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### Abstract

Superconducting technology is ideal for producing selective and sensitive base-station receive filters for wireless communications. Although superconducting filters improve the performance of wireless communications systems, filter nonlinearities limit their potential applications under real-world operating conditions that involve high-power interfering signals. These limitations are examined numerically and experimentally. The intermodulation response of a minimum-phase filter is found to be largely independent of its topology, and to depend on the bandwidth and the steepness of the amplitude response at the band edge. The effect of the small-signal frequency response on the intermodulation response can be separated from those that depend on resonator geometry and material properties. As a result, the intermodulation performance is quantified by a set of figures of merit that measure the quality of a filter design and the material and resonator properties.

I am indebted to D. Ambriz and J. Fuller for their technical support and assistance during the measurements. I would like to thank G. Matthaei for helpful discussions and useful comments on the manuscript, and R. Hammond and B. Willemsen for fruitful discussions. This work was supported in part by the Defense Advanced Research Project Agency Defense Sciences Office DARPA Order No. J607 issued by DARPA/CMD under Contract No. MDA972-00-C-0010.

I. INTRODUCTION

II. THEORY OF WEAKLY NONLINEAR FILTERS

III. INTERMODULATION RESPONSE OF BANDPASS FILTERS

A. Intermodulation response of closely separated signals

B. Intermodulation response of widely separated signals

IV. INTERMODULATION SCALING LAWS

A. Low-pass filters

B. Bandpass filters

C. Figures of merit

V. COMPARISON WITH EXPERIMENTAL RESULTS

VI. CONCLUSIONS

### Key Topics

- Superconductivity
- 39.0
- Superconductivity models
- 21.0
- Bandpass filters
- 15.0
- Materials properties
- 15.0
- Chebyshev filters
- 10.0

## Figures

Schematic view of a two-port network with internal branches and nodes.

Schematic view of a two-port network with internal branches and nodes.

(a) -resonator bandpass filter with source and load impedances . Optional coupling between two nonadjacent resonators is denoted by a dashed line. Also shown are various resonator structures that are used to model nonlinear bandpass filters: (b) series-type resonator, (c) shunt-type resonator, and (d) -type resonator.

(a) -resonator bandpass filter with source and load impedances . Optional coupling between two nonadjacent resonators is denoted by a dashed line. Also shown are various resonator structures that are used to model nonlinear bandpass filters: (b) series-type resonator, (c) shunt-type resonator, and (d) -type resonator.

(a) Transducer gain and (b) group delay of , 4, 8, and 16 resonator Chebyshev bandpass filters with a 0.1-dB passband ripple and a 5-MHz equal-ripple bandwidth as a function of frequency.

(a) Transducer gain and (b) group delay of , 4, 8, and 16 resonator Chebyshev bandpass filters with a 0.1-dB passband ripple and a 5-MHz equal-ripple bandwidth as a function of frequency.

Intermodulation response of , 4, 8, and 16 resonator Chebyshev bandpass filters at the lower third-order mixing frequency as a function of the same frequency for a constant input tone separation of 0 kHz. The other parameters are (, 2), , and . The filters have a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

Intermodulation response of , 4, 8, and 16 resonator Chebyshev bandpass filters at the lower third-order mixing frequency as a function of the same frequency for a constant input tone separation of 0 kHz. The other parameters are (, 2), , and . The filters have a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

Intermodulation response of a four-resonator Chebyshev bandpass filter at the lower third-order mixing frequency as a function of the same frequency for a constant input tone separation of 0 kHz. The other parameters are (, 2), , and (, …, 4). Also shown are the IMD power contributions of individual resonators. The filter has a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

Intermodulation response of a four-resonator Chebyshev bandpass filter at the lower third-order mixing frequency as a function of the same frequency for a constant input tone separation of 0 kHz. The other parameters are (, 2), , and (, …, 4). Also shown are the IMD power contributions of individual resonators. The filter has a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

