4th International Conference on Power Engineering, Energy and Electrical Drives
Istanbul, Turkey, 13-17 May 2013
Determination of the Effect of Conductive Disk Thickness on Braking Torque for a Low Power Eddy Current Brake Mehmet Onur GULBAHCE
Derya Ahmet KOCABAS
Ahmet Kubilay ATALAY
Department of Electrical Engineering Istanbul Technical University Istanbul, Turkey
[email protected]
Department of Electrical Engineering Istanbul Technical University Istanbul, Turkey
[email protected]
Department of Electrical Engineering Istanbul Technical University Istanbul, Turkey
[email protected]
Abstract— In Eddy current brakes, mechanical energy is converted into thermal energy and the system is mainly an electromechanical conversion device. A rotating conductive disk is placed in front of calculated number of pole surfaces that create a magnetic field which is unchanged according to time. Since the disk rotates, a relatively changing magnetic field passes through the conductive disk causing Eddy currents to be induced inside the disk. These induced Eddy currents produce an opposing magnetic field forcing the rotating disk to slow down by means of consuming mechanical energy. Mathematical analysis of the effects of Eddy currents is almost impossible due to the complexity of both the magnetic problem and geometry. There is no obtained certain relationship which can explain output data in terms of input data since the relation includes too many variables including disk areas, disk thickness, disk radius, speed, etc. In this study, different eddy current brake designs are analysed where all design constraints were kept unchanged apart from conductive disk thickness to determine the effect of change. All mentioned designs are analysed by commercial software using finite element method (FEM). Torque vs. speed, total power dissipation vs. speed characteristics for low, medium and high speed regions and change of critical speed and maximum braking torque according to conductive disk thickness are also obtained. Keywords-component; eddy currents; eddy current brakes; electromagnetic brakes; finite elements method
I.
INTRODUCTION
Eddy current brakes are electromechanical energy conversion systems which convert mechanical energy into thermal energy. Mainly this helps reducing shaft speed in some applications by consuming the stored kinetic energy and also these brakes can be used for loading a motor in load test by producing an opposing magnetic field and braking force. When a conductive material is exposed to a time varying magnetic field, Eddy currents are induced in the material due to the change in magnetic flux. Mainly, stationary magnetic field producing poles are placed onto a yoke which is placed in front of a conductive material shaped as a disk that is connected to a shaft of a motor to be loaded. The windings surrounding the poles are fed by direct current resulting in a
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time independent magnetic field in the air gap. But rotating conductive disk facing this field is exposed to a time varying magnetic field depending on rotation resulting in induction of Eddy currents [1]. Nature of the changing magnetic field enforces Eddy currents to complete their path as a loop inside the conductive disk. Consumed mechanical energy expression resembles that of the energy created by a current passing through a simple resistance [2,3]. Two interacting magnetic fields weakening each other causes a braking force and a braking torque. Unlike the conventional brake systems, the Eddy current brakes are frictionless and less damage occurs in the materials used in the design of the brake. Also system response of the brake is more sustainable and less maintenance is needed [4]. Since all electromechanical conversion systems can all be expressed in terms of differential equations, it is possible to obtain a set of partial differential equations to describe the system mathematically. Serious effort was performed to obtain a mathematical model to solve these equations [5], but none of the studies in the literature [2,3] managed to obtain a simple equation representing the relation between the input and output data for all speed regions. Those models or studies [2,3] were only successful for only one speed region where the rest of the curve seems to be chaotic. Depending on the complexity of the whole system and the equations, simplifying numerical methods in engineering like FEM can be used. A great number of commercial software based on this method can be found. In this paper, a software is used provided by the national distributor company. The output torque and angular speed depend on too many design and operational parameters like conductive disk dimensions, the number and the dimensions of poles, excitation current, air gap distance and the specific properties of the used materials, etc. Most of the designs including permanent magnets which can be found in the literature are experienced and it was seen that it can provide a limited adjustable load torque [5]. In the whole study of the authors, effects of changes in design parameters are all analysed to extend the limits of the
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designed Eddy current brake. Here in this study, only the effect of the change in conductive disk thickness is analysed and four different Eddy current brakes having different conductive disk thickness where other remaining design parameters are all kept unchanged. Different speed inputs up to 7500 rpm are used to obtain speed characteristics for all low, medium and high speed regions and more than 40 simulations are performed. II.
GENERAL THEORY OF EDDY CURRENT BRAKE
Eddy current brakes are composed of a stationary magnetic flux source like permanent magnet or electromagnet and a conductive disk connected to a rotating mechanical energy source. Depending on rotation, the conductive material exposed to a time varying magnetic flux density which can be expressed by Lenz’s law where E is electric field intensity and B is magnetic flux density. JG JG ∂B ∇× E = − (1) ∂t Occurrence of electric field causes current, because of Ohm’s law where J is current density and these currents are eddy currents which complete their path along the material as a loop. JG JG J =σ ⋅E (2) The action between eddy currents and magnetic flux density creates a braking force (F) and induced braking torque can be expressed as below, where r is the radius of the disk;
III.
