# Fractional-Order LCL Filters: Principle, Frequency Characteristics, and Their Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}and L

_{2}, and one capacitor, C, is a pivotal circuit extensively utilized in power electronic converters, such as in a grid-connected inverter, as shown in Figure 1a.

_{rp}denotes resonant frequency. When ω < ω

_{rp}and ω > ω

_{rp}, the asymptotic slopes of the logarithmic magnitude-frequency characteristic curve of an integer-order LCL filter are −20 dB/dec and −60 dB/dec, respectively [39]. The integer-order LCL filter plays a key role in filtering high-order harmonics in the AC side voltage and improving the quality of the current injected into the grid [37,38,39]. Compared with L filters, integer-order LCL filters can achieve a better high-frequency harmonic attenuation effect. However, the integer-order LCL filter has an inherent resonant peak, which will destroy the stability of the system when the system is not designed and controlled properly [37,38,39].

^{α}L filter [40]. Theoretical analysis and numerical simulations show that the LC

^{α}L filter provides a wider bandwidth to mitigate higher-order resonant frequencies than its integer-order counterpart. The circuit and mathematical model of a fractional-order LCL filter are preliminarily given in [41]. However, the detailed theoretical derivation and analysis of the frequency characteristics are missing.

- (1)
- This paper pioneered a method for the theoretical analysis of the frequency characteristics of fractional-order LCL filters, summarized their five critical properties, and systematically revealed their principles and frequency characteristics.
- (2)
- It is found that the necessary and sufficient condition for resonance in the magnitude-frequency characteristic curve of fractional-order LCL filters is that the sum of the orders of the fractional-order inductor and the fractional-order capacitor is equal to 2. This provides a theoretical basis for effectively avoiding resonance in fractional-order LCL filters.
- (3)
- This paper fills the gap in the research on the frequency characteristics of general fractional systems with $2\alpha +\beta $-order (where $\alpha ,\beta \in (0,2)$).

