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In this paper a comparison of a sixth-order active band pass R-filter output response with the output response of a sixth-order band pass RC-filter at different quality factors (Q = 2, 5, 7, 8 and 10) was carried out at a fixed frequency of 10 KHz. The architecture used in the design is the multiple feedbacks for both filter networks. The simulated response characteristics show that both filters (R- and RC-filters) have their mid-band gains increasing with Q, while their bandwidths monotonically decreased with Q-values. The bandwidths are in the range of 22.23 dB to 62.97 dB and –55.49 dB to –50.81 dB (Q = 2 to 10) for R- and RC-filters respectively. At higher Q-values, R-filter showed better selectivity with a smaller bandwidth (400 Hz) at the edge of the pass band, when compared to 450 Hz for the RC-filter. The roll-off rate around –58.9 dB/decade for the R-filter appears to be that of a third-order filter response, while the RC-filter has its response in the range –106 to –132 dB/decade which is in the neighbourhood of an ideal sixth-order response (roll-off of 120 db/decade). A shift in the center frequency with Q was observed for the RC-filter only.

In modern electronic circuits, unwanted signals are a major challenge to contend with. This is due to interferences in the form of noise and harmonics. These unwanted signals pose problem to certain specified desired bands of frequencies. In many state-of-the-art equipment or systems, such as receivers, EEG and FDM etc., high quality factors and fast roll-off rate filter networks are used to select/reject or separate/combine signals at different frequencies [

The development of capacitor-less filter (R-filter) network has eliminated these bulky components (thereby reducing cost of production) and has also enhanced the stability of the filters. The building block of the R-filter is the internally compensated operational amplifiers (Op-Amps) [

Although, several papers have reported on the evaluation of the second-and third-order active R-filters [

The architecture that was used to implement both the sixth order active R-band pass filter and the sixth order active RC-filter is the multiple feed-back topologies. This topology was realized by cascading second order band pass filters of

for RC- and R-band pass filters respectively. All operational amplifiers (Op-Amps) are of the

or in terms of quality factor,

where

The quality factor,

From Equations (2) and (3), _{ }are related to the quality factor

The design parameters are that the filter should have a constant resonant frequency of 10 KHz at quality factors of 2, 5, 7, 8 and 10, respectively. First, we consider the design with a center frequency of 10 KHz and

Using Equation (4) and choosing

Similar procedure was used to determine the resistor values_{ }for high quality factor

Stage 1 of

or in terms of quality factor,

Quality Factor, Q | Designed Resistor Values (Ω) | Experimental Resistor Values (Ω) | ||||
---|---|---|---|---|---|---|

R_{A} | R_{B} | R_{2} | R_{A} | R_{B} | R_{2} | |

2 | 33.0 K | 398.13 | 6.37 K | 33.0 K | 390 | 6.2 K |

5 | 82.0 K | 159.41 | 15.91 K | 82.0 K | 160 | 15.0 K |

7 | 120.0 K | 113.78 | 22.28 K | 120.0 K | 120 | 22.0 K |

8 | 130.0 K | 99.53 | 25.46 K | 130.0 K | 100 | 22.0 K |

10 | 4.7 K | 80.94 | 31.83 K | 4.7 K | 82 | 33.0 K |

where

The second-order band pass R-filter network in

But from Equation (6a),

Therefore, from Equations (8a) and (8b) the gain,

For cascaded band-pass R-filter with multiple feedbacks, the gain can be expressed as:

From Equations (7) and (8), it can be seen that the resonance frequency is dependent on the voltage divider network formed by

First, we consider the design of second-order band pass R-filter (stage 1) with resonant frequency of 10 KHz, _{ }as 10 KΩ

Similar calculations for the component values were carried out using Equation (7) to (11) for higher

Quality Factor, Q | Designed Resistor Values (Ω) | Experimental Resistor Values (Ω) | ||||||
---|---|---|---|---|---|---|---|---|

R_{1} | R_{2} | R_{3} | R_{4} | R_{1} | R_{2} | R_{3} | R_{4} | |

2 | 97.7 K | 1.0 K | 10.00 K | 10.00 K | 97.6 K | 1.0 K | 10.00 K | 10.00 K |

5 | 97.7 K | 1.0 K | 10.00 K | 2.50 K | 97.6 K | 1.0 K | 10.00 K | 2.50 K |

7 | 93.1 K | 953.0 | 9.77 K | 1.63 K | 91.0 K | 953.0 | 9.76 K | 1.62 K |

8 | 93.1 K | 953.0 | 10.00 K | 1.43 K | 91.0 K | 953.0 | 10.00 K | 1.43 K |

10 | 93.1 K | 953.0 | 12.00 K | 1.33 K | 91.0 K | 953.0 | 12.00 K | 1.33 K |

Stage | R-Filter | ||
---|---|---|---|

Band-Width (Hz) | Mid-Band Gain (dB) | Roll-Off (dB/decade) | |

1 | 1300 | 18.68 | −19.63 |

2 | 750 | 37.72 | −39.26 |

3 | 600 | 56.78 | −58.91 |

(stage 1) to the third section (stage 3). The band-width was however found to decrease monotonically with each additional section. From the result of the roll-off presented in

Thus, for the three cascading stages (

The magnitude response of the cascaded R- and RC-band pass filters (for

Quality Factor, Q | R-Filter | RC-Filter | ||||
---|---|---|---|---|---|---|

Band-Width (Hz) | Mid-Band Gain (dB) | Roll-Off (dB/decade) | Band-Width (Hz) | Mid-Band Gain (dB) | Roll-Off (dB/decade) | |

2 | 2600 | 22.23 | −58.95 | 2300 | −55.49 | −106.21 |

5 | 1200 | 44.52 | −58.96 | 1250 | −55.70 | −122.89 |

7 | 700 | 53.36 | −58.91 | 950 | −57.70 | −130.72 |

8 | 600 | 56.78 | −58.91 | 900 | −56.70 | −132.29 |

10 | 400 | 62.97 | −58.90 | 450 | −50.81 | −132.99 |

remained fixed for the R-filter network (

We have successfully designed and compared the band-pass responses of higher-order R- and RC-filter net works and found that in addition to the advantages of miniaturization, ease of design and high frequency per- formance, the R-filter network provides better selectivity, greater stop band attenuation and steeper cut-off at the edge of the pass band especially at higher Q-values. Also, no relative shift of the centre frequency was observed with R-filter network unlike the RC-filter network which had a relative shift of its centre frequency with Q. The low roll-off rate recorded for the three cascading sections of the R-filter indicate a third order configuration ra- ther than the proposed sixth order. Nevertheless, the three cascaded network can be modified to provide higher roll-off when desired.