# An Analysis of the Performance of Trolleybus Brushes Developed from Recycled Materials
## 1 Introduction  
Electro-mobility of public transport in cities has become an important issue in recent years, since passenger and freight transport consume more than a quarter of global primary energy and is responsible for significant greenhouse emissions [1; 2]. Trolleybuses are commonly perceived as a valuable alternative to conventional buses in sustainable urban transportation systems [3; 4; 5; 6]. However, the very presence of sliding current-carrying contact zone causes non-trivial problems. The traction brushes made of a mixture of graphite and copper filings have to be replaced on a weekly basis under standard operating conditions, because of their wear [7]. The issue of how to assure a proper electric contact has been the subject of numerous research works and monographs [8; 9; 10; 11]. It should be stressed that the problem is extremely complex, taking into account the roughness of the contacting surfaces, random occurrence of contact spots, elastic and plastic deformations, the presence of dirt, grease, soot burnouts, as well as possible vibrations of the pantograph-catenary system. The interdisciplinary nature of the problem attracts researchers on materials science and tribology, physicists, and mechanical and electrical engineers. [PERSON] and colleagues [7] focused on anisotropic electrical and mechanical properties of graphite. The authors have noticed that the friction and wear behavior of copper-graphite are influenced by the magnitude of electrical current. The authors have also examined the effect of contact polarity on the surfaces of contact, on the transferred layers, on the contact electrical potential and on the contact wear. One of the most commonly used theories concerning the electrical contact resistance between two metal surfaces was developed more than fifty years ago by [PERSON] (the year of publication of his monograph [8] is considered in this context (fourth edition, most often cited); however, [PERSON] refers in his paper [9] to the results obtained by [PERSON] presented in 1929 in a newsletter issued by a commercial company). Briefly speaking, [PERSON] assumed the true points of contact (referred to as \(a\)-points) between two jagged surfaces might be approximated as ellipses. Next he computed thecontact resistance from the relationship \(R_{S}=\frac{\rho}{4a_{c}}f(\gamma)\) where \(\gamma=\sqrt{a/b}\) is a geometric quantity (the ratio of two semi-axes of the ellipse), \(\rho\) is material specific resistivity whereas \(a_{c}\) is the radius of equivalent circular spot with area equal to that of the elliptical \(a\)-spot. Next, the theory was extended to cover the case of similar \(a\)-spots, cf. Figure 1, leading to the relationship  
\[R=\rho\Big{(}\frac{1}{2 na}+\frac{1}{2\alpha}\Big{)}\]  
where \(a\) is the mean \(a\)-spot radius (obtained from the averaging \(\sum_{i}a_{i}/n\) (\(a_{i}\) is the radius of \(i\)-th \(a\)-spot, \(n\)-the total number of \(a\)-spots), whereas \(\alpha\) is the radius of the cluster of \(a\)-spots, referred to as the Holm radius. The Holm theory has been in use for several decades and a number of its refinements have been suggested to take into account different factors. It should be remarked that there is an annual international conference on electrical contacts, called the Holm conference [13]. Generally, the theory is valid for contacts of much larger size than the mean free path of the electrons. [PERSON] et al. [14] suggested an expression for the contact resistance that in the limiting cases reduces itself to the Holm resistance and the Sharvin resistance (which is relevant for nano-electronic devices). [PERSON] and [PERSON] [15] derived, from the first principles, a general electrical contact resistance theory for conductive rough surfaces. The analysis was based on fractal geometry for the surface topography description, elastic-plastic deformation of contacting asperities, and size-dependent constriction resistance of microcontacts.