## 2. The Circuit and Mathematical Models of a Fractional-Order LCL Filter

## 3. The Frequency Characteristics and Analysis of a Fractional-Order LCL Filter

#### 3.1. The Resonance Characteristics of a Fractional-Order LCL Filter

**Property**

**1.**

**Property**

**2.**

#### 3.2. The Corner Frequency and Logarithmic Magnitude-Frequency Characteristics of a Fractional-Order LCL Filter

**Property**

**3.**

#### 3.3. The Phase-Frequency Characteristics of a Fractional-Order LCL Filter

**Property**

**4.**

#### 3.4. The Phase Crossover Frequency and Gain Margin of a Fractional-Order LCL Filter

**Property**

**5.**

#### 3.5. The Gain Crossover Frequency and Phase Margin of a Fractional-Order LCL Filter

## 4. Simulation Results and Analysis

#### 4.1. The Frequency Characteristic Simulation Curves of a Fractional-Order LCL Filter

#### 4.2. The Analysis of the Frequency Characteristics of a Fractional-Order LCL Filter

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Podlubny, I. Fractional Differential Equations; Academic: New York, NY, USA, 1999. [Google Scholar]
- Monje, C.A.; Chen, Y.; Vinagre, B.M.; Xue, D.; Feliu, V. Fractional-Order Systems and Controls: Fundamentals and Applications; Springer: London, UK, 2010. [Google Scholar]
- Petras, I. Fractional Calculus. In Fractional-Order Nonlinear Systems; Higher Education: Beijing, China, 2011. [Google Scholar]
- Radwan, A.G.; Soliman, A.M.; Elwakil, A.S. Design equations for fractional-order sinusoidal oscillators: Four practical circuit examples. Int. J. Circuit Theory Appl.
**2008**, 36, 473–492. [Google Scholar] [CrossRef] - Elwakil, A.S. Fractional-order circuits and systems: An emerging interdisciplinary research area. IEEE Circuits Syst. Mag.
**2010**, 10, 40–50. [Google Scholar] [CrossRef] - Radwan, A.G.; Salama, K.N. Fractional-order RC and RL circuits. Circuits Syst. Signal Process.
**2012**, 31, 1901–1915. [Google Scholar] [CrossRef] - Radwan, A.G.; Fouda, M.E. Optimization of Fractional-Order RLC Filters. Circuits Syst. Signal Process.
**2013**, 32, 2097–2118. [Google Scholar] [CrossRef] - Xu, J.; Li, X.; Meng, X.; Qin, J.; Liu, H. Modeling and analysis of a single-phase fractional-order voltage source pulse width modulation rectifier. J. Power Sources
**2020**, 479, 228821. [Google Scholar] [CrossRef] - Xu, J.; Li, X.; Liu, H.; Meng, X. Fractional-order modeling and analysis of a Three-phase Voltage Source PWM Rectifier. IEEE Access
**2020**, 8, 13507–13515. [Google Scholar] [CrossRef] - Freeborn, T.J.; Maundy, B.; Elwakil, A. Fractional resonance-based RL
_{β}C_{α}filters. Math. Probl. Eng.**2013**, 2013, 726721. [Google Scholar] [CrossRef] - Diao, L.; Zhang, X.; Chen, D. Fractional-order multiple RL alpha C beta circuit. Acta Phys. Sin.
**2014**, 63, 038401. [Google Scholar] [CrossRef] - Westerlund, S.; Ekstam, L. Capacitor theory. IEEE Trans. Dielectr. Elect. Insul.
**1994**, 1, 826–839. [Google Scholar] [CrossRef] - Westerlund, S. Dead Matter Has Memory; Causal Consulting: Kalmar, Sweden, 2002. [Google Scholar]
- Valsa, J.; Vlach, J. RC models of a constant phase element. Int. J. Circuit Theory Appl.
**2013**, 41, 59–67. [Google Scholar] [CrossRef] - Sarafraz, M.S.; Tavazoei, M.S. Passive realization of fractional-order impedances by a fractional element and RLC components: Conditions and procedure. IEEE Trans. Circuits Syst. I Reg. Pap.
**2017**, 64, 585–595. [Google Scholar] [CrossRef] - Semary, M.S.; Fouda, M.E.; Hassan, H.N.; Radwan, A.G. Realization of fractional-order capacitor based on passive symmetric network. J. Adv. Res.
**2019**, 18, 147–159. [Google Scholar] [CrossRef] - Tsirimokou, G.; Psychalinos, C.; Elwakil, A.S. Emulation of a constant phase element using operational transconductance amplifiers. Analog Integr. Circuits Signal Process.
**2015**, 85, 413–423. [Google Scholar] [CrossRef] - Bertsias, P.; Psychalinos, C.; Radwan, A.G.; Elwakil, A.S. High-Frequency Capacitorless Fractional-Order CPE and FI Emulator. Circuits Syst. Signal Process.
**2018**, 37, 2694–2713. [Google Scholar] [CrossRef] - Jiang, Y.; Zhang, B. High-Power Fractional-Order Capacitor With 1 < alpha < 2 Based on Power Converter. IEEE Trans. Ind. Electron.
**2018**, 65, 3157–3164. [Google Scholar] - Kapoulea, S.; Psychalinos, C.; Elwakil, A.S.; Radwan, A.G. One-terminal electronically controlled fractional-order capacitor and inductor emulator. AEU-Int. J. Electron. Commun.
**2019**, 103, 32–45. [Google Scholar] [CrossRef] - Jesus, I.S.; Machado, J.A.T. Development of fractional order capacitors based on electrolyte processes. Nonlinear Dyn.
