# Adaptive Bias Current Control of an Active Magnetic Bearing Under Motion-Induced Eddy Currents

Timon Dewitte, Ruben De Croo, Frederik De Belie

GHENT UNIVERSITY

Technologiepark-Zwijnaarde 131

Ghent, Belgium

Tel.: +32 / 9 - 264 34 18

E-Mail: ruben.decroo@ugent.be

***Index Terms***—active magnetic bearing, AMB, adaptive control, augmented Kalman filter, bias current.

***Abstract***—This paper investigates the impact of motion-induced eddy currents on the dynamic performance of active magnetic bearings (AMBs) with a solid rotor. Since these eddy currents weaken the magnetic field in the air gap, the force exerted on the rotor is reduced which affects the dynamic behaviour of the AMB. An analytical model of the change in magnetic field distribution is presented and compared with simulation results using the finite element method for different rotor speeds. An adaptive control approach comprising a parameter estimation algorithm and a PI controller is implemented to mitigate these effects. This is based on an augmented Kalman filter which estimates the bias current corresponding to the weakened magnetic field. Based on this estimate, the actual bias current is adapted in order to counteract the reduction in magnetic field. Experimental results of the adaptive algorithm are presented and compared to simulation results.

## I. INTRODUCTION

In recent years, the interest in magnetic bearings has increased due to its compelling advantages compared to conventional bearings. Magnetic bearings allow rotational movement with low friction and no wear [1]. For these reasons, they are used in turbomachinery, compressors and pumps operating at high rotational speeds [2].

Magnetic bearings can be categorized in three groups according to their operational principle: the passive magnetic bearing(PMB), the hybrid magnetic bearing (HMB) and the active magnetic bearing (AMB). Passive magnetic bearings use permanent magnets in order to levitate the rotor. However, they tend to be difficult in construction and offer low damping and limited load

carrying capacity [3]. Hybrid magnetic bearings combine both permanent magnets and electromagnets to levitate the rotor, whereas AMBs only use electromagnets for levitation. Active magnetic bearings have the advantage that their dynamic behaviour can be changed according to the specific requirement of the application [4].

Active magnetic bearings are nonlinear and inherently unstable systems making them challenging to control. Therefore, achieving the desired performance and stability of an AMB often depends on accurate modeling. However, many existing models do not include the effect of eddy currents. As AMBs typically operate at high rotational speeds, motion-induced eddy currents can become significant, particularly in AMBs with solid rotors [5].

Several studies have been conducted yet about the behaviour of eddy currents and their influence on the magnetic actuator performance. The effects of eddy currents on a solid-core electromagnetic actuator such as an AMB was studied by [6]. They obtained a model relating the magnetic flux to the control current in the presence of induced eddy currents. These results are extended by [7]. In frequency domain, the relation between the flux and control current is obtained by introducing a complex permeability. Additionally, by modeling the effects of the eddy currents as a parasitic auxiliary winding, the transfer function relating the flux and control current could be obtained. These works are extended by [8]. By dropping the assumption that the magnetic actuator is semi-infinite, representing the fact that one of the dimensions of the actuator is way larger than the other, a two-dimensional analysis of the eddy currents is performed eventually resulting in an equivalent magnetic circuit incorporating the effects of the eddy currents. The effects of eddy currents were also studied by [9], [10]. They obtained a fractional order model to represent the
dynamics of the AMB in the presence of eddy current effects.

Previous works studied how the eddy currents induced by the control signal influence the dynamics. However, during rotation of the rotor, two types of eddy currents are present in the rotor: eddy currents due to the time-varying control signal and motion-induced eddy currents. Therefore, the goal of this paper is to study the effect of the motion-induced eddy currents on the dynamic behaviour of an AMB with a solid rotor and to design an adaptive controller which mitigates the effects of the motion-induced eddy currents.

## II. MODELING OF THE AMB

### A. Mechanical model of the AMB

The representation of the AMB under consideration is shown in Fig. 1 in which the right-handed stationary reference frame is defined. The phases are oriented with an angle of 45 degrees with respect to the vertical. This AMB is part of a larger setup composed of a conventional bearing and an AMB as shown in Fig. 2, where the conventional roller bearing is modeled infinitely stiff and is therefore represented by a hinge. The system parameters are defined in Table. I.