Intermodulation response of , 4, 8, and 16 resonator Chebyshev bandpass filters at the lower mixing frequency , which is held constant at 1870 MHz, as a function of the lower input tone frequency . The other parameters are (, 2), , and . The filters have a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

Intermodulation response of , 4, 8, and 16 resonator Chebyshev bandpass filters at the lower mixing frequency , which is held constant at 1870 MHz, as a function of the lower input tone frequency . The other parameters are (, 2), , and . The filters have a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

Intermodulation response of a four-resonator Chebyshev bandpass filter at the lower mixing frequency , which is held constant at 1870 MHz, as a function of the lower input tone frequency . The other parameters are (, 2), , and (, …, 4). Also shown are the power contributions of individual resonators. The filter has a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

Intermodulation response of a four-resonator Chebyshev bandpass filter at the lower mixing frequency , which is held constant at 1870 MHz, as a function of the lower input tone frequency . The other parameters are (, 2), , and (, …, 4). Also shown are the power contributions of individual resonators. The filter has a passband ripple of 0.1 dB and an equal-ripple bandwidth of 5 MHz.

The peak IMD power at the -order mixing frequency as a function of steepness for Chebyshev (filled circles) and Butterworth (open circles) low-pass filters with the nonlinearity exponent , 5, 7, and 9. The solid line denotes the power-law dependence , where . The passband ripple of the Chebyshev filters is 0.1 dB. For a given , the peak IMD power is scaled with a factor , which is the peak IMD power of a one-resonator low-pass Butterworth filter.

The peak IMD power at the -order mixing frequency as a function of steepness for Chebyshev (filled circles) and Butterworth (open circles) low-pass filters with the nonlinearity exponent , 5, 7, and 9. The solid line denotes the power-law dependence , where . The passband ripple of the Chebyshev filters is 0.1 dB. For a given , the peak IMD power is scaled with a factor , which is the peak IMD power of a one-resonator low-pass Butterworth filter.

Distribution functions of IMD power levels of low-pass filters due to individual resonators at the third-order mixing frequency where the total third-order IMD power has its largest value. Shown are distributions for (a) Chebyshev filters with various passband ripple values and for (b) Butterworth filters as a function of , where is the number of resonators in a filter and labels the resonators starting from the one closest to the input . The nonlinearity exponent and the number of resonators per filter is , 32, 63, and 128.

Distribution functions of IMD power levels of low-pass filters due to individual resonators at the third-order mixing frequency where the total third-order IMD power has its largest value. Shown are distributions for (a) Chebyshev filters with various passband ripple values and for (b) Butterworth filters as a function of , where is the number of resonators in a filter and labels the resonators starting from the one closest to the input . The nonlinearity exponent and the number of resonators per filter is , 32, 63, and 128.

The peak IMD power at the -order mixing frequency as a function of steepness for Chebyshev (filled symbols) and Butterworth (open symbols) bandpass filters with the nonlinearity exponent and 5 and the fractional bandwidth (circles) and 0.01 (squares). The passband ripple of the Chebyshev filters is 0.1 dB. For a given , the peak IMD power is scaled with a factor , which is the peak IMD power of a one-resonator bandpass Butterworth filter with . The solid line denotes the power-law dependence , where and .

The peak IMD power at the -order mixing frequency as a function of steepness for Chebyshev (filled symbols) and Butterworth (open symbols) bandpass filters with the nonlinearity exponent and 5 and the fractional bandwidth (circles) and 0.01 (squares). The passband ripple of the Chebyshev filters is 0.1 dB. For a given , the peak IMD power is scaled with a factor , which is the peak IMD power of a one-resonator bandpass Butterworth filter with . The solid line denotes the power-law dependence , where and .

Figure of merit characterizing the third-order intermodulation response of Butterworth and Chebyshev bandpass filters as a function of the steepness parameter for various values of the passband ripple . The nonlinearity exponent , (dashed line), and 0.01 (solid line).

