任意阶标度分形格分抗与非正则格型标度方程

Scaling fractal-lattice franctance approximation circuits of arbitrary order and irregular lattice type scaling equation

Carlson分形格电路是分抗的理想逼近情形,但仅具有负半阶运算性能,逼近效益随着电路节次数的增加逐渐降低.虽然可嵌套得到-1/2~n阶(n为大于或等于2的整数)分抗逼近电路,但结构复杂,无法实现任意分数阶运算.通过类比拓展Carlson分形格电路,获得具有高逼近效益的任意实数阶微积算子的分抗逼近电路——标度分形格分抗,并用非正则格型标度方程进行数学描述.分别探讨非正则格型标度方程的近似求解和真实解.通过调节电阻递进比α与电容递进比β的取值,可构造出具有任意运算阶的标度分形格分抗逼近电路.标度拓展极大地提高了标度分形格分抗电路的逼近效益.随着标度因子的增加,负半阶标度分形格分抗的逼近效益逐渐增大并明显高于Carlson分形格分抗.设计了基于五节Carlson分形格分抗与负半阶标度分形格分抗的半阶微分运算电路,并对周期三角波和周期方波信号进行半阶微分运算,实验测试结果与理论分析一致.

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 approxi- mation 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 ap- proximation 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.