#
Validation of Fractional-Order Lowpass Elliptic Responses of (1 + α)-Order Analog Filters^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Approximated Lowpass Elliptic Response

#### 2.1. Coefficient Determination

#### 2.2. Symmetrical Fittings

#### 2.3. Asymmetrical Fittings

#### 2.4. Stability

## 3. Fitted Frequency Range Comparison

## 4. Circuit Simulations

## 5. Summary of Optimization Procedure

- Select desired second-order elliptic magnitude characteristics to approximate with fractional-order filter:
- (a)
- Passband ripple (5 dB in this work).
- (b)
- Stopband attenuation ($-50$ dB in this work).

- Select target frequency band to use for procedure fitting.
- Apply optimization solver to objective function to evaluate coefficients for target filter order (1 + $\alpha $).
- Evaluate stability of fractional-order filter with solved coefficients using W-plane transformations.
- Select appropriate circuit topology to realize fractional-order transfer function.
- Calculate necessary component values to realize target coefficients for desired center frequency.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Simulated magnitude responses of $(1+\alpha )$ lowpass fractional order transfer function for $\alpha =0.1$ to $0.9$ in steps of $0.1$ with coefficients selected to approximate elliptic response; and (

**b**) details of passband ripple.

**Figure 2.**Coefficients of fractional-order transfer function in Equation (7) to approximate second-order elliptic characteristics applying least squares fitting to three symmetrical frequency ranges.

**Figure 3.**DC and high-frequency gain of fractional-order transfer function using coefficients to approximate elliptic characteristics.

**Figure 4.**Coefficients of fractional-order transfer function in Equation (7) to approximate second-order elliptic characteristics applying least squares fitting to asymmetrical frequency ranges with: (

**a**) lower limits of $\omega ={10}^{-5}$ rad/s; and (

**b**) upper limits of $\omega ={10}^{5}$.

**Figure 5.**DC and high-frequency gain of fractional-order transfer function using coefficients from asymmetrical fittings using frequency ranges with: (

**a**) lower limits of $\omega ={10}^{-5}$ rad/s; and (

**b**) upper limits of $\omega ={10}^{5}$ to approximate elliptic characteristics.

**Figure 6.**Minimum phase angle of roots of Equation (10) using coefficients from symmetrical frequency fittings.

**Figure 7.**Minimum phase angle of roots of Equation (10) using coefficients from asymmetrical frequency fittings with: (

**a**) lower limits of $\omega ={10}^{-5}$ rad/s; and (

**b**) upper limits of $\omega ={10}^{5}$.

**Figure 8.**Simulated magnitude responses of (

**a**) $(1+0.75)$ and (

**b**) (1 + 0.25) lowpass fractional order transfer function using coefficients from different symmetrical frequency fitting ranges.

**Figure 9.**Simulated magnitude responses of $(1+0.25)$ lowpass fractional order transfer function using coefficients from different asymmetrical frequency fitting ranges with: (

**a**) lower limits of $\omega ={10}^{-5}$ rad/s; and (

**b**) upper limits of $\omega ={10}^{5}$.

**Figure 10.**Simulated magnitude responses of $(1+0.75)$ lowpass fractional order transfer function using coefficients from different asymmetrical frequency fitting ranges with: (

**a**) lower limits of $\omega ={10}^{-5}$ rad/s; and (

**b**) upper limits of $\omega ={10}^{5}$.

**Figure 11.**Circuit topologies to realize fractional-order low-pass notch filter response given by Equation (7) when ${a}_{3}$ is: (

**a**) positive; and (

**b**) negative. Note that in both topologies ${\mathrm{C}}_{1}$ is a fractional-order capacitor with impedance ${Z}_{{C}_{1}}=1/{s}^{\alpha}{C}_{1}$.

**Figure 12.**Foster-I circuit topology to realize a fifth order approximation of a fractional-order capacitor.

**Figure 13.**(

**a**) Experimental setup to measure magnitude responses of approximated $(1+\alpha )$ order filters; and (

**b**) breadboard implementation of topology using fifth order approximation of a fractional-order capacitor (outlined using the dashed box).

**Figure 14.**Theoretical (solid), SPICE simulated (dashed), and experimental (dashed-dotted) magnitude responses of (

**a**) $(1+\alpha )=1.8$ and (

**b**) $1.2$ order approximated elliptic filter responses.

**Figure 15.**Transient responses (dashed) of experimentally realized (

**a**) $(1+\alpha )=1.2$ and (

**b**) $1.8$ order approximated elliptic filter responses when applying a 800 Hz square wave input (solid).

$\mathit{\alpha}$ | ${\mathit{C}}_{1}$ (F s${}^{\mathit{\alpha}-1}$) | ${\mathit{C}}_{2}$ (nF) | ${\mathit{R}}_{1}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{2}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{3}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{4}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{5}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{6}$ ($\mathbf{\Omega}$) |
---|---|---|---|---|---|---|---|---|

$0.8$ | 62n | 10 | 1k | 680 | $90.8$k | $8.08$k | 492 | 562 |

$0.2$ | $46.9\mu $ | $6.8$ | $4.7$k | 1k | $3.17$k | 65k | 428 | $2.38$k |

**Table 2.**Component values to realize fractional-order capacitors with $\alpha =0.8$ and $0.2$ using fifth order Foster I topology centered at 10 kHz.

$\mathit{\alpha}$ | ${\mathit{R}}_{0}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{\mathit{a}}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{\mathit{b}}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{\mathit{c}}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{\mathit{d}}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{\mathit{e}}$ ($\mathbf{\Omega}$) | ${\mathit{C}}_{\mathit{a}}$ (nF) | ${\mathit{C}}_{\mathit{b}}$ (nF) | ${\mathit{C}}_{\mathit{c}}$ (nF) | ${\mathit{C}}_{\mathit{d}}$ (nF) | ${\mathit{C}}_{\mathit{e}}$ (nF) |
---|---|---|---|---|---|---|---|---|---|---|---|

$0.8$ | $58.8$ | $65.1$ | $326.4$ | $1.47$k | $7.06$k | $84.23$k | $12.8$ | $16.2$ | $22.6$ | $29.7$ | $15.7$ |

$0.2$ | $931.5$ | $374.9$ | $573.6$ | $837.2$ | $1.23$k | $1.93$k | $1.28$ | $5.29$ | $22.85$ | $98.5$ | $393.8$ |

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**MDPI and ACS Style**

Kubanek, D.; Freeborn, T.J.; Koton, J.; Dvorak, J. Validation of Fractional-Order Lowpass Elliptic Responses of (1 + *α*)-Order Analog Filters. *Appl. Sci.* **2018**, *8*, 2603.
https://doi.org/10.3390/app8122603

**AMA Style**

Kubanek D, Freeborn TJ, Koton J, Dvorak J. Validation of Fractional-Order Lowpass Elliptic Responses of (1 + *α*)-Order Analog Filters. *Applied Sciences*. 2018; 8(12):2603.
https://doi.org/10.3390/app8122603

**Chicago/Turabian Style**

Kubanek, David, Todd J. Freeborn, Jaroslav Koton, and Jan Dvorak. 2018. "Validation of Fractional-Order Lowpass Elliptic Responses of (1 + *α*)-Order Analog Filters" *Applied Sciences* 8, no. 12: 2603.
https://doi.org/10.3390/app8122603