The magnetic flux density B can be derived by the definition of the magneticvector potential B=∇×A .

Experimental Testing and Discussions

The mathematical expression for the magnetization M isrepresented by the Fourier series.

The magnetic flux density B and the magnetic intensity H satisfyMaxwell’s equations.

The magnetic vector potential A=∑An(r)e-jnp(θ or α)iz is defined as∇×A = B.

Where Inp and Knp are the modified Bessel functions of the firstand second type of order np.

Conclusion

Abstract

Torque Calculation and Experimental Testing of Eddy Current Brake with Permanent Magnets considering Thermal Behavior

Objectives

Kyung-Hun Shin1, Hyung-Il Park2, Han-Wook Cho3, and Jang-Young Choi1

1Electrical Engineering, Chungnam National University, Daejeon 34134, Korea2Advanced Brake Engineering Team, Hyundai Mobis, 17-2 Mabuk-ro, Giheung-gu, Yongin 16891, Korea

3Electric, Electronic and Comm. Eng. Edu., Chungnam National University, Daejeon 34134, Korea

This paper presents a torque analysis and testing of an eddy current brake (ECB) with permanent magnets (PMs) using analytical magnetic field calculations. The magnetic field solutions are obtained by considering the eddy current reaction using a magnetic vector potential and a two-dimensional(2D) polar coordinate system. The magnetic torque is also determined from these solutions. A 2D finite element method (FEM) is employed to validate the analytical magnetic calculations. The braking torque measurement is experimentally verified to validate the analytical study. From theexperimental study, we obtain analytical and FEM results of braking torque with thermal behavior, and these are in good agreement with the experimental results.

Presented at the MT25 Conference , 2017 August 27 - September 1, RAI - Amsterdam; Program I.D. number: Tue-Af-Po2.06-21

Structure of an ECM with PMs and Analysis Model

In this study, the torque calculation of an ECB with parallel magnetized PMs was analyzed using magnetic field calculations. The magnetic field solutions were obtained using a magnetic vector potential and a 2D polar coordinate system. The magnetic torque was also derived

from these solutions. 2D FEM and experimental testing were employed to validate the results of the analysis. The induced eddy currents heated and increased the resistivity of the conducting cylinder. Therefore, we used the temperature measured from the experimental testing to calculate the conductivities of the conductor to achieve an exact

agreement of the results. From experimental study, we obtained modified analytical and FEM results of braking torque with thermal behavior, and these were in good

agreement with the experimental results.

Electromagnetic Analysis of an ECB

The rotor is rotated based on the (r, q, z) coordinate system, and the stator is fixed based on the (r, a, z) coordinate system.

Here, q and a are angular components, and their relationship is given by q = a + wrt, where wr (rad/s) is the angular speed of the rotor.

We consider the analytical model to be composed of three regions: the conducting cylinder (I), the air gap (II), and the PMs (III).

Fig. 3 shows the flux line distributions of the ECB with parallelly magnetized PMs obtained from FEM calculations at 0 rpm and 3000 rpm

The torque measurement results and the analytical and FEM results are shown in Fig. 8.

We can see that the measured values decrease compared with the predicted values because the testing setup was equipped with only a forced air-cooling system, which is incapable of efficiently removing the heat generated by the induced eddy currents, and the analytical and FEM analyses donot consider the thermal behavior.

We used the measured temperature obtained from the experimental testing to calculate the conductivities of the conductor for exact agreement of theresults, as shown in Fig. 9.

Finally, we obtained modified analytical and FEM results of braking torque with thermal behavior, and these are in good agreement with theexperimental results, as shown in Fig. 10.

FIG. 1. Side and front views of ECB FIG. 2. Simplified multilayer analytical model

qa

FIG. 3. Flux line distributions obtained by FEM

20

2

20 0

(

0

(

I

II

III

A M)

A

A H) J

2 2

0

2 22

2 2

2 2

2 2

2 2

20

( ) (

( ) ( )1 ( )( )

( ) ( )1 ( )( ) 0

)1 ( )( )

0,

I IIn nn

II IIIIn nn

rn n

III IIIIIIn n

n n

n al r

A r A r npA r

r r r r r

A r A r npA r

r r r r

jnpM M

A r A r npA r

r r r r

her jnpw e

q

w

0

0

( )

)(

B H M

B H M

( ) rn nn

jnpM eM qq

r θM i i

02( )

( )

)

1 (

(

)rn nI np I np jnp

n nn

II np II np jnpn n

n

III III jnpn np n n np n

n

jnpM Mr C r D r e

r C r D r e

r C

rn

I r D K r e

pq a

a

q

Iz

IIz

IIIz

A i

A i

A i

Fig. 4. Comparison of analytical and FEM results of flux density of the initial model presented in Fig. 3 at (a) 0 rpm and (b) 3000 rpm.

Fig. 5. Comparison of analytical and FEM results of braking torque of the initial model as a function of rotor speed.

*0

, ,a a

SR Rq q

III III

r θF B B

The magnetic force (F) produced by the interaction between the conductingcylinder and the PMs is derived using the Maxwell stress tensor [12].

The torque (T) can be calculated by T = Ra F.

Fig. 6. Comparison of analytical and FEM results of current density.

The analytically calculated current density using J = -al ∂A/∂t can also be ingood agreement with the FEM results because the flux density and torque resultsfrom the analytical analysis agree with the FEM results.

TABLE I SPECIFICATIONS OF THE TESTING APPARATUS

Symbol Designation Value Unit

tm Thickness of the PMs 8 mm Ro Outer radius of the PMs 19 mm Ri Inner radius of the PMs 11 mm Br Residual magnetic flux density 1.3 T tc Thickness of the conducting cylinder 3 mm Ra Outer radius of the conducting cylinder 27 mm Rg Inner radius of the conducting cylinder 24 mm al Conductivity of the conducting cylinder 1.4e7 S/m la Active length of the rotor 40 mm P Number of pole-pairs 2

Fig. 9. Variation of measured temperature rise with conductivity in conducting cylinder.

Fig. 8. Analytical, FEM, and experimental results of braking torque

of the ECB model.

Fig. 10. Modified analytical, FEM, and experimental results of braking

torque with thermal behavior.