By expressing Euler's equations around the hinge point, following equations are obtained:

$$ \begin{cases} I_y \ddot{\theta}_y = -\frac{mgl_c}{\sqrt{2}} \cos(\theta_y) + (F_{x,1} - F_{x,4})l \cos(\theta_y) \\ I_x \ddot{\theta}_x = -\frac{mgl_c}{\sqrt{2}} \cos(\theta_x) + (F_{y,2} - F_{y,3})l \cos(\theta_x) \end{cases} \quad (1) $$

<table><caption>TABLE I: AMB parameters</caption><thead><tr><th>Parameters</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Mass of the rotor [kg]</td><td>m</td><td>1.8</td></tr><tr><td>Gravitational constant [$\frac{m}{s^2}$]</td><td>g</td><td>9.81</td></tr><tr><td>Distance to AMB [m]</td><td>l</td><td>0.26</td></tr><tr><td>Distance to center of gravity [m]</td><td>l<sub>c</sub></td><td>0.22</td></tr><tr><td>Inertia around x-axis [$kgm^2$]</td><td>I<sub>x</sub></td><td>0.1</td></tr><tr><td>Inertia around y-axis [$kgm^2$]</td><td>I<sub>y</sub></td><td>0.1</td></tr><tr><td>Width of side leg [mm]</td><td>a</td><td>11</td></tr><tr><td>Axial length of the AMB [mm]</td><td>b</td><td>30</td></tr><tr><td>Nominal air gap thickness [mm]</td><td>δ<sub>0</sub></td><td>1</td></tr></tbody></table>

Since the air gap thickness $\delta_0$ between the rotor and the pole shoes of the stator is way smaller than the length $l$, the angles $\theta_x$ and $\theta_y$ are small such that (1) can be simplified to:

$$ \begin{cases} \frac{I_y \ddot{x}}{l} = -\frac{mgl_c}{\sqrt{2}} + (F_{x,1} - F_{x,4})l \\ \frac{I_x \ddot{y}}{l} = -\frac{mgl_c}{\sqrt{2}} + (F_{y,2} - F_{y,3})l \end{cases} \quad (2) $$

<figure><img src="image_1.png" /><figcaption>Fig. 1: AMB representation with the associated reference frame</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>Fig. 2: Mechanical representation of the AMB system</figcaption></figure>

### B. Equivalent magnetic circuit

To relate the forces exerted on the rotor to the phase currents, the equivalent magnetic circuit is developed. The equivalent magnetic circuit can be derived by applying Ampère's law along curve $C_1$ shown in Fig. 3. From symmetry, applying Ampère's law along curve $C_2$ will yield the same results. By assuming the magnetization characteristics of the rotor and stator to be linear with high relative permeability $\mu_r$, the magnetic potential drop in the stator and rotor can be neglected, Ampère's law then results in:

$$ Ni = \frac{B_{\delta,m}\delta}{\mu_0} + \frac{B_{\delta,s}\delta}{\mu_0} \quad (3) $$

where $B_{\delta,m}$ and $B_{\delta,s}$ denote the magnetic flux density underneath the middle leg and side leg of the pole shoe and $\mu_0$ is the permeability of air.

When the leakage flux is neglected, the magnetic flux $\phi$ through the magnetic circuit remains constant. By relating the magnetic flux density to the magnetic flux underneath the legs of the pole shoes, by using the area of the pole shoes in contact with the air gap, (3) can be expressed as:
$$Ni = \frac{\phi\delta}{\mu_0 s_m b} + \frac{\phi\delta}{\mu_0 s_s b} \quad (4)$$

With $s_m$ half the arc length of the middle leg of the pole shoe and $s_s$ the full arc length of the side leg of the pole shoe.

