# Implementation of a Fractional-Order Electronically Reconfigurable Lung Impedance Emulator of the Human Respiratory Tree

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{1−$\alpha $}. In the case that $-1<\alpha <0$ the expression in (1) represents a fractional-order inductor with $k\equiv {L}_{\alpha}$ being the pseudo-inductance in Henry/sec

^{1−$\alpha $}. When $\alpha =0,-1,1$, the element is a resistor, capacitor, and inductor, respectively.

^{0.05}to 238 nFarad/s

^{0.42}). Also, the variation in the dispersion coefficient from 0.6 to nearly double the value at 1.1 indicates the importance of re-configurability of any proposed circuit model in order to reflect the measured values of any particular individual, which is the focus of this work.

## 2. Emulation of Fractional-Order Capacitors

## 3. Circuit Implementation

^{th}- integration stage.

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AMS | Austria Mikro Systeme |

CFE | Continued Fraction Expansion |

CMOS | Complementary Metal-Oxide-Semiconductor |

CPE | Constant Phase Element |

FBD | Functional Block Diagram |

IC | Integrated Circuits |

MOS | Metal-Oxide-Semiconductor |

OTA | Operational Transconductance Amplifier |

PFE | Partial Fraction Expansion |

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**Figure 1.**Fractional-order electrical equivalent of the impedance of the lung in the human respiratory tree [16].

**Figure 2.**Functional block diagram for emulating a fractional-order capacitor of order $\alpha =(0,1)$ [30].

**Figure 3.**Functional block diagram for emulating a fractional-order capacitor of order $\alpha =(0,2)$ [31].

**Figure 4.**FBD of an Inverse-Follow-the-Leader-Feedback topology for approximating a fractional-order differentiator using the Continued Fraction Expansion tool [30].

**Figure 5.**FBD for approximating a fractional-order differentiator using the Partial Fraction Expansion tool [34].

**Figure 6.**Implementation of the fractional-order differentiation stage, required for the implementation of the functional block diagram in Figure 3.

**Figure 8.**Implementation of the $V/I$ converter stage using an appropriately configured multiple- output OTA.

**Figure 9.**Implementation of a floating resistor using an appropriately configured multiple-output OTA.

**Figure 11.**Layout design of the proposed emulator of the lung impedance model of the human respiratory tree in Figure 1 (fractional-order differentiator in red polygon, integer-order differentiator in green polygon, $V/I$ converter in yellow polygon, and active resistor in blue polygon).

**Figure 14.**Statistical histograms about the sensitivity of the (

**a**) magnitude, and (

**b**) phase of the lung impedance, at the center frequency of the approximation (${f}_{0}$ = 2.5 Hz).

**Table 1.**Values of the elements of the fractional-order electrical equivalent of the impedance of the lung in the human respiratory tree [16].

Case | $\mathit{R}\phantom{\rule{0.166667em}{0ex}}(\mathbf{M}\mathsf{\Omega})$ | ${\mathit{C}}_{\mathit{\alpha}}$ (nF/s${}^{1-\mathit{\alpha}}$) | Order of Element $(\mathit{\alpha})$ |
---|---|---|---|

#1 | 6.44 | 238 | 0.5808 |

#2 | 4.292 | 90.4 | 0.7978 |

#3 | 6.02 | 135 | 0.802 |

#4 | 9.54 | 206 | 0.81 |

#5 | 4.89 | 178 | 0.9349 |

#6 | 5.93 | 17.5 | 0.95 |

#7 | 6.06 | 26 | 1.134 |

**Table 2.**Values of the elements of the fractional-order emulator in Figure 3.

Case | ${\mathit{C}}_{\mathit{\alpha}}$ (nF/s${}^{1-\mathit{\alpha}}$) | Order of Element $(\mathit{\alpha})$ | Order of Differentiator $(\mathit{q})$ | ${\mathit{g}}_{\mathit{m},\mathbf{VI}}$ ($\mathsf{\mu}$S) |
---|---|---|---|---|

