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Although Carlson fractal-lattice fractance approximation circuit belongs to the ideal approximation, it can only have operational performance of fractional operator of negative half-order. When series of this circuit increases, the approximation benefit decreases. Even though the fractance approximation circuit of -1/2n (n is an integer greater than or equal to 2) order can be obtained by using nested structures, the structure of this kind of circuit is complicated and fractional operation of arbitrary order cannot be achieved by this circuit. The Liu-Kaplan fractal-chain fractance class, which can be regarded as scaling extension circuits of the Oldham fractal-chain fractance class, has high approximation benefit and can realize operational performance of arbitrary fractional order. Based on analogy, arbitrary order scaling fractal-lattice franctance approximation circuits of high approximation benefit and corresponding lattice type scaling equation can be achieved through respectively making scaling extension to the Carlson fractal-lattice franctance approximation circuit and its normalized iterating equation. There exists the possibility to verify the validity of this scaling extension and scaling fractal-lattice fractance approximation circuits with operational performance of arbitrary order in different ways, including the transmission parameter matrix algorithm, the iterating matrix algorithm and the coefficient vector iterating algorithm. Arbitrary order scaling fractal-lattice franctance approximation circuits can be realized by adjusting both the resistance progressive-ratio and the capacitance progressive-ratio parameters. The approximation benefit of scaling fractal-lattice franctance approximation circuit of arbitrary order is determined by both the scaling factor and the circuit series. The introduced extension benefit function is to be used in performance analyses. Besides, performance comparisons have been made between the Carlson fractal-lattice franctance approximation circuit of five series and the scaling fractal-lattice franctance approximation circuit of negative half-order. With the increasing of the value of the scaling factor, approximation efficiency of the scaling fractal-lattice franctance approximation circuits gradually increases, which are higher than those of the Carlson fractal-lattice franctance approximation circuits. The Carlson fractal-lattice franctance approximation circuit and the scaling fractal-lattice franctance approximation circuit of five series are designed to be used in the active differential operational circuit of half-order to construct experimental testing systems. The approximation performances of both circuits are investigated from the aspects of order-frequency characteristic and F-frequency characteristic. The approximation performance of the scaling fractal-lattice franctance approximation circuit outperforms that of the Carlson fractal-lattice franctance approximation circuit. As the successful application case, the active differential operational circuit designed by the scaling fractal-lattice franctance approximation circuit is used to do the half-order calculus of triangular and square wave signals. This paper is merely an incipient work on scaling fractal-lattice franctance approximation circuits of arbitrary order and irregular lattice type scaling equations.
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Keywords:
- fractional calculus /
- scaling extension /
- scale factor /
- fractal franctance approximation circuits
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[1] Yuan X 2015 Mathematical Principles of Fractance Approximation Circuits (Beijing: Science Press) pp3-15 (in Chinese) [袁晓 2015 分抗逼近电路之数学原理(北京:科学出版社) 第315页]
[2] He Q Y, Yuan X 2016 Acta Phys. Sin 65 160202 (in Chinese) [何秋燕, 袁晓 2016 65 160202]
[3] He Q Y, Yu B, Yuan X 2017 Chin. Phys. B 26 040202
[4] Shang D S, Chai Y S, Cao Z X, Lu J, Sun Y 2015 Chin. Phys. B 24 109
[5] Shen J X, Cong J Z, Chai Y S, Shang D S, Shen S P, Zhai K, Tian Y, Sun Y 2016 Phys. Rev. Appl. 6 021001
[6] Carlson G E 1960 M. S. Dissertation (Kansas State: Kansas State University)
[7] Pu Y F, Yuan X 2016 IEEE Access 4 1
[8] Yuan X, Feng G Y 2015 Proceedings of the 26th Academic Annual Conference of Circuits and Systems Branch, Chinese Institute of Electronics Chang Sha, China, October 23-26, 2015 p295 [袁晓, 冯国英 2015 中国电子学会电路与系统分会第二十六届学术年会论文集 中国长沙, 2015 年10月23日26日 第295页]
[9] Yuan Z, Yuan X 2017 Acta Electron. Sin. 45 2511 (in Chinese) [袁子, 袁晓 2017 电子学报 45 2511]
[10] Han Q, Liu C X, Sun L, Zhu D R 2013 Chin. Phys. B 22 020502
[11] Wang F Q, Ma X K 2013 Chin. Phys. B 22 030506
[12] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 110505]
[13] Yang J H, Zhu H 2013 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2013 62 024501]
[14] Zhou H W, Wang C P, Duan Z Q, Zhang M, Liu J F 2012 Sci. Sin.: Phys. Mech. Astron. 42 310 (in Chinese) [周宏伟, 王春萍, 段志强, 张淼, 刘建锋 2012 中国科学: 物理学 力学 天文学 42 310]
[15] Wu F, Liu J F, Bian Y, Zhou Z W 2014 J. Sichuan Univ. (Engineering Science Edition) 46 22 (in Chinese) [吴斐, 刘建锋, 边宇, 周志威 2014 四川大学学报 (工程科学版) 46 22]
[16] Yu Y J, Wang Z H 2015 Acta Phys. Sin. 64 238401 (in Chinese) [俞亚娟, 王在华 2015 64 238401]
[17] Li H H, Chen D Y, Zhang H, Wang F F, Ba D D 2016 Mech. Syst. Signal Process. 80 414
[18] Li H H, Chen D Y, Zhang H, Wu C Z, Wang X Y 2017 Appl. Energ. 185 244
[19] Xu B B, Wang F F, Chen D Y, Zhang H 2016 Energ. Convers. Manag. 108 478
[20] Tao L, Yuan X, Yi Z, Liu P P 2015 Sci. Tech. Eng. 15 81 (in Chinese) [陶磊, 袁晓, 易舟, 刘盼盼 2015 科学技术与工程 15 81]
[21] Liu P P , Yuan X, Tao L, Yi Z 2016 J. Sichuan Univ. (Nat. Sci. Ed.) 53 353 (in Chinese) [刘盼盼, 袁晓, 陶磊, 易舟 2016 四川大学学报 (自然科学版) 53 353]
[22] Yu B, Yuan X, Tao L 2015 J. Electr. Inf. Technol. 37 21 (in Chinese) [余波, 袁晓, 陶磊 2015 电子与信息学报 37 21]
[23] Yuan X, Chen X D, Li Q L, Zhang S P, Jiang Y D, Yu J B 2002 Acta Electron. Sin. 30 769 (in Chinese) [袁晓, 陈向东, 李齐良, 张蜀平, 蒋亚东, 虞厥邦 2002 电子学报 30 769]
[24] Yuan X, Zhang H Y, Yu J B 2004 Acta Electron. Sin. 32 1658 (in Chinese) [袁晓, 张红雨, 虞厥邦 2004 电子学报 32 1658]
[25] Zhao Y Y, Yuan X, Teng X D, Wei Y H 2004 J. Sichuan Univ. (Eng. Sci. Ed.) 36 94 (in Chinese) [赵元英, 袁晓, 滕旭东, 魏永豪 2004 四川大学学报(工程科学版) 36 94]
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