<figure><img src="image_3.png" /><figcaption>Fig. 3: Magnetic representation of the AMB system</figcaption></figure>

The reluctance of each air gap can thus be written as:

$$\left\{ \begin{aligned} \Re_{\delta,m} &= \frac{\delta}{\mu_0 s_m b} \\ \Re_{\delta,s} &= \frac{\delta}{\mu_0 s_s b} \end{aligned} \right. \quad (5)$$

From Maxwell's tensor, the force per unit area exerted on the AMB as function of the field strength $H$ is expressed as:

$$\tau = \frac{H^2 \mu_0}{2} \quad (6)$$

Then, by integrating this expression over the pole surface and using (4) to determine the magnetic field strength in each air gap, the force exerted along the x-direction by a pole as function of the phase current is obtained and is given by:

$$F_x = \frac{ab\mu_0(\frac{\Re_{\delta,m}}{\Re_{\delta,m}+\Re_{\delta,s}})^2 N^2 i^2}{\delta^2} + \frac{ab\mu_0(\frac{\Re_{\delta,s}}{\Re_{\delta,m}+\Re_{\delta,s}})^2 N^2 i^2}{\delta^2} \quad (7)$$

where parameter $a$ represents the width of the legs of the pole shoes as defined in Fig. 3. This equation simplifies significant if the force coefficient $k$ is introduced:

$$F_x = k \frac{i^2}{\delta^2} \quad (8)$$

Taking into account that the air gap thickness underneath the pole shoe located along the positive x-axis differs from the air gap thickness at the pole shoe along the negative x-axis and that both depend on the rotor position $x$, they are expressed as:

$$\left\{ \begin{aligned} \delta_1 &= \delta_0 - x \\ \delta_4 &= \delta_0 + x \end{aligned} \right. \quad (9)$$

With $\delta_1$ the air gap thickness at the pole shoe along the positive x-axis, and thus at phase 1, and $\delta_4$ the air gap thickness at the pole shoe along the negative x-axis, and thus at phase 4.

Then, by considering (9) and substituting the expression for the force generated by a pole in (2), the following is obtained:

$$\left\{ \begin{aligned} \frac{I_y \ddot{\bar{\bar{x}}}}{l} &= -\frac{mgl_c}{\sqrt{2}} + kl\left(\frac{i_1^2}{(\delta_0 - x)^2} - \frac{i_4^2}{(\delta_0 + x)^2}\right) \\ \frac{I_x \ddot{\bar{\bar{y}}}}{l} &= -\frac{mgl_c}{\sqrt{2}} + kl\left(\frac{i_2^2}{(\delta_0 - y)^2} - \frac{i_3^2}{(\delta_0 + y)^2}\right) \end{aligned} \right. \quad (10)$$

These equations represent the dynamic model of a current-controlled AMB. The dynamic model of the AMB along the x-axis in its state-space form is written as:

$$\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} -\frac{mgl_c l}{\sqrt{2} I_y} + \frac{kl^2}{I_y} \left( \frac{i_1^2}{(\delta_0 - x_1)^2} - \frac{i_4^2}{(\delta_0 + x_1)^2} \right) \end{bmatrix} \quad (11)$$

With:

$$\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} x \\ \dot{x} \end{bmatrix} \quad (12)$$

The state-space model along the y-axis is identical due to the symmetry of the AMB.

### III. EDDY CURRENT MODELING

#### A. Three-layer model

In this section, the analytical expression to quantify the field weakening by the motion-induced eddy currents is derived. To derive this expression, a three-layer model of the AMB is used as shown in Fig. 4. This model is obtained by unrolling the cylindrical AMB in Cartesian coordinates. Three distinct regions can be identified in the AMB model. Region I represents the stator with a relative permeability $\mu_r$. Region II represents the air gap of the AMB. Region III represents the rotor with the same relative permeability $\mu_r$ as the stator and is moving with a linear velocity $V$. At the interface between the air gap and the stator, there is a current sheet producing the magnetic fields in the AMB.