JG JG JG F = J ×B
(3)
JG JG T = F ⋅r
(4)
Eddy current brakes include stationary poles creating the magnetic field which are excited by excitation current or permanent magnets and moving conducting disk. In practise eddy current brakes have non-conducting and conducting regions to be modelled by using magnetic scalar potential ϑ and magnetic vector potential A where HT is total magnetic field intensity [6]. In non-conducting regions (5) is used.
(5)
This total magnetic field intensity has two components including φ as reduced magnetic scalar potential and HS as the magnetic field intensity of source. H T = −∇.φ + H S
(6)
The relation between HS and JS, source current density is as shown below.
POWERENG 2013
∇ × HS = JS
(7)
The Laplacian type equation is below where μ is the permeance; ∇.μ.∇.ϑ = 0
(8)
In eddy current or conducting regions B, magnetic flux density and E, electric field intensity can be expressed as below where V is scalar electric potential, ϖ is speed [6]. B = ∇× A
E=−
∂A − ∇.V + ϖ × ∇ × A ∂t
(9) (10)
In order to obtain an equation where V is omitted, equation 2,10 and Ampère’s law can be used;
∇×
1
μ
§ ∂A · ×∇× A = σ ¨− − ϖ × ∇ × A + ∇.V ¸ © ∂t ¹
(11)
Eq. 12 is attained by obtaining the divergence of (11). § ∂A · ∇.σ ¨ − − ϖ × ∇ × A + ∇.V ¸ = 0 © ∂t ¹
(12)
Scalar electric potential can be expressed in terms of A and ϖ as in (13). V = A.ϖ
FINITE ELEMENTS EQUATIONS
H T = −∇.ϑ
Istanbul, Turkey, 13-17 May 2013
(13)
By substituting (13) into (11),
§ ∂A · ×∇× A = σ ¨ − − (ϖ .∇ ) A − ( A.∇ )ϖ − A × ( ∇×ϖ ) ¸ (14) μ © ∂t ¹ is obtained. ∇×
1
IV.
3D TRANSIENT MAGNETIC MODEL
In this study, an Eddy current brake is analysed to determine the effect of the change in conductive disk thickness on braking torque. The brake has a conductive disk which is made of pure aluminium and also steel 1010 was chosen as magnetic material for the poles and the yoke, since the magnetic characteristic of the material is more convenient for this application. Four designs having different conductive disk thicknesses are used to obtain the effect. Rest of the design parameters are all kept unchanged. The design parameters which are constant during the analysis are given in TABLE I. To increase the effect of the brake, poles are placed beside both air gaps on either sides of the disk. The problem solved by considering skin and proximity effect, motion induced eddy currents and time diffusion of
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4th International Conference on Power Engineering, Energy and Electrical Drives
Symbol
Quantity
Value
I
Excitation Current
10A
N
Number of Turns per pole
300
p
Number of Pole Pairs
8
τ
Pole Width
90o
g
Air-gap Width
2mm
rmag
Electromagnet Radius
20mm
R
Conductive Disk Radius
380mm
μr σ
Relative permeability of
1
Conductive Disk Conductivity of Disk
58000000Ω-1
Conductive disk
pole
coil
Istanbul, Turkey, 13-17 May 2013
Figure 2. The mesh of the model
V.
RESULTS
In transient solution, mechanical damping torque and magnetic flux vary with the movement of the conductive disk. Used speed levels are 50, 100, 250, 500, 750, 1000, 1500, 2000, 3000, 5000 and 7500 rpm. Chosen conductive disk thickness values are 0.5, 1, 1.5 and 2 cm. Since the aim of the study is to obtain a characteristic for different speed values, numerous simulations are needed. Just to inform the reader about the results of one simulation, authors prefer to give some samples for only one of the 44 simulations before representing the combined data. The chosen results are from the output data which is calculated for 1500 min-1 for the thickness of 1 cm. All figures including Fig.3-6 present distributions of different operational quantities. Magnetic flux density inside the conductive disk is given in Fig. 3 where the maximum value is settled to be 0,5T inside the aluminium disk. Vectorial representations of magnetic field intensity inside the conductive disk, eddy current density inside the conductive disk and magnetic flux density over the pole surface are given in Fig. 4, 5, 6 respectively.
Figure 1. ¼ geometry including the poles visible for only one side of the 3D transient magnetic model
magnetic fields. To simulate this problem, 3D transient optimised design parameters of Eddy current brake magnetic model was used. Only ¼ of the geometry is modelled so as to solve the problem quicker. One side of 3D transient magnetic model of the Eddy current brake is given below in Fig. 1 [8]. In order to obtain an accurate solution, a fine mesh is required. The model is meshed depending on the region. Moving eddy current regions need more mesh than nonconducting region as in Fig. 2 [8].