**2009**, 56, 45–55. [Google Scholar] [CrossRef] - Krishna, M.S.; Das, S.; Biswas, K.; Goswami, B. Fabrication of a fractional order capacitor with desired specifications: A study on process identification and characterization. IEEE Trans. Electron Devices
**2011**, 58, 4067–4073. [Google Scholar] [CrossRef] - Haba, T.C.; Ablart, G.; Camps, T.; Olivie, F. Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos Solitons Fractals
**2005**, 24, 479–490. [Google Scholar] [CrossRef] - Mondal, D.; Biswas, K. Packaging of single-component fractional order element. IEEE Trans. Device Mater. Reliab.
**2013**, 13, 73–80. [Google Scholar] [CrossRef] - Agambayev, A.; Patole, S.P.; Farhat, M.; Elwakil, A.; Bagci, H.; Salama, K.N. Ferroelectric fractional-order capacitors. ChemElectroChem
**2017**, 4, 2807–2813. [Google Scholar] [CrossRef] - Agambayev, A.; Patole, S.; Bagci, H.; Salama, K.N. Tunable fractional-order capacitor using layered ferroelectric polymers. AIP Adv.
**2017**, 7, 095202. [Google Scholar] [CrossRef] - Freeborn, T.J.; Maundy, B.; Elwakil, A.S. Measurement of supercapacitor fractional-order model parameters from voltage-excited step response. IEEE J. Emerg. Sel. Top. Circuits Syst.
**2013**, 3, 367–376. [Google Scholar] [CrossRef] - Machado, J.A.T.; Galhano, A.M.S.F. Fractional order inductive phenomena based on the skin effect. Nonlinear Dyn.
**2012**, 68, 107–115. [Google Scholar] [CrossRef] - Tripathy, M.C.; Mondal, D.; Biswas, K.; Sen, S. Experimental studies on realization of fractional inductors and fractional-order bandpass filters. Int. J. Circuit Theory Appl.
**2015**, 43, 1183–1196. [Google Scholar] [CrossRef] - Adhikary, A.; Choudhary, S.; Sen, S. Optimal design for realizing a grounded fractional order inductor using GIC. IEEE Trans. Circuits Syst. I Reg. Pap.
**2018**, 65, 2411–2421. [Google Scholar] [CrossRef] - Wang, F.; Ma, X. Transfer function modeling and analysis of the open-loop buck converter using the fractional calculus. Chin. Phys. B
**2013**, 22, 030506. [Google Scholar] [CrossRef] - Wang, F.; Ma, X. Modeling and Analysis of the Fractional Order Buck Converter in DCM Operation by using Fractional Calculus and the Circuit-Averaging Technique. J. Power Electron.
**2013**, 13, 1008–1015. [Google Scholar] [CrossRef] - Wei, Z.; Zhang, B.; Jiang, Y. Analysis and Modeling of Fractional-Order Buck Converter Based on Riemann-Liouville Derivative. IEEE Access
**2019**, 7, 162768–162777. [Google Scholar] [CrossRef] - Tan, C.; Liang, Z. Modeling and simulation analysis of fractional-order Boost converter in pseudo-continuous conduction mode. Acta Phys. Sin.
**2014**, 63, 070502. [Google Scholar] - Yang, N.; Liu, C.; Wu, C. Modeling and dynamics analysis of the fractional-order Buck-Boost converter in continuous conduction mode. Chin. Phys. B
**2012**, 21, 080503. [Google Scholar] [CrossRef] - Chen, X.; Chen, Y.; Zhang, B.; Qiu, D. A Modeling and Analysis Method for Fractional-Order DC-DC Converters. IEEE Trans. Power Electron.
**2017**, 32, 7034–7044. [Google Scholar] [CrossRef] - Twining, E.; Holmes, D.G. Grid current regulation of a three-phase voltage source inverter with an LCL input filter. IEEE Trans. Power Electron.
**2003**, 18, 888–895. [Google Scholar] [CrossRef] - Wang, X.; Ruan, X.; Liu, S.; Tse, C.K. Full feedforward of grid voltage for grid-connected inverter with LCL filter to suppress current distortion due to grid voltage harmonics. IEEE Trans. Power Electron.
**2010**, 25, 3119–3127. [Google Scholar] [CrossRef] - Wu, W.; He, Y.; Tang, T.; Blaabjerg, F. A New Design Method for the Passive Damped LCL and LLCL Filter-Based Single-Phase Grid-Tied Inverter. IEEE Trans. Ind. Electron.
**2013**, 60, 4339–4350. [Google Scholar] [CrossRef] - El-Khazali, R. Fractional-Order LCαL Filter-Based Grid Connected PV Systems. In Proceedings of the 2019 IEEE 62nd International Midwest Symposium on Circuits and Systems (MWSCAS), Dallas, TX, USA, 4–7 August 2019; pp. 533–536. [Google Scholar]
- Sun, E. Modeling and Simulation of Fractional Active Power Filter. Master’s Thesis, School of Electrical Engineering, Dalian University of Technology, Dalian, China, 2017. (In Chinese with English abstract). [Google Scholar]
- Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the IEEE International Conference on Neural Networks, Perth, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948. [Google Scholar]
- Price, K.V.; Storn, R. Differential evolution: A simple evolution strategy for fast optimization. Dr. Dobb’s J.
**1997**, 22, 18–24. [Google Scholar]

