Scaling fractal-lattice franctance approximation circuits of arbitrary order and irregular lattice type scaling equation
Authors: Yu Bo  , He Qiu-Yan  , Yuan Xiao  ,   College of Physics and Engineering, Chengdu Normal University, Chengdu 611130, China;,   College of Electronics and Information Engineering, Sichuan University, Chengdu 610064, China
Published: December 2020

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.