Magnetic field simulation in a magnetorheological fluid brake taking magnetic hysteresis into account Piotr Sujka Pozna´n University of Technology, Institute of Industrial Electrical Engineering Piotrowo 3A, 60-965 Pozna´n, Poland, e-mail: [email protected]

Abstract— In the paper results of magnetic field simulation in magnetorheological fluid brake taking magnetic hysteresis into account was presented. A hysteresis phenomenon was taken into account by using a Jiles-Atherton model. An influence of magnetic hysteresis on distribution of magnetic field in the region with magnetorheological fluid was studied.

I. INTRODUCTION Magnetorheological fluid is a colloidal suspension of magnetically polarised particles with diameters ranging from 0.5 to 10µm in a carrier fluid, mostly synthetic oil with a low evaporation rate or water. The main feature of the fluid is dramatic change of viscosity and consequently, of shear stress upon the application of magnetic field. The fluids are more and more often used in many devices like brakes or vibration dampers [4], [5]. Cross-section of the magnetoreological fluid brake built at Pozna´n University of Technology is shown in fig. 1. This

II. JILES-ATHERTON MODEL In this model the magnetization in the magnetic material is decomposed into two terms — reversible and irreversible components [2] M = Mrev + Mirr . (1) The irreversible component can be expressed as the differential equation Man − Mirr dMirr = , (2) dH kδ − α(Man − Mirr ) where the anhysteretic magnetization ¶ µ a He − . Man = Msat coth a He

(3)

In above equations: Msat — saturation magnetization k, a, α — parameters which are determined from measured hysteresis loops, He = H + αM — effective field, δ = sgn dH dt — directional parameter. The reversible component can be expressed as Mrev = c(Man + Mirr ), (4) where c is the parameter obtained from measured hysteresis. Replacing (4) in (1) and differentiating with respect to H gives dM Man − Mirr dMan = (1 − c) +c . dH kδ − α(Man − Mirr ) dH

(5)

The differential equation (5) can be solved by using common methods (for example Euler or Runge-Kutta). III. FIELD-CIRCUIT MODEL

Fig. 1.

Magnetorheological fluid brake in cross-section

is a disk-rotor brake system. Magnetic field is excited by a ring coil placed in a stator. The diameter and the thickness of the rotor are 80 mm and 8 mm, respectively. Elements of a magnetic circuit are made of carbon steel St45. This material has relatively wide hysteresis loop. It is expected that an influence of the hysteresis of this material on working of the brake is large, especially when the winding is not supplied and the device was magnetized previously. The paper proposes a field-circuit mathematical model of electromagnetic phenomenon in the brake with magnetorheological fluid taking magnetic hysteresis into account. The JilesAtherton model was used.

The magnetic circuit of the magnetorheological fluid brake has axial symmetry. Therefore a cylindrical coordinate system r, z, ϑ was applied. The differential equations describing the magnetic field in the brake can be formulated in the form [3] µ ¶ µ ¶ 1 ∂ϕ ∂ 1 ∂ϕ ∂ + = ∂r µ0 l ∂r ∂z µ0 l ∂z ∂Mr ∂Mz γ dϕ = −J + − + (6) ∂z ∂r l dt where: µ0 — the magnetic pemeability of the vacum, l = 2πr, ϕ = 2πrAϑ , Aϑ — the magnetic vector potential J = i/s — the current density in the windings, i — the winding current, s — the cross-sectional area of the wire, Mr Mz — the components of the magnetization vector obtained from the Jiles-Atherton model. The last term on the right side of the equation (6) represents eddy currents induced in the area with conductivity γ.

In general, electromagnetic field in the brake is voltageexcited. This means that the currents i in the windings are not known in advance i.e. prior to the electromagnetic field calculation. Therefore, it is necessary to consider the equations of the electric circuit of the device. The set of equations can be written as dΨ u = Ri + , (7) dt where: u — the vector of supply voltages, i — the vector of loop currents, R — the vector of loop resistances, Ψ — the flux linkage vector. In order to solve equations (6) and (7) the finite element method and a "step-by-step" procedure were used [1]. The finite element and time discretization lead to the following system of non-linear algebraic matrix equations · ¸· ¸ S n + (∆t)−1 G −N ϕn = in −N T −∆tR · ¸ θ n + (∆t)−1 Gϕn−1 = . (8) −∆tun − N T ϕn−1

Fig. 3. Distribution of the magnetic field lines for U = 0V after supply cycle U = 10V

In above equations: n subscript — the number of time steps, ∆t — the time step, S — the magnetic stiffness matrix, ϕ — the vector of the nodal potentials, N T — the matrix that transforms the potentials ϕ into flux linkages with the windings, G — the matrix of conductances of elementary rings formed by the mesh, θ — the vector the magnetomotive force in regions with the ferromagnetic material obtained from the Jiles-Atherton model. Equation (8) is non-linear. In order to solve this equation the Newton iterative method was used. IV. RESULTS The presented method of field simulation has been used for analysis of influence of the hysteresis phenomenon on distribution of the magnetic flux density in the brake with magnetorheological fluid. The transient and steady states have been analysed. The magnetic field distribution obtained for

Fig. 4.

Magnetic flux density B(r) in a gap with magnetorheological fluid

to be demagnetized. The result of demagnetizing the magnetic circuit by supply the negative voltage is shown in fig. 4. V. CONCLUSION In the paper field-circuit model of magnetic phenomena in disk-rotor magnetorheological fluid brake has been presented. A hysteresis phenomenon has been taken into account by using the Jiles-Atherton model. An influence of magnetic hysteresis on distribution of magnetic field has been studied. After magnetization, magnetic flux density in a gap with magnetorheological fluid has non-zero values, even when the brake was demagnetized with negative voltage impulse. It is not possible to demagnetize magnetic circuit using direct current. REFERENCES

Fig. 2.

Distribution of the magnetic field lines for U = 10V

U = 10 V is shown in fig. 2. Fig. 3 shows the magnetic lines distribution after turn-off supply voltage. There is magnetic field in the fluid for U = 0 V. This field produces the breaking torque in the brake. In order to reduce this parasitic breaking torque the magnetic circuit has

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