformed to that of FIG. 6B. Recall that the arrangement of FIG. 3C can be rewritten as shown in FIG. 3D as mentioned hereinbefore; by putting such relationship in FIG. 6B, the arrangement of FIG. 6B can be transformed to that of FIG. 6C. 
     Furthermore, from the fact the polar network of FIG. 2 can be achieved through direct transformation from the block diagram of FIG. 3D, it is possible to construct such a circuit arrangement as shown in FIG. 7, by putting the relationship between FIG. 3D and FIG. 2 in the arrangement of FIG. 6C which constitutes the even-order polar leapfrog filter according to the embodiment of the present invention, which is equivalent to the arrangement of FIG. 1 as will readily be appreciated from the above discussion. The circuits indicated at 12 and 13 in FIG. 7 correspond to the integrator 12 and differentiator 13 of FIG. 2, respectively; thus, it will be readily apparent that the circuit 21 which comprises the integrator 12 and differentiator 13, corresponds to the polar networks described above with reference to FIG. 2. In FIG. 7, reference numerals 20, 22 and 23 indicate integrators comprising variable transconductance amplifiers G 1 , G i+1 , and G n  having the output terminals thereof grounded through capacitors L 1 , L i+1  and C n  respectively. Each of the integrators 20, 22 and 23 corresponds to the integrator 12, and only the integrator 20 is constructed in the form of self negative feedback type. Furthermore, negative feedback is provided in the form of leapfrog from the output side integrator 23 to the immediately preceding integrator 22, from the integrator 22 to the immediately preceding polar network 21, and from the polar network 21 to the input side integrator 20. In this case, the sum of the number of the integrators and the number of the polar network is even and equal to the order of the filter, and it is possible to increase the order of the polar leapfrog filter by additionally providing the combination of the polar network 21 and integrator 22 which are connected to each other as indicated by dotted lines. 
     FIG. 8 illustrates the frequency characteristics of the polar leapfrog filter shown in FIG. 7, pass band of which is changed as indicated by (a), (b) and (c) as the transconductance gm of the variable transconductance amplifier constituting the integrator is changed to 1, 0.5 and 2 mS (Simens or Mho) by changing the operating current flowing through the differential transistor pair of the variable transconductance amplifier. 
     While the present invention has been illustrated and described with respect to specific embodiments thereof, it is to be understood that the present invention is by no means limited thereto but encompasses all changes and modifications which will become possible within the scope of the appended claims.