An Eddy Current Braking System

Transcription

1 An Eddy Current Braking System Lee Barnes, John Hardin, and Charles A. Gross Electrical Engineering Auburn University Dewain Wasson Research and Development Diversified Products Corp. Abstract This paper describes details of a computer model of an eddy-current brake for application in an exercise device. This paper reports on the design of a computer analysis package (EBRAKE) which calculates disk torque as a function of disk speed, for a given system geometry and coil current. n trod u c ti on Theoiy One continuing area of interest in Power Engineering involves the analysis of eddy - current brakes. The application under investigation in this study involved an exercise bicycle for Diversified Products Corporation. The problem facing DP was that of absorbing energy input by the user into the bicycle. Presently, caliper brake pads are used in many bikes to achieve the resistance, but this resistance remains constant with increasing speed. Electromagnetic braking, on the other hand, provides resistance proportional to the speed. The principle behind the operation of an eddy - current brake relates to basic electromagnetic induction theory. When electric conducting material moves through a magnetic field, currents are induced in the material. Since the path for the current is not well - defined, the current tends to flow in short circular paths around the vicinity of the field. The short circular paths resemble eddies in a stream, and thus are named "eddy - currents". Normally the production of eddy - currents is undesirable, because they contribute to losses in many applications, such as rotating machines and transformers. However, it is precisely these losses on which the brake capitalizes. The eddy - currents react with the magnetic field to produce a force which is counter and in proportion to the relative motion of the material. The general theory relating to eddy-current braking is relatively well understood. Consider the system shown in figure 1. A description of the variables used is provided in table 1: Figure. Diagram of variables. Eddy Current Brake with The theory is based on Faraday's induction law, i.e. when a conducting element moves through a constant magnetic field, a current is induced [l]. This current reacts with the magnetic field to produce a force which /93 $ EEE 58

2 opposes the motion of the element. The basic expression for the force is given by: Force = Volume * 0 * Bi *velocity where 0 is the conductivity of the disk in S/m, volume is the volume of the magnetic footprint in m3, velocity is the translational velocity of the disk through the magnetic field in d s, and Bo is the magnetic field intensity in Wb/m2 generated by the core [2]. Some authors have refined this formula by accounting for the fact that the magnetic field esists outside the shadow area of the core itself [3], and the plate/disk which moves through the field is not infinite [4]. Using these modifications yields the following expression [3 A : F = abtob;ac*velocity Table 1. Var r Description of variables. Radius of the disk Description R Distance from the center of the disk 10 m the center of the magnetic footprint. a Cross-sectional width of the core. Number of turns of the coil around the N core. m T where 1 1 circular core with the same cross-sectional area as the core being used. Having found an expression for the force, the torque can be found by multiplying force by lever arm R. Most of the variables found in this expression are direct a core values specific to disk and core materials (0). a and c are calculated using the formulas previously introduced. Only Bo remains to be found. t may be calculated by air gap 59

3 ~ the air gap. This concept is represented in the graph in figure 3. Since no closed form expression for the Although these materials can be saved to disk once generated, one can not create them 'on-the-fly,' that is, not during the editinglcreation of a case. Note that the data in Table 1 is largely in English units--the computer prompts for this unit and then does the necessary conversion to S for calculations. The numeric results of the computer program as well as a printout of the graphical information follow: 1 / Fo a Eo Ni F [A-W] Figure 3. Graphical representation of magnetic eq u ival e n t ci rc u i t. Table li. Computer nput Data. ; ; t Lengthofcore cross-section Variable 1 Description Vdue 1 Units Width of core cross-section Mean core length Excitation current 0.5 A reluctance of the core is available, the measured characteristic is actually used. EBWE uses a ten-point interpolation for the 4-F curve and employs the Newton- Raphson method for the root of U Disk conducti\ily--aluminum 35 MSm disk t 1 Dirk thickness R Disl m&us n Mapet locahon n 8 Width of a r gap 01 in The reluctance of the air gap is given by 'airgap =8 p,*b*a The solution of equation (A) yields the point of intersection between the nonlinear curve and the load line. This value for Ho is then substituted into the interpolation to find Bo. N Number oftuma 1900 Table ll. Computer Output and Lab Output. Disk Speed EBRAKE Disk Torque Measured Disk [RPM1 [tn-lb] Torque Computer results The data found in Table 1 describes the specific system investigated. This data was measured and derived from the lab tests. EBRAKE utilizes a library feature for disk and core materials whereby one may enter conductivity for various disk materials and a family of points to be used in the interpolation of the core properties. Later, during a specific case run, all that is required is to choose one of the materials defined earlier

4 Case Title: Test of Program Case Desc: Wabash Maqnet Torque-Speed Characteristic Parameters Disk nfo Thickness Radius Conductivity : flagnet location : flagnet nfo Length Uidth Core length : # of turns : Current Air Gap No-Load Torque : Flux Density : fmml [cml tfis/ml n [cml ;= ot 3 U [cml 0 c Ccml Ccml CA Cml 10.0 [in-lbl [kg Disk Data Crank Data Speed CRPMl Fl=Print to Epson F2=Print to HP Esc=Exi t Figure 4. Graphical computer output. Experimental results Conclusion The experimental work was performed by Diversified Products. The coil excitation circuit was monitored by a digital multimeter to measure the current. The resulting torque was measured in the following way. A torque transducer was situated in the drive shaft between the driving force (a small motor) and the crank of the bike. The crank was coupled through some gear/pulley ratio to the disk itself. The transducer was a strain gage which measured the amount of torque necessary to keep the crank tuming at the same angular velocity as the motor shaft. The output of the transducer was fed into a digital multimeter and time-averaged over several periods to arrive at the values indicated. By considering the physical properties of the disk and magnet, it is possible to arrive at a mathematical model for the eddy-current brake. This model has been used to provide industry with a design tool [EBRAKE]. By using the program, one may determine the relative effect of changing various parameters in the system without costly prototype testing. Acknowledgements The authors would like to thank Diversified Products Corporation for their assistance and cooperation throughout this project. 61

5 References [ 11 H. D. Wiederick, N. Gauthier, D.A. Campbell, and P. Rochon, "Magnetic Braking: Simple Theory and Experiment," American Joumal of Phvsics. Vol. 55, No. 6, June [2] Martin A. Plonus, Applied Electromagnetics, New York: McGraw-Hill, [3] Mark A. Heald, "Magnetic Braking: mproved Theory," American Joumal of Physics, Vol. 56, No. 6, June [4] J. H. Wouterse, "Critical Torque and Speed of Eddy- Current Brake with Widely Separated Soft ron Poles," EE Proceedinvs. Part B. Electric Power Amlications, Vol. 138, No. 4, July