Figure of merit characterizing the third-order intermodulation response of Butterworth and Chebyshev bandpass filters as a function of the steepness parameter for various values of the passband ripple . The nonlinearity exponent , (dashed line), and 0.01 (solid line).

(a) Transducer gain and (b) the group delay of an eight-resonator PCS -band filter made of YBCO high- superconducting material as a function of frequency. The filter has one cross coupling between the third and the sixth resonators producing two transmission zeros. The symbols denote the data measured at 77 K and lines are the theoretical fit.

(a) Transducer gain and (b) the group delay of an eight-resonator PCS -band filter made of YBCO high- superconducting material as a function of frequency. The filter has one cross coupling between the third and the sixth resonators producing two transmission zeros. The symbols denote the data measured at 77 K and lines are the theoretical fit.

Two-tone third-order intermodulation responses of an eight-resonator PCS -band filter made of YBCO high- superconducting material. (a) The IMD power at the lower (filled circles and line) and higher (open circles and dashed line) third-order mixing frequencies as a function of the same frequencies for a fixed tone separation of 50 kHz, (b) the IMD power at the lower mixing frequency , which is held constant at 1870 MHz, as a function of the lower tone frequency (the filled and open circles denote the IMD powers obtained by exchanging the input and output ports), and (c) the IMD power at the lower mixing frequency as a function of the same frequency, holding the lower tone frequency constant at 1872 MHz. The power of the input tones is 0 dBm. The circles denote the data measured at 77 K and the lines are the theoretical prediction assuming a third-order nonlinearity. The characteristic power used in the theoretical calculation is 86.8 dBm for all but the first and the last resonators for which it is 86.6 dBm.

Two-tone third-order intermodulation responses of an eight-resonator PCS -band filter made of YBCO high- superconducting material. (a) The IMD power at the lower (filled circles and line) and higher (open circles and dashed line) third-order mixing frequencies as a function of the same frequencies for a fixed tone separation of 50 kHz, (b) the IMD power at the lower mixing frequency , which is held constant at 1870 MHz, as a function of the lower tone frequency (the filled and open circles denote the IMD powers obtained by exchanging the input and output ports), and (c) the IMD power at the lower mixing frequency as a function of the same frequency, holding the lower tone frequency constant at 1872 MHz. The power of the input tones is 0 dBm. The circles denote the data measured at 77 K and the lines are the theoretical prediction assuming a third-order nonlinearity. The characteristic power used in the theoretical calculation is 86.8 dBm for all but the first and the last resonators for which it is 86.6 dBm.

Figure of merit as a function of the measured peak IMD power at the third-order mixing frequency for various cellular and PCS bandpass filters made of TBCCO (filled circles) and YBCO (open circles) high- superconducting materials. A dashed line shows a trajectory of filters for which . The intermodulation responses are measured at 77 K using two-tone experiments where the input tones have an equal power of and their frequency separation is kept to a constant value of 50 kHz. The analysis assumes that the governing nonlinearity is of the third order.

Figure of merit as a function of the measured peak IMD power at the third-order mixing frequency for various cellular and PCS bandpass filters made of TBCCO (filled circles) and YBCO (open circles) high- superconducting materials. A dashed line shows a trajectory of filters for which . The intermodulation responses are measured at 77 K using two-tone experiments where the input tones have an equal power of and their frequency separation is kept to a constant value of 50 kHz. The analysis assumes that the governing nonlinearity is of the third order.

Two-resonator bandpass filter where resonators described by impedances (, 2) are coupled together by an immittance inverter . The source and load impedances are .

Two-resonator bandpass filter where resonators described by impedances (, 2) are coupled together by an immittance inverter . The source and load impedances are .

## Tables

Six cellular and PCS high- superconducting bandpass filters used in this study.

Six cellular and PCS high- superconducting bandpass filters used in this study.

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