**Figure 1.**The main circuit topology and Bode plot of an integer-order LCL filter. (

**a**) A grid-connected inverter with an integer-order LCL filter. (

**b**) The Bode plot of an integer-order LCL filter.

**Figure 2.**The main circuit topology and the model structure of a fractional-order LCL filter. (

**a**) The main circuit topology of a fractional-order LCL filter. (

**b**) The model structure of a fractional-order LCL filter.

**Figure 3.**The waveforms of $|\mathrm{cos}(\alpha +\beta )\pi /2{|}^{1/(\alpha +\beta )}$ and $|\mathrm{sin}(\alpha +\beta )\pi /2{|}^{1/(\alpha +\beta )}$ versus $\alpha +\beta $.

**Figure 4.**The change diagram of the digital solution of ${\omega}_{\mathrm{c}}$ obtained by the differential evolution algorithm versus $\alpha -\beta $ plane.

**Figure 5.**The frequency characteristic curves of a fractional-order LCL Filter when $\alpha $ is constant and $\beta $ changes. (

**a**) The frequency characteristics when $\alpha =0.8$. (

**b**) The frequency characteristics when $\alpha =1.0$. (

**c**) The frequency characteristics when $\alpha =1.2$.

**Figure 6.**The frequency characteristic curves of a fractional-order LCL Filter when $\beta $ is constant and $\alpha $ changes. (

**a**) The frequency characteristics when $\beta =0.8$. (

**b**) The frequency characteristics when $\beta =1.0$. (

**c**) The frequency characteristics when $\beta =1.2$.

**Table 1.**The frequency characteristic indicators of a fractional-order LCL when $\alpha $ is constant and $\beta $ changes.

Serial Number | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{\omega}}_{\mathbf{g}}$ (rad/s) | $20\mathbf{lg}{\mathit{K}}_{\mathbf{g}}$ (dB) | ${\mathit{\omega}}_{\mathbf{c}}$ (rad/s) | $\mathit{\gamma}$ (°) | ${\mathit{\omega}}_{\mathbf{t}}$ (rad/s) |
---|---|---|---|---|---|---|---|

1 | 0.8 | 0.6 | 5,257,083 | 53.39 | 8059 | 107.98 | 2,024,102 |

2 | 0.8 | 0.8 | 508,310 | 28.80 | 8075 | 107.93 | 329,599 |

3 | 0.8 | 1.0 | 98,864 | 9.06 | 8186 | 107.76 | 87,883 |

4 | 0.8 | 1.2 | 28,867 | −79.03 | 33,194 | −72.00 | 28,867 |

5 | 0.8 | 1.4 | NaN | Inf | 13,749 | 241.70 | 11,092 |

6 | 1.0 | 0.6 | 429,574 | 47.38 | 1334 | 90.00 | 329,598 |

7 | 1.0 | 0.8 | 92,922 | 27.10 | 1334 | 89.99 | 87,883 |

8 | 1.0 | 1.0 | 28,867 | −75.58 | 29,510 | 90.00 | 28,867 |

9 | 1.0 | 1.2 | NaN | Inf | 1345 | 90.16 | 11,092 |

10 | 1.0 | 1.4 | NaN | Inf | 1379 | 91.43 | 4772 |

11 | 1.2 | 0.6 | 87,883 | 45.96 | 402 | 71.99 | 87,882 |

12 | 1.2 | 0.8 | 28,867 | −57.73 | 28,950 | −108.0 | 28,867 |

13 | 1.2 | 1.0 | NaN | Inf | 402 | 72.01 | 11,092 |

14 | 1.2 | 1.2 | NaN | Inf | 402 | 72.07 | 4772 |

15 | 1.2 | 1.4 | NaN | Inf | 403 | 72.33 | 2487 |

**Table 2.**The frequency characteristic indicators of a fractional-order LCL when $\beta $ is constant and $\alpha $ changes.

