Inductorless lowpass filter utilizing frequency dependent negative resistors

Claim 2

In a given lowpass filter of even degree having a first load resistor connected in parallel with a first capacitor thereof that is connected to ground; the given filter being realized by replacing in the given filter each given inductor with an associated resistor, each given resistor with an associated capacitor, and each given capacitor with an associated frequency dependent negative resistor (FDNR) to produce an equivalent inductorless network wherein corresponding associated and given elements have the same numerical values of admittance except that admittance values of the associated elements are multiplied by the complex frequency parameter p, the equivalent network having an FDNR connected in parallel with a second load capacitor that is connected to ground; the method of terminating the equivalent network in a second load resistor comprising the steps of replacing the parallel combination of the second capacitor and the FDNR in the equivalent network with a positive immittance converter having input and output ports and having an output admittance across the output ports that is 90* out of phase with respect to an admittance connected across the input ports thereof, connecting the parallel combination of the second resistor and a third capacitor across the input port of the converter, and connecting the output port of the converter to the equivalent network at the previous points of connection thereto of the replaced second capacitor and FDNR.

Claim 3

The method according to claim 1 wherein the operation of the converter is defined by the relationship Yout(p) pkYin(p) wherein Yin(p) is the admittance connected across the input port, Yout(p) is the admittance produced across the output port, k is a constant determined by the design of the converter, p j Chi , j Square Root -1, and omega is the radian frequency.

Claim 4

The method according to claim 2 including the steps of selecting the second resistor to have an admittance Y1(p) satisfying the relationship Y1(p) (1/pk) Y2(p) wherein Y2(p) is the admittance of the second capacitor, and selecting the third capacitor to have an admittance Y3(p) satisfying the relationship Y3(p) (1/pk) Y4 (p) is the admittance of the FDNR.

Claim 5

In a given lowpass filter having a first load resistor connected thereto and to ground; the given filter being realized by replacing in the given filter each given inductor with an associated resistor, each given resistor with an associated capacitor, and each giVen capacitor with an associated frequency dependent negative resistor (FDNR) to produce an equivalent inductorless network wherein corresponding associated and given elements have the same numerical values of admittance except that admittance values of the associated elements are multiplied by the complex frequency parameter p, the equivalent network having a first load capacitor that is connected to ground; apparatus replacing and simulating the first capacitor for terminating the equivalent network in a resistive load impedance comprising a second resistor having a pair of terminals across which the output of the filter-network is taken, a positive immittance converter having a pair of input terminals connected to associated terminals of said second resistor and having a pair of output terminals, one of each of said input and output terminals being connected to ground, said converter being characterized in that the voltage between the other input and output terminals is negligible compared to that developed across said second resistor and causing the admittance across the output terminals to be 90* out of phase with an admittance connected across the input terminals thereof, and means for connecting the output terminals of said converter to the equivalent network at the original point of connection thereto of the first capacitor.

Claim 6

In a given lowpass filter having a first load resistor connected thereto and to ground; the given filter being realized by replacing in the given filter each given inductor with an associated resistor, each given resistor with an associated capacitor, and each giVen capacitor with an associated frequency dependent negative resistor (FDNR) to produce an equivalent inductorless network wherein corresponding associated and given elements have the same numerical values of admittance except that admittance values of the associated elements are multiplied by the complex frequency parameter p, the equivalent network having a first load capacitor that is connected to ground; apparatus replacing and simulating the first capacitor for terminating the equivalent network in a resistive load impedance comprising a second resistor having a pair of terminals across which the output of the filter-network is taken, a positive immittance converter having a pair of input terminals connected to associated terminals of said second resistor and having a pair of output terminals, one of each of said input and output terminals being connected to ground, said converter being characterized in that the voltage between the other input and output terminals is negligible compared to that developed across said second resistor and causing the admittance across the output terminals to be 90* out of phase with an admittance connected across the input terminals thereof, and means for connecting the output terminals of said converter to the equivalent network at the original point of connection thereto of the first capacitor.

Claim 7

Apparatus according to claim 4 wherein the operation of said converter satisfies the relationship Yout(p) pkYin(p) wherein Yout(p) is the admittance produced across the output terminals thereof, p j omega , j Square Root -1, omega is the radian frequency, k is the proportionality constant determined by said converter, and Yin(p) is the admittance connected across the input terminals of said converter.

Claim 8

Apparatus according to claim 5 wherein the admittance Y1(p) of said resistor satisfies the relationship Y1(p) (1/pk) Y2(p) where Y2(p) is the admittance of the first capacitor.

Claim 9

Apparatus according to claim 4 wherein the given lowpass filter is of even degree and has a second capacitor connected in parallel with the first load resistor, the equivalent network having a frequency dependent negative resistor (FDNR) connected in parallel with the first load capacitor; and including instead of the FDNR a third capacitor connected in parallel with the second resistor.

Claim 10

Apparatus according to claim 7 wherein the operation of said converter satisfies the relationship Yout(p) pkYin(p) wherein Yout(p) is the admittance produced across the output terminals thereof, p j omega , j Square Root - 1, omega is the radian frequency, k is the proportionality constant determined by said converter, and Yin(p) is the admittance connected across the input terminals of said converter; the admittance Y1(p) of said second resistor satisfies the relationship Y1(p) (1/pk) Y2(p) where Y2(p) is the admittance of the first capacitor; and the admittance Y3(p) of said third capacitor satisfies the relationship Y3(p) (1/pkpk) Y4(p) where Y4(p) is the admittance of the FDNR.