#1 | 238 | 0.5808 | 0.5808 | 1.18 |

#2 | 90.4 | 0.7978 | 0.7978 | 0.81 |

#3 | 135 | 0.802 | 0.802 | 1.23 |

#4 | 206 | 0.81 | 0.81 | 1.92 |

#5 | 178 | 0.9349 | 0.9349 | 2.34 |

#6 | 17.5 | 0.95 | 0.95 | 0.24 |

#7 | 26 | 1.134 | 0.134 | 0.59 |

Case | Order of Differentiator $(\mathit{q})$ | ${\mathit{K}}_{0}$ | ${\mathit{K}}_{1}$ | ${\mathit{K}}_{2}$ | ${\mathit{\tau}}_{1}$(ms) | ${\mathit{\tau}}_{2}$(ms) |
---|---|---|---|---|---|---|

#1 | 0.5808 | 0.146 | 1 | 6.86 | 5.17 | 114.3 |

#2 | 0.7978 | 0.048 | 1 | 20.69 | 2.30 | 85.14 |

#3 | 0.802 | 0.047 | 1 | 21.29 | 2.25 | 84.65 |

#4 | 0.81 | 0.044 | 1 | 22.5 | 2.15 | 83.71 |

#5 | 0.9349 | 0.012 | 1 | 81.9 | 0.71 | 70.09 |

#6 | 0.95 | 0.009 | 1 | 109.6 | 0.54 | 68.56 |

#7 | 0.134 | 0.668 | 1 | 1.5 | 12.9 | 209.5 |

Case | Order of Differentiator $(\mathit{q})$ | ${\mathit{K}}_{0}$ | ${\mathit{K}}_{1}$ | ${\mathit{K}}_{2}$ | ${\mathit{\tau}}_{1}$(ms) | ${\mathit{\tau}}_{2}$(ms) |
---|---|---|---|---|---|---|

#1 | 0.5808 | 6.86 | −6.12 | −0.59 | 5.43 | 108.9 |

#2 | 0.7978 | 20.69 | −20.24 | −0.4 | 2.37 | 82.78 |

#3 | 0.802 | 21.29 | −20.85 | −0.39 | 2.31 | 82.34 |

#4 | 0.81 | 22.5 | −22.07 | −0.38 | 2.21 | 81.50 |

#5 | 0.9349 | 81.9 | −81.73 | −0.16 | 0.71 | 69.37 |

#6 | 0.95 | 109.6 | −109.4 | −0.12 | 0.54 | 68.02 |

#7 | 0.134 | 1.5 | −0.51 | −0.32 | 13.8 | 195.7 |

Case | Differentiator | $\mathit{V}/\mathit{I}$ Converter | Resistor | |||
---|---|---|---|---|---|---|

Order $(\mathit{q})$ | ${\mathit{I}}_{\mathit{B}1}$(pA) | ${\mathit{I}}_{\mathit{B}2}$(pA) | ${\mathit{I}}_{\mathit{B}}$(pA) | ${\mathit{I}}_{\mathit{B},\mathbf{VI}}$(mA) | ${\mathit{I}}_{\mathit{B},\mathit{R}}$(mA) | |

#1 | 0.5808 | 51.7 | 36.1 | 350 | 66 | 8.7 |

#2 | 0.7978 | 118.5 | 47.5 | 350 | 45.6 | 13.1 |

#3 | 0.802 | 121.6 | 47.7 | 350 | 69.1 | 9.3 |

#4 | 0.81 | 127.1 | 48.2 | 300 | 107.8 | 5.9 |

#5 | 0.9349 | 395.5 | 56.7 | 700 | 131.3 | 11.5 |

#6 | 0.95 | 520 | 57.8 | 600 | 13.5 | 9.5 |

#7 | 0.134 | 20.3 | 20.1 | 600 | 33.2 | 9.3 |

Transistor | Figure 6 | Figure 7 | Figure 8 | Figure 9 |
---|---|---|---|---|

($\mathsf{\mu}$m/$\mathsf{\mu}$m) | ($\mathsf{\mu}$m/$\mathsf{\mu}$m) | ($\mathsf{\mu}$m/$\mathsf{\mu}$m) | ($\mathsf{\mu}$m/$\mathsf{\mu}$m) | |