Serial Number | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{\omega}}_{\mathbf{g}}$ (rad/s) | $20\mathbf{lg}{\mathit{K}}_{\mathbf{g}}$ (dB) | ${\mathit{\omega}}_{\mathbf{c}}$ (rad/s) | $\mathit{\gamma}$ (°) | ${\mathit{\omega}}_{\mathbf{t}}$ (rad/s) |
---|---|---|---|---|---|---|---|

1 | 0.6 | 0.8 | 19,437,846 | 50.51 | 16,538 | 124.85 | 2,024,630 |

2 | 0.8 | 0.8 | 508,310 | 28.80 | 8075 | 107.93 | 329,599 |

3 | 1.0 | 0.8 | 92,922 | 27.10 | 1334 | 89.99 | 87,883 |

4 | 1.2 | 0.8 | 28,867 | −57.73 | 28,950 | −108.0 | 28,867 |

5 | 1.4 | 0.8 | NaN | Inf | 171 | 54.00 | 11,092 |

6 | 0.6 | 1.0 | 686,670 | 13.13 | 368,962 | 56.77 | 329,598 |

7 | 0.8 | 1.0 | 98,864 | 9.06 | 8186 | 107.76 | 87,882 |

8 | 1.0 | 1.0 | 28,867 | −75.58 | 29,510 | −90.00 | 28,867 |

9 | 1.2 | 1.0 | NaN | Inf | 402 | 72.01 | 11,092 |

10 | 1.4 | 1.0 | NaN | Inf | 171 | 54.01 | 4772 |

11 | 0.6 | 1.2 | 107,916 | −7.69 | 133,810 | −19.98 | 87,883 |

12 | 0.8 | 1.2 | 28,867 | −93.42 | 33,192 | −72.00 | 28,867 |

13 | 1.0 | 1.2 | NaN | Inf | 1345 | 90.16 | 11,092 |

14 | 1.2 | 1.2 | NaN | Inf | 402 | 72.07 | 4771 |

15 | 1.4 | 1.2 | NaN | Inf | 170.7 | 54.04 | 2487 |

Properties | Integer-Order LCL Filters | Fractional-Order LCL Filters | Notes on Fractional-Order LCL Filters |
---|---|---|---|

Variables | Three Variables $({L}_{1},{L}_{2},C)$ | Five Variables $({L}_{1},{L}_{2},C,\alpha ,\beta )$ | $\alpha $ is the order of the fractional-order inductors, and $\beta $ is the order of the fractional-order capacitor. |

Range of $\alpha $ and $\beta $ | $\alpha =\beta =1$ | $\alpha ,\beta \in (0,2)$ | An integer-order LCL filter is the special case of a fractional-order LCL filter when $\alpha =\beta =1$. |

The transfer function, ${G}_{\mathrm{gi}}(s)$ | $\frac{1}{{L}_{1}{L}_{2}Cs}\frac{1}{{s}^{2}+A}$ | $\frac{1}{{L}_{1}{L}_{2}C{s}^{\alpha}}\frac{1}{{s}^{\alpha +\beta}+A}$ | |

Resonance peak | Exists a resonance peak | Exists a resonance peak when $\alpha +\beta =2$ | The necessary and sufficient condition for the existence of a resonance peak is $\alpha +\beta =2$. |

Resonant frequency, ${\omega}_{\mathrm{rp}}$ | $\sqrt{\left({L}_{1}+{L}_{2}\right)/\left({L}_{1}{L}_{2}C\right)}$ | $\sqrt{\left({L}_{1}+{L}_{2}\right)/\left({L}_{1}{L}_{2}C\right)}$ | ${\omega}_{\mathrm{rp}}$ is determined by the values of ${L}_{1}$, ${L}_{2}$, and $C$, and is independent of $\alpha $ and $\beta $. |

Corner frequency, ${\omega}_{\mathrm{t}}$ | $\sqrt{\left({L}_{1}+{L}_{2}\right)/\left({L}_{1}{L}_{2}C\right)}$ | $\begin{array}{l}\{\begin{array}{l}{\omega}_{\mathrm{t}1}=\sqrt[\alpha +\beta ]{\left|A\mathrm{cos}[\left(\alpha +\beta \right)\pi /2]\right|}\\ \alpha +\beta \in (0,0.5]\cup [1.5,2.5]\cup [3.5,4)\end{array}\\ \{\begin{array}{l}{\omega}_{\mathrm{t}2}=\sqrt[\alpha +\beta ]{\left|A\mathrm{sin}[\left(\alpha +\beta \right)\pi /2]\right|}\\ \alpha +\beta \in (0.5,1.5)\cup (2.5,3.5)\end{array}\end{array}$ | According to the different value range of $\alpha +\beta $, there are two calculation formulas of ${\omega}_{\mathrm{t}}$. ${\omega}_{\mathrm{t}}$ is affected by both $\alpha $ and $\beta $. |

