# Simplified Calculation Model for Contact Resistance Based on Fractal Rough Surfaces Method
## 1 Introduction  
Electrical contacts are widely used in the connection of various types of electrical equipment. The corresponding ECR is an important indicator to evaluate the stability and reliability of the electrical connection. In an in-service contact situation, any contact surface is not perfectly smooth but relatively rough at the microscopic scale, including processing texture, corrugation, and other geometric features. With two rough surfaces contacted, the real contact area is much smaller than the macroscopic contact area, and the violent current contraction occurs at several discrete contact spots. The shape and distribution of the micro-contact spots and the local shrinkage of the conductive path affect the current distribution, which in turn affects the ECR. Therefore, it is important to model the rough surface and predict discrete contact point areas in the calculation of ECR. The surface micromorphology modeling method based on numerical simulation has undergone a long period of research. The method uses specific roughness feature parameters with autocorrelation functions to form a Gaussian or non-Gaussian simulated rough surface. From the early Greenwood-Williamson model to the latter random theory of rough surfaces and the elliptical contact statistical model, the construction methods of rough surfaces cannot overcome the dependence on the detection accuracy of measuring instruments [1]. The essential characteristic parameters \(D\) and \(G\) for characterizing surface morphology were proposed by [PERSON] et al., which provided a new thought for subsequent morphological studies [2]. [PERSON] found that the parameter \(D\) is varied with a negative exponential function of the conventional roughness parameter \(R_{a}\)[3]. [PERSON] studied anisotropic surfaces based on fractal theory and found that the fractal parameters are sensitive to the direction of surface scratches [4]. [PERSON] pointed out that surface micromorphology has non-smooth and multi-scale characteristics. It is also explainedthat the microform of the part can be characterized by fractal dimensions with different parameters [5]. [PERSON] constructed 3D rough surfaces based on modified two-variable Weierstrass-Mandelbrot (W-M) fractal functions [6]. [PERSON] compared the advantages and disadvantages of different rough surface simulation methods, which designed a simulation method that effectively converted the autocorrelation function and nonlinear equation system [7], but cannot effectively simulate non-Gaussian surfaces. Rough surfaces are usually constructed to study contact properties and current transport properties. When two rough surfaces are contacted, the actual contact occurs through a portion of the asperities, that is, several discrete contact areas on the contact surface. The earliest model of rough surface contact was constructed by [PERSON] in 1957 [8], in which the real rough contacting surface is simplified into a multilayer hemisphere. It was found that the real contact area is proportional to the contact load. However, the model neglected the effect of the actual surface morphology, which led to difficulties in application and errors of contact resistance. Based on the classical Hertz contact theory, [PERSON] and [PERSON] established an elastic contact model [9], which lays the theoretical foundation for statistical surface contact properties. Some scholars have further extended the application of G-W models such as contact stiffness [10; 11], sliding contact [12], and contact damping [13; 14]. Although the elastic-plastic deformation and multi-scale of asperities are considered, it is not clear how the topography of the contact surface depends on the resolution of the measuring instrument. Another method is the contact model by fractal geometry theory. Based on the two-dimensional W-M function and the island area distribution law, [PERSON] and [PERSON] [2] built a classical M-B fractal contact model and revealed an analytical solution of the real contact area with the contact load.