#
Minimum MOS Transistor Count Fractional-Order Voltage-Mode and Current-Mode Filters^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Fractional-Order Filters

## 3. Proposed Implementations

#### 3.1. Voltage-Mode Filters

#### 3.2. Current-Mode Filters

## 4. Simulation and Comparison Results

#### 4.1. Voltage-Mode Filters

#### 4.2. Current-Mode Filters

#### 4.3. Comparison Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MOS | Metal-Oxide-Semiconductor |

CMOS | Complementary Metal-Oxide-Semiconductor |

IC | Integrated Circuit |

RC | Resistor Capacitor |

OP-AMP | Operational Amplifier |

OTA | Operational Transconductance Amplifier |

CCII | Second-generation Current Conveyor |

CFOA | Current Feedback Operational Amplifier |

CM | Current-Mirrors |

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**Figure 1.**Realization of voltage-mode fractional-order (

**a**) low-pass, (

**b**) high-pass, and (

**c**) band-pass filters using OTAs as active elements.

**Figure 2.**Proposed voltage-mode fractional-order (

**a**) low-pass and high-pass, and (

**b**) band-pass filter topologies.

**Figure 3.**Realization of current-mode fractional-order (

**a**) low-pass, (

**b**) high-pass, and (

**c**) band-pass filters using OTAs as active elements.

**Figure 4.**Proposed current-mode fractional-order (

**a**) low-pass, (

**b**) high-pass, and (

**c**) band-pass filters.

**Figure 5.**RC network for approximating the fractional-order capacitors in the proposed fractional-order filter topologies.

**Figure 6.**Simulated frequency responses of the proposed voltage-mode low-pass and high-pass filters in Figure 2a (

**a**) gain, and (

**b**) phase.

**Figure 7.**Demonstration of the electronic tuning capability of the filter structure in Figure 2a.

**Figure 8.**Statistical plots about the sensitivity of the half-power frequency in the case of the voltage-mode (

**a**) low-pass, and (

**b**) high-pass filters of order $\alpha =0.5$.

**Figure 9.**Frequency responses of the band-pass filter in Figure 2b for various orders.

**Figure 11.**Demonstration of the electronic tuning capability of the current-mode low-pass and high-pass filters.

**Figure 12.**Statistical plots about the sensitivity of the half-power frequency in the case of the current-mode (

**a**) low-pass, and (

**b**) high-pass filters of order $\alpha =0.5$.

**Table 1.**Values of passive elements of the network in Figure 5.

Element | $85.4\phantom{\rule{3.33333pt}{0ex}}\mathbf{nF}/{\mathbf{sec}}^{0.7}$ | $37.4\phantom{\rule{3.33333pt}{0ex}}\mathbf{nF}/{\mathbf{sec}}^{0.5}$ | $16.3\phantom{\rule{3.33333pt}{0ex}}\mathbf{nF}/{\mathbf{sec}}^{0.3}$ |
---|---|---|---|

${R}_{p}$ | 12 M$\mathsf{\Omega}$ | 30 M$\mathsf{\Omega}$ | 90 M$\mathsf{\Omega}$ |

${R}_{1}$ | 1.6 M$\mathsf{\Omega}$ | 457.7 k$\mathsf{\Omega}$ | 133.1 k$\mathsf{\Omega}$ |

${R}_{2}$ | 6.4 M$\mathsf{\Omega}$ | 3.8 M$\mathsf{\Omega}$ | 3.2 M$\mathsf{\Omega}$ |

${R}_{3}$ | 10.6 M$\mathsf{\Omega}$ | 8.9 M$\mathsf{\Omega}$ | 10.8 M$\mathsf{\Omega}$ |

${R}_{4}$ | 11 M$\mathsf{\Omega}$ | 13.4 M$\mathsf{\Omega}$ | 23.1 M$\mathsf{\Omega}$ |

${C}_{1}$ | 472 pF | 1.1 nF | 2.1 nF |

${C}_{2}$ | 972.3 pF | 1.4 nF | 1.4 nF |

${C}_{3}$ | 2.4 nF | 2.5 nF | 1.8 nF |

${C}_{4}$ | 13.5 nF | 9 nF | 4.3 nF |

**Table 2.**Frequency characteristics of the low-pass filter in Figure 2a.

Parameter | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ |
---|---|---|---|

${f}_{h,lp}$ (Hz) | 0.76 (0.7) | 2.6 (2.7) | 5.2 (5.3) |

phase @ ${f}_{h,lp}$ (deg) | −8.5 (−8.3) | −14.8 (−15) | −24 (−23.9) |

slope (dB/Oct.) | −1.6 (−1.8) | −2.8 (−3) | −4 (−4.2) |

**Table 3.**Frequency characteristics of the high-pass filter in Figure 2a.

Parameter | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ |
---|---|---|---|