Slope of the logarithmic magnitude-frequency characteristic, $\mathrm{d}L(\omega )/\mathrm{d}\mathrm{lg}\omega $ | $\{\begin{array}{l}-20\mathrm{dB}/\mathrm{dec},\omega {\omega}_{\mathrm{t}}\\ -60\mathrm{dB}/\mathrm{dec},\omega {\omega}_{\mathrm{t}}\end{array}$ | $\{\begin{array}{l}-20\alpha \mathrm{dB}/\mathrm{dec},\omega L{\omega}_{\mathrm{t}}\\ -20\left(2\alpha +\beta \right)\mathrm{dB}/\mathrm{dec},\omega {\omega}_{\mathrm{t}}\end{array}$ | The slope is only determined by $\alpha $ when $\omega <<{\omega}_{\mathrm{t}}$, while is affected by both $\alpha $ and $\beta $ when $\omega >>{\omega}_{\mathrm{t}}$. The range of slope is (0 dB/dec, −40 dB/dec) and (0 dB/dec, −120 dB/dec) when $\omega <<{\omega}_{\mathrm{t}}$ and $\omega >>{\omega}_{\mathrm{t}}$, respectively. |

Center frequency, ${\omega}_{\mathrm{o}}$ | $\sqrt{\left({L}_{1}+{L}_{2}\right)/\left({L}_{1}{L}_{2}C\right)}$ | $\sqrt[\alpha +\beta ]{\left({L}_{1}+{L}_{2}\right)/\left({L}_{1}{L}_{2}C\right)}$ | ${\omega}_{\mathrm{o}}$ is affected by both $\alpha $ and $\beta $. |

Phase-frequency characteristic, $\angle {G}_{\mathrm{gi}}(j\omega )$ | $\{\begin{array}{l}-\pi /2,\omega {\omega}_{\mathrm{o}}\\ -3\pi /2,\omega {\omega}_{\mathrm{o}}\end{array}$ | $\{\begin{array}{l}-\pi \alpha /2,\omega {\omega}_{\mathrm{o}}\\ -\pi \left(\alpha +\beta /2\right),\alpha +\beta \in (0,2],\omega {\omega}_{\mathrm{o}}\\ -\pi \left(\alpha +\beta /2\right)+2\pi ,\alpha +\beta \in (2,4),\omega {\omega}_{\mathrm{o}}\end{array}$ | $\angle {G}_{\mathrm{gi}}(j\omega )$ is only determined by $\alpha $ when $\omega <<{\omega}_{\mathrm{o}}$, while is affected by both $\alpha $ and $\beta $ when $\omega >>{\omega}_{\mathrm{o}}$. The high-frequency phase curve changes from $-\pi \alpha /2$ to the more lagging direction when $\alpha +\beta \in (0,2]$, and to the opposite direction when $\alpha +\beta \in (2,4)$. |

Phase crossover frequency, ${\omega}_{\mathrm{g}}$ | $\sqrt{\left({L}_{1}+{L}_{2}\right)/\left({L}_{1}{L}_{2}C\right)}$ | $\sqrt[\alpha +\beta ]{\frac{A\mathrm{sin}\left[\left(1-\alpha /2\right)\pi \right]}{-\mathrm{sin}\left[\left(\alpha +\beta /2\right)\pi \right]}}$ | If and only if $\alpha +\beta /2\in \left(1,2\right)$, the phase-frequency characteristics of a fractional-order LCL filter has a ${\omega}_{\mathrm{g}}$. |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, J.; Zeng, E.; Li, X.; He, G.; Liu, W.; Meng, X.
Fractional-Order LCL Filters: Principle, Frequency Characteristics, and Their Analysis. *Fractal Fract.* **2024**, *8*, 38.
https://doi.org/10.3390/fractalfract8010038

**AMA Style**

Xu J, Zeng E, Li X, He G, Liu W, Meng X.
Fractional-Order LCL Filters: Principle, Frequency Characteristics, and Their Analysis. *Fractal and Fractional*. 2024; 8(1):38.
https://doi.org/10.3390/fractalfract8010038

**Chicago/Turabian Style**

Xu, Junhua, Ermeng Zeng, Xiaocong Li, Guopeng He, Weixun Liu, and Xuanren Meng.
2024. "Fractional-Order LCL Filters: Principle, Frequency Characteristics, and Their Analysis" *Fractal and Fractional* 8, no. 1: 38.
https://doi.org/10.3390/fractalfract8010038