Mathematical Problems in Engineering

Volume 2015, Article ID 436721, 10 pages

http://dx.doi.org/10.1155/2015/436721

## Parameter Analysis on Torque Stabilization for the Eddy Current Brake: A Developed Model, Simulation, and Sensitive Analysis

^{1}Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan, Hubei 430070, China^{2}Hubei Collaborative Innovation Center for Automotive Components Technology, Wuhan, Hubei 430070, China^{3}School of Automotive Engineering, Wuhan University of Technology, Wuhan, Hubei 430070, China^{4}School of Economics, Wuhan University of Technology, Wuhan, Hubei 430070, China

Received 10 March 2015; Revised 15 May 2015; Accepted 17 May 2015

Academic Editor: Emiliano Mucchi

Copyright © 2015 Quan Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Eddy current brake (ECB) is an attractive contactless brake whereas it suffers from braking torque attenuation when the rotating speed increases. To stabilize the ECB’s torque generation property, this paper introduces the concept of anti-magneto-motive force to develop the ECB model on the fundamental of magnetic circles. In the developed model, the eddy current demagnetization and the influence of temperature which make the braking torque attenuation are clearly presented. Using the developed model of ECB, the external and internal characteristics of the ECB are simulated through programming by *MATLAB*. To find the sensibility of the influences on ECB’s torque generation stability, the stability indexes are defined and followed by a sensibility analysis on the internal parameters of an ECB. Finally, this paper indicates that (i) the stability of ECB’s torque generating property could be enhanced by obtaining the optimal combination of “demagnetization speed point and the nominal maximum braking torque.” (ii) The most remarkable influencing factor on the shifting the demagnetization speed point of ECB was the thickness of the air-gap. (iii) The radius of pole shoe’s cross section area and the distance from the pole shoe center to the rotation center are both the most significant influences on the nominal maximum braking torque.

#### 1. Introduction

Eddy current brake (ECB) is an attractive auxiliary braking device for vehicles, which could be directly controlled by wire. Comparing with some other auxiliary brake, ECB enjoys the following advantages [1–3]: (a) the ECB is easy to be controlled, and it nearly does not suffer the braking delay, for it is directly controlled by the current applied in the wire. (b) The ECB enjoys better performances in low-speed domain than either regenerative brake or hydraulic retarder. (c) The ECB does not require the internal combustion engine (ICE) to provide the negative pressure as the pneumatic source (nevertheless, the hydraulic retarder and the engine brake need the internal combustion engine to provide negative pressure when they are working). Overall, the ECB should be a promising auxiliary brake for future vehicles, as vehicles tend to be more electric drove, especially for the vehicles without internal combustion engine like pure electric vehicles and fuel cell battery electric vehicles.

For the ECB, the applied current generates magnetic flux in the core, and an eddy current is induced around pole shoe of the core when the magnetic flux goes through a rotating conductive disk. The braking force is generated by the interaction between eddy current and magnetic flux. According to the theoretic works in [4, 5], the braking torque of an ECB was simply expressed as a function of the angular speed of a disk and the applied current when ignoring the demagnetization. If the angular speed is a constant, the braking force is proportional to the applied current, and vice versa. Actually, the braking torque would not be continuously increasing while the angular speed or applied current rises. According to experimental results on ECB’s torque-speed property [2, 6–8], at the very beginning, the braking torque increases rapidly while the rotating speed ascends. Nevertheless, after the braking torque peaked at its maximum value, the braking torque tends to drop significantly if the rotating speed goes on increasing. The reason of this phenomenon is that the demagnetization effect cut down the ECB’s torque generation property while, at the same time, the ECB’s torque stabilization is destroyed. Specifically, for excitation eddy current brake, to prevent the demagnetization effect not only means to improve the braking stability but also to save the energy. To explore the electromagnetic property of the ECB, Smythe [1] firstly implies magnetic potential theory to model the ECB, and he described the demagnetization effects by deriving the Maxwell’s equations. Wouterse [3] indicates structure index C based on the numerical analysis and experiment validation to simplified Smythe’s model. Simeu and George’s firstly [4] introduced an ECB based on magnetic circle theory and his model is widely used in control algorithm research on ECB [5, 9]. However, Simeu’s model can only express how ECB retards the rotator in low-speed region, but it cannot fully describe the ECB’s property in high-speed region for it ignored the eddy current demagnetization. Recently, with the development of computational science, many have tried to model ECB with FEM [10–12], although the FEA results could be a reference for engineering design, it is still reasonable to develop current ECB’s theoretical model for detecting the influences on ECB’s braking torque stability [6–8]. Hence, this paper presents a modelling method based on magnetic circle theory, which is easy to find the relations of the internal parameters to the braking torque stabilization. Using this developed ECB model, the sensitivity of the influences of the braking torque stabilization could be easily found through simulation and related sensitivity analysis.

This paper is organized as follows. Section 2 presents the basic construction of a typical ECB while the parameters for modeling are present. Section 3 introduces the modeling assumptions and the mathematical modeling works; in this section, by introducing the concept of antimagnetic force, a developed ECB model is derivate in detail. Section 4 includes the simulation results on both external parameters and internal parameters of the ECB based on the developed model. In Section 5, the braking torque stability index including the demagnetization speed point (demagnetization speed point) and the nominal maximum braking torque is introduced, followed by a sensibility analysis on internal parameters to braking torque stability index.

#### 2. The Principle of ECB

The configuration of a typical ECB is presented in Figure 1. A typical ECB consisted of rotator, shaft, coil, and iron core. For easy fabrication, this paper presents an ECB that is designed with two pairs of coil-core systems which are circumferentially equispaced around the rotator. The rotator is a copper made disc, with the radius of and the thickness of . In each coil-core system, two U-shaped iron cores are symmetry arranged aside the rotator. The zone between these two U-shaped iron cores calls air-gap; the thickness of the air-gap is . At the heat of each iron core, the coils are winding into turns and each coil is series-wound connected. The pole shoe is at the top of the U-shaped iron-core, and in this paper, the pole shoe’s cross section area is roundness with radius of and the area of .