Fractal dimensionality and scale invariance in an AC electric circuits and transmission lines

Abstract Fractal geometry is fascinating due to its ability to describe the complex geometries naturally occurring in real world. The fractal or Hausdorff dimension d is used to describe the laws of scale invariance, which are exemplified by functions satisfying a mathematical law of the form f ( s x ) = s d f ( x ), being s the scale factor. Closely related is the problem of determining the fixed point of physical systems. Mathematically, the fixed point corresponds to a point in the domain of a function which is mapped to itself. In dynamical systems, it typically corresponds to the value to which the response of the system converge and stabilize. The present contribution aims to present the general concepts of scale invariance and fixed or critical points, employing such aspects to the problem of complex impedance association, both in series and in ladder configuration. The complex impedance, and transfer function of transmission lines and flow graphs related to the convergence the iterative equations to a fixed points are discussed in more details.

摘要：分形几何（Fractal geometry）因其能够刻画现实世界中自然存在的复杂几何形态而极具魅力。分形维数或豪斯多夫维数（Hausdorff dimension）$d$用于描述标度不变性（scale invariance）的规律，这类规律可由满足形如$f(sx) = s^d f(x)$的数学法则的函数加以例证，其中$s$为缩放因子（scale factor）。与之密切相关的是物理系统不动点（fixed point）的求解问题。从数学层面而言，不动点指的是函数定义域内被映射至自身的点。在动力学系统（dynamical systems）中，不动点通常对应系统响应收敛并稳定下来的数值。本研究旨在阐述标度不变性以及不动点或临界点的核心概念，并将这些概念应用于串联与梯形两种结构下的复阻抗（complex impedance）关联问题。本文还将进一步详细讨论复阻抗、传输线（transmission lines）的传递函数（transfer function），以及与迭代方程（iterative equations）收敛至不动点相关的流图（flow graphs）。