${f}_{h,hp}$ (Hz) | 142 (145) | 38 (37.3) | 18 (18.7) |

phase @ ${f}_{h,hp}$ (deg) | −172 (−172) | −165 (−165) | −156 (−156) |

slope (dB/Oct.) | 1.5 (1.8) | 2.7 (3) | 3.9 (4.2) |

**Table 4.**Frequency characteristics of the band-pass filter in Figure 2b.

Parameter | $(0.5,\phantom{\rule{3.33333pt}{0ex}}0.7)$ | $(0.5,\phantom{\rule{3.33333pt}{0ex}}0.5)$ | $(1,\phantom{\rule{3.33333pt}{0ex}}0.5)$ |
---|---|---|---|

${f}_{peak}$ (Hz) | 6.6 (6.6) | 10 (10) | 16 (16.2) |

$gain@{f}_{peak}$ (dB) | −9.8 (−9.8) | −10.7 (−10.7) | −7.9 (−7.8) |

${f}_{low}$ (Hz) | 1 (1) | 1 (1) | 5 (5.2) |

${f}_{high}$ (Hz) | 36.3 (37) | 93 (96) | 72 (70) |

**Table 5.**Frequency characteristics of the low-pass filter in Figure 4a.

Parameter | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ |
---|---|---|---|

${f}_{h,lp}$ (Hz) | 0.76 (0.7) | 2.6 (2.7) | 5.2 (5.3) |

phase @ ${f}_{h,lp}$ (deg) | −8.5 (−8.3) | −14.8 (−15) | −24 (−23.9) |

slope (dB/Oct.) | −1.6 (−1.8) | −2.8 (−3) | −4 (−4.2) |

**Table 6.**Frequency characteristics of the high-pass filter in Figure 4b.

Parameter | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.5$ | $\mathit{\alpha}=0.7$ |
---|---|---|---|

${f}_{h,hp}$ (Hz) | 143 (145) | 36.4 (37.3) | 18.2 (18.7) |

phase @ ${f}_{h,hp}$ (deg) | 8 (8.3) | 14.7 (15) | 23.9 (23.9) |

slope (dB/Oct.) | 1.7 (1.8) | 2.9 (3) | 4.1 (4.2) |

Parameter | $(0.5,\phantom{\rule{3.33333pt}{0ex}}0.7)$ | $(0.5,\phantom{\rule{3.33333pt}{0ex}}0.5)$ | $(1,\phantom{\rule{3.33333pt}{0ex}}0.5)$ |
---|---|---|---|

${f}_{peak}$ (Hz) | 9.2 (9.4) | 9.8 (10) | 9 (9.2) |

$gain@{f}_{peak}$ (dB) | −12.7 (−12.7) | −13.7 (−13.7) | −10.9 (−10.8) |

${f}_{low}$ (Hz) | 1.7 (1.9) | 1.4 (1.5) | 4 (4.1) |

${f}_{high}$ (Hz) | 33 (35.5) | 62 (66) | 29.3 (31.6) |

**Table 8.**Comparison between the MOS transistor count and dc power dissipation for the proposed voltage-mode filters.

Implementation | Number of Transistors | Power Dissipation | ||||
---|---|---|---|---|---|---|

Low-Pass | High-Pass | Band-Pass | Low-Pass | High-Pass | Band-Pass | |

OTA-C | 18 | 36 | 27 | $8{V}_{DD}{I}_{0}$ | $16{V}_{DD}{I}_{0}$ | $12{V}_{DD}{I}_{0}$ |

Proposed | 3 | 3 | 3 | $2{V}_{DD}{I}_{0}$ | $2{V}_{DD}{I}_{0}$ | $2{V}_{DD}{I}_{0}$ |

**Table 9.**Comparison between the MOS transistor count and dc power dissipation for the proposed current-mode filters.

Implementation | Number of Transistors | Power Dissipation | ||||
---|---|---|---|---|---|---|

Low-Pass | High-Pass | Band-Pass | Low-Pass | High-Pass | Band-Pass | |

OTA-C | 18 | 18 | 27 | $8{V}_{DD}{I}_{0}$ | $8{V}_{DD}{I}_{0}$ | $12{V}_{DD}{I}_{0}$ |

Proposed | 7 | 7 | 7 | $6{V}_{DD}{I}_{0}$ | $6{V}_{DD}{I}_{0}$ | $6{V}_{DD}{I}_{0}$ |

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**MDPI and ACS Style**

Bertsias, P.; Psychalinos, C.; Elwakil, A.S.; Maundy, B.
Minimum MOS Transistor Count Fractional-Order Voltage-Mode and Current-Mode Filters. *Technologies* **2019**, *7*, 85.
https://doi.org/10.3390/technologies7040085

**AMA Style**

Bertsias P, Psychalinos C, Elwakil AS, Maundy B.
Minimum MOS Transistor Count Fractional-Order Voltage-Mode and Current-Mode Filters. *Technologies*. 2019; 7(4):85.
https://doi.org/10.3390/technologies7040085

**Chicago/Turabian Style**

Bertsias, Panagiotis, Costas Psychalinos, Ahmed S. Elwakil, and Brent Maundy.
2019. "Minimum MOS Transistor Count Fractional-Order Voltage-Mode and Current-Mode Filters" *Technologies* 7, no. 4: 85.
https://doi.org/10.3390/technologies7040085