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## Fractional-order multiple RLαCβ circuit

Diao Li-Jie, Zhang Xiao-Fei, Chen Di-Yi
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• #### 摘要

RLC并联电路是一种非常重要的单元电路，本文尝试着系统地分析和总结分数阶RLαCβ并联电路的基本特征和规律. 对比整数阶RLC并联电路，电感的分数阶阶次α和电容的分数阶阶次β作为两个新的关键参数，使得分数阶RLαCβ并联电路在设计上有了更多自由度、更大的柔性和新意. 同时，它们的引入也增加了许多新的现象和规律. 本文首先分析了分数阶RLαCβ并联电路的两个基本特性：导纳和相位. 进而分析了分数阶条件下分数阶RLαCβ并联电路所特有的纯虚阻抗的问题. 并且，分析了LC电路中特有的现象之一——谐振，和五个参数对谐振的影响. 进一步地，阻抗和相位关于各参数的敏感性分析也得到了详细地研究. 数值分析和理论分析两者互相印证，彼此支持.

#### Abstract

Since the RLC circuit is a basic circuit, attention is directed to the generalization of the fundamentals of fractional multiple RLαCβ circuit. Compared with the conventional multiple RLC circuit, the effects of fractional orders, α and β, is the key factor for extra freedom, more flexibility and novelty. First, we study the basic features including the admittance and phase. Then, the conditions for fractional-order multiple RLαCβ circuit to act as pure imaginary impedances are derived, which are unrealizable in the conventional case. As a peculiar phenomenon–resonance, the relationships among resonance frequency, fractional order and LC are studied in detail. In addition, sensitivity analysis including some interesting rules is illustrated. Finally, numerical simulations are carried out to validate the above studies.

#### Authors and contacts

###### 1. Department of Electrical Engineering, Northwest A & F University University, Yangling 712100, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 50879072), the National High Technology Research and Development Program of China (Grant No. 2011BAD29B08), the Fundamental Research Funds for the Central Universities of China (Grant No. Z109021310), and the National Innovative Experimental Project for College Students.

#### 参考文献

 [1] Pang X, Liu C X 2013 Acta Phys. Sin. 62 150504 (in Chinese) [庞霞, 刘崇新 2013 物理学报 62 150504] [2] Jia H Y, Chen Z Q, Xue W 2013 Acta Phys. Sin. 62 140503 (in Chinese) [贾红艳, 陈增强, 薛薇 2013 物理学报 62 140503] [3] Li X, Zhang Z D, Bi Q S 2013 Acta Phys. Sin. 62 220502 (in Chinese) [李旭, 张正娣, 毕勤胜 2013 物理学报 62 220502] [4] Chen D Y, Wu C, Iu H H C, Ma X Y 2013 Nonlinear Dynam. 73 1671 [5] Wang F Q, Ma X K 2013 Chin Phys. B 22 030506 [6] Zhang X, Bao B C, Wang J P, Ma Z H, Xu J P 2012 Acta Phys. Sin. 61 160503 (in Chinese) [张希, 包伯成, 王金平, 马正华, 许建平 2012 物理学报 61 160503] [7] Wu X L, Xiao G C, Lei B 2013 Acta Phys. Sin. 62 050503 (in Chinese) [吴旋律, 肖国春, 雷博 2013 物理学报 62 050503] [8] Liang Y, Yu D S, Chen H 2013 Acta Phys. Sin. 62 158501 (in Chinese) [梁燕, 于东升, 陈昊 2013 物理学报 62 158501] [9] Wen S P, Zeng Z G, Huang T W, Chen Y R 2013 Phys. lett. A 377 34 [10] Ahamed A I, Lakshmanan M 2013 Int. J. Bifurcat. Chaos 23 1350098 [11] Sheng Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 150503] [12] Li C P, Chen Y Q, Kurths J 2013 Philos. T. Roy. Soc. A 371 20130037 [13] Chen W, Zhang J J, Zhang J Y 2013 Fract. Calc. Appl. Anal. 16 76 [14] Li L X, Peng H P, Luo Q, Yang Y X, Liu Z 2013 Acta Phys. Sin. 62 020502 (in Chinese) [李丽香, 彭海朋, 罗群, 杨义先, 刘喆 2013 物理学报 62 020502] [15] Chen D Y, Zhang R F, Sprott J C, Chen H T, Ma X Y 2012 Chaos 22 023130 [16] Sakthivel R, Ganesh R, Ren Y, Anthoni S M 2013 Commun. Nonlinear Sci. 18 3498 [17] Huang J F, Liu R Y, Lai W C, Shin C W, Hsu C M 2012 Chin. Phys. B 21 084210 [18] Wang Z, Huang X, Li Y X, Song X N 2013 Chin. Phys. B 22 010504 [19] Nagahara M, Yamamoto Y 2013 IEEE T. Signal Proces. 61 4473 [20] Radwan A G, Salama K N 2012 Circ. Syst. Signal Pr. 31 1901 [21] Galvao R K H, Hadjiloucas S, Kienitz K H, Paiva H M 2013 IEEE T. Circuits-I 60 624