M1n-M2n | 8/15 | 1/5 | 1/1 | 2/1 |

M3n-M4n | 40/15 | 5/5 | 5/1 | 10/1 |

M5n-M7n | 25/15 | 25/5 | 15/15 | 10/5 |

M1p-M2p | 10/5 | 0.5/5 | 25/10 | 5/15 |

Case | Order $(\mathit{q})$ | Differentiator | $\mathit{V}/\mathit{I}$ Converter | Resistor | |
---|---|---|---|---|---|

$\mathbf{Gain}$ | $\mathit{arg}(\mathit{Z})\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | $1/{\mathit{g}}_{\mathit{m},\mathbf{VI}}\phantom{\rule{0.166667em}{0ex}}($M$\mathsf{\Omega})$ | $\mathit{R}\phantom{\rule{0.166667em}{0ex}}($M$\mathsf{\Omega})$ | ||

#1 | 0.5808 | 1 (1) | 53.7 (52.3) | 0.85 (0.84) | 6.45 (6.44) |

#2 | 0.7978 | 1 (1) | 71.8 (71.8) | 1.22 (1.23) | 4.27 (4.29) |

#3 | 0.802 | 1 (1) | 71.7 (72.18) | 0.81 (0.81) | 6.01 (6.02) |

#4 | 0.81 | 1 (1) | 72.7 (72.9) | 0.51 (0.52) | 9.54 (9.54) |

#5 | 0.9349 | 1 (1) | 83.7 (84.14) | 0.43 (0.43) | 4.89 (4.89) |

#6 | 0.95 | 1 (1) | 84.2 (85.5) | 0.41 (0.42) | 5.92 (5.93) |

#7 | 0.134 | 1 (1) | 12.5 (11.5) | 1.69 (1.69) | 6.04 (6.06) |

**Table 8.**Simulated and theoretical (in parentheses) values of impedance magnitude and phase of the model in Figure 1, at the center frequency of the approximation (${f}_{0}$ = 2.5 Hz).

Case | Order of Element $(\mathit{\alpha})$ | $\mid \mathit{Z}\mid \phantom{\rule{0.166667em}{0ex}}(\mathit{M}\mathsf{\Omega})$ | $\mathit{arg}(\mathit{Z})\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ |
---|---|---|---|

#1 | 0.5808 | 6.9 (6.9) | −5.64 (−6.13) |

#2 | 0.7978 | 4.70 (4.71) | −13.12 (−14.7) |

#3 | 0.802 | 6.25 (6.24) | −6.26 (−7.3) |

#4 | 0.81 | 9.68 (9.66) | −2.68 (−3) |

#5 | 0.9349 | 4.96 (4.94) | −5.35 (−4.96) |

#6 | 0.95 | 7.83 (7.42) | −32.5 (−34) |

#7 | 1.134 | 5.92 (5.96) | −17.1 (−16.2) |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Kaskouta, E.; Kapoulea, S.; Psychalinos, C.; Elwakil, A.S.
Implementation of a Fractional-Order Electronically Reconfigurable Lung Impedance Emulator of the Human Respiratory Tree. *J. Low Power Electron. Appl.* **2020**, *10*, 18.
https://doi.org/10.3390/jlpea10020018

**AMA Style**

Kaskouta E, Kapoulea S, Psychalinos C, Elwakil AS.
Implementation of a Fractional-Order Electronically Reconfigurable Lung Impedance Emulator of the Human Respiratory Tree. *Journal of Low Power Electronics and Applications*. 2020; 10(2):18.
https://doi.org/10.3390/jlpea10020018

**Chicago/Turabian Style**

Kaskouta, Elpida, Stavroula Kapoulea, Costas Psychalinos, and Ahmed S. Elwakil.
2020. "Implementation of a Fractional-Order Electronically Reconfigurable Lung Impedance Emulator of the Human Respiratory Tree" *Journal of Low Power Electronics and Applications* 10, no. 2: 18.
https://doi.org/10.3390/jlpea10020018