Figure 3. Magnetic flux density distribution of the conductive disk
The time step in transient magnetic problems is very crucial to increase the accuracy of the calculated eddy current. Finite element algorithm uses nodal solution, so time step and mesh size play an important role in obtaining a correct solution. The time step is chosen 0.5ms for a stable and correct solution. To obtain a full characteristic, the four models are analysed for 11 different speed values and more than 40 simulations are completed. The separated output data for all speed levels are processed by curve fitting in order to attain the combined characteristics. All figures including output data is given in Results.
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Figure 4. Magnetic field intensity vector of the conductive disk
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4th International Conference on Power Engineering, Energy and Electrical Drives
Istanbul, Turkey, 13-17 May 2013
Total Power Dissipation (W) 0 -800 -1600 -2400 -3200 Figure 5. Eddy current density vector of the conductive disk
-4000 Speed (rpm)
-4800
0 2000 4000 6000 8000 Figure 8. Total power dissipation versus mechanical speed Maximum Braking Torque (N.m.) -10
-15
-20
-25
-30 Figure 6. Magnetic flux density vector of poles surface
In Fig. 7-8. the fitted curves for the combination of the data obtained from 44 simulations including all low, medium and high speed regions for the Eddy current brake are given. Fig.7 represents braking torque change according to mechanical speed. Total power dissipation related to shaft speed is given in Fig. 8. In Fig. 9, maximum braking torque related to disk thickness is shown obtained from the curves given in Fig. 7. In Fig.10, critical speed change according to conductive disk thickness is given which is also obtained from Fig.7.
1000
Disk Thickness (cm) 0,5 1 1,5 2 Figure 9. Maximum Braking Torque versus Disk Thickness Critical Speed (rpm)
700
400
Braking Torque (N.m.) 0 -8
100 0,5
1
Disk Thickness (cm) 1,5
2
Figure 10. Critical Speed versus Disk Thickness
-16 -24
VI.
-32 -40 -48 0
Speed (rpm) 1000 2000 3000 4000 5000 6000 7000 Figure 7. Braking torque versus mechanical speed
POWERENG 2013
CONCULUSION
In this paper, main design parameters for a specially designed Eddy current brake are mentioned and main operational principle is re-introduced. The relation between output data and conductive disk thickness which is one of major design parameters is obtained. A special brake design is analysed by FEM. Four different disk thicknesses are used in simulation for different 11 speed levels between 50-7500 rpm. Naturally 44 different simulation results are gathered together to obtain braking torque vs speed, total power dissipation vs
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4th International Conference on Power Engineering, Energy and Electrical Drives
Istanbul, Turkey, 13-17 May 2013
speed curves. Two main characteristic curves which are maximum braking torque and critical speed vs disk thickness are obtained after processing the data. All results and characteristics confirmed that decreasing the conductive disk thickness result in an increase in the load capacity, total power dissipation and the critical speed value. It also causes a decrease in maximum braking torque. This proves that as the disk thickness increases, the disk resistance increases which causes the critical speed to be decreased and torque-speed curve to be narrowed. It is shown that a specific brake to be designed needs a critical conductive disk thickness value to be chosen. In practice it is not enough to calculate the physical dimensions, but also the durability of the disk material needs to be high enough to resist physical quantities which are mentioned above. ACKNOWLEDGMENT Authors are thankful to Figes A.S. for providing the trial version of their commercial software for this study. REFERENCES [1]
[2]
[3]
[4] [5]
[6]
[7]
[8]
E. J. Davies, “General Theory of Eddy Current Couplings and Brakes,” Proceedings of the Institution of Electrical Engineers, vol. 113, no.5, pp. 825-837, May 1966 Schieber, D., “Braking Torque on Rotating Sheet in Stationary Magnetic Field,” Proceedings of the Institution of Electrical Engineers, vol. 121, no. 2, pp. 117–122.,1974 Wiederick, H. D., Gauthier, N., Campbell, D. A., and Rochon, P., “Magnetic Braking: Simple Theory and Experiment,” American Journal of Physics, vol. 55, no. 6, 500–503., 1987 H. Sodano, J. Bae, “Eddy current damping in structures,” The Shock and Vibration Digest, vol. 36, no.6, pp. 469-478, November 2004. Gay, S. E., Ehsani, M., “ Analysis and experimental testing of permanenet magnet eddy current brake”, Vehicle Power and Propulsion, 2005, IEEE Conference, 7-9 September, 2005. M. V. K. Chari, “Finite Element Solution of the Eddy-Current Problem in magnetic Structures,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-93, no. 1, pp. 62-72 D. Rodger, H. C. Lai, and P.K. Vong, “ Finite element model of eddy current brakes”, Computation in Electromagnetics, 2002, CEM 2002,The Fourth International Conference on (Ref. No. 2002/063) Gulbahce, M. O., Kocabas, D. A., Habir I. “ Finite Element Analysis of a Small Power Eddy Current Brake”, Mechatronica, 2012, IEEE Conference, 5-7 December, 2012.
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