#### 施引文献

•  [1] Pang X, Liu C X 2013 Acta Phys. Sin. 62 150504 (in Chinese) [庞霞, 刘崇新 2013 物理学报 62 150504] [2] Jia H Y, Chen Z Q, Xue W 2013 Acta Phys. Sin. 62 140503 (in Chinese) [贾红艳, 陈增强, 薛薇 2013 物理学报 62 140503] [3] Li X, Zhang Z D, Bi Q S 2013 Acta Phys. Sin. 62 220502 (in Chinese) [李旭, 张正娣, 毕勤胜 2013 物理学报 62 220502] [4] Chen D Y, Wu C, Iu H H C, Ma X Y 2013 Nonlinear Dynam. 73 1671 [5] Wang F Q, Ma X K 2013 Chin Phys. B 22 030506 [6] Zhang X, Bao B C, Wang J P, Ma Z H, Xu J P 2012 Acta Phys. Sin. 61 160503 (in Chinese) [张希, 包伯成, 王金平, 马正华, 许建平 2012 物理学报 61 160503] [7] Wu X L, Xiao G C, Lei B 2013 Acta Phys. Sin. 62 050503 (in Chinese) [吴旋律, 肖国春, 雷博 2013 物理学报 62 050503] [8] Liang Y, Yu D S, Chen H 2013 Acta Phys. Sin. 62 158501 (in Chinese) [梁燕, 于东升, 陈昊 2013 物理学报 62 158501] [9] Wen S P, Zeng Z G, Huang T W, Chen Y R 2013 Phys. lett. A 377 34 [10] Ahamed A I, Lakshmanan M 2013 Int. J. Bifurcat. Chaos 23 1350098 [11] Sheng Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 150503 (in Chinese) [申永军, 杨绍普, 邢海军 2012 物理学报 61 150503] [12] Li C P, Chen Y Q, Kurths J 2013 Philos. T. Roy. Soc. A 371 20130037 [13] Chen W, Zhang J J, Zhang J Y 2013 Fract. Calc. Appl. Anal. 16 76 [14] Li L X, Peng H P, Luo Q, Yang Y X, Liu Z 2013 Acta Phys. Sin. 62 020502 (in Chinese) [李丽香, 彭海朋, 罗群, 杨义先, 刘喆 2013 物理学报 62 020502] [15] Chen D Y, Zhang R F, Sprott J C, Chen H T, Ma X Y 2012 Chaos 22 023130 [16] Sakthivel R, Ganesh R, Ren Y, Anthoni S M 2013 Commun. Nonlinear Sci. 18 3498 [17] Huang J F, Liu R Y, Lai W C, Shin C W, Hsu C M 2012 Chin. Phys. B 21 084210 [18] Wang Z, Huang X, Li Y X, Song X N 2013 Chin. Phys. B 22 010504 [19] Nagahara M, Yamamoto Y 2013 IEEE T. Signal Proces. 61 4473 [20] Radwan A G, Salama K N 2012 Circ. Syst. Signal Pr. 31 1901 [21] Galvao R K H, Hadjiloucas S, Kienitz K H, Paiva H M 2013 IEEE T. Circuits-I 60 624
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##### 出版历程
• 收稿日期:  2013-10-14
• 修回日期:  2013-10-21
• 刊出日期:  2014-02-05

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