The following assumptions are made in the three-layer model:

* The magnetic material characteristics of the stator and rotor are linear and saturation is neglected

* The stator is assumed smooth

* Dimensions in the z-direction are assumed large such that variations in the z-direction can be neglected and the AMB can be modeled to extend to infinity along the z-direction

* Both the transient and steady-state control currents $i_c$ are assumed small compared to the bias current $I_b$ and are therefore neglected
<figure><img src="image_4.png" /><figcaption>Fig. 4: Three layer model of the AMB</figcaption></figure>

To obtain the analytical expression for the magnetic field in the AMB from the three-layer model, the derivation starts from the magnetic diffusion equation given by:

$$ \nabla^2 \mathbf{B} = \mu \sigma \nabla \frac{\partial \mathbf{B}}{\partial t} - \mu \sigma \nabla \times (\mathbf{V} \times \mathbf{B}) \quad (13) $$

where $\sigma$ is the conductivity and $\mu$ the permeability. All bold variables represent three-dimensional vectors. Taking into account the assumptions mentioned above, this expression can be simplified to:

$$ \left\{ \begin{array}{l} \frac{\partial^2 B_x}{\partial x^2} + \frac{\partial^2 B_x}{\partial y^2} = -\mu \sigma V_x \frac{\partial B_y}{\partial y} \\ \frac{\partial^2 B_y}{\partial x^2} + \frac{\partial^2 B_y}{\partial y^2} = \mu \sigma V_x \frac{\partial B_y}{\partial x} \end{array} \right. \quad (14) $$

These coupled partial differential equations can be uncoupled by using Maxwell's equation which states that the divergence of the magnetic flux density always equals zero:

$$ \nabla B = \frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} = 0 \quad (15) $$

By substituting (15) in (14) the following set of decoupled partial differential equations is obtained:

$$ \left\{ \begin{array}{l} \frac{\partial^2 B_x}{\partial x^2} + \frac{\partial^2 B_x}{\partial y^2} = \mu \sigma V_x \frac{\partial B_x}{\partial x} \\ \frac{\partial^2 B_y}{\partial x^2} + \frac{\partial^2 B_y}{\partial y^2} = \mu \sigma V_y \frac{\partial B_y}{\partial y} \end{array} \right. \quad (16) $$

This set of equations holds true for all three regions. However, for region I and II, the velocity $V_x$ equals zero while it is non-zero for region III since the rotor is moving with respect to the stationary reference frame.

This set of partial differential equations can be reduced to a set of ordinary differential equations by exploiting the periodicity along the x-axis. Due to the periodicity, solutions can be written as a decomposition of spatial harmonics as given by:

$$ \left\{ \begin{array}{l} B_x(x, y) = \sum_{n=-\infty}^{\infty} \tilde{B}_{x,n}(y) e^{j \frac{2\pi}{T} nx} \\ B_y(x, y) = \sum_{n=-\infty}^{\infty} \tilde{B}_{y,n}(y) e^{j \frac{2\pi}{T} nx} \end{array} \right. \quad (17) $$

With $T$ the fundamental period of the solution. In the next derivations the term $\frac{2\pi}{T}n$ will be written short hand $k_n$. By the linearity of the differential equations, they can be solved for each periodic component separately. The resulting sets of ordinary differential equations for each periodic component are given by:

$$ \left\{ \begin{array}{l} -k_n^2 \tilde{B}_{x,n}^I + \frac{d^2 \tilde{B}_{x,n}^I}{dy^2} = 0 \\ -k_n^2 \tilde{B}_{y,n}^I + \frac{d^2 \tilde{B}_{y,n}^I}{dy^2} = 0 \end{array} \right. \quad (18) $$

$$ \left\{ \begin{array}{l} -k_n^2 \tilde{B}_{x,n}^{II} + \frac{d^2 \tilde{B}_{x,n}^{II}}{dy^2} = 0 \\ -k_n^2 \tilde{B}_{y,n}^{II} + \frac{d^2 \tilde{B}_{y,n}^{II}}{dy^2} = 0 \end{array} \right. \quad (19) $$

$$ \left\{ \begin{array}{l} -k_n^2 \tilde{B}_{x,n}^{III} + \frac{d^2 \tilde{B}_{x,n}^{III}}{dy^2} = j k_n \mu_0 \mu_r \sigma V_x \tilde{B}_{x,n}^{III} \\ -k_n^2 \tilde{B}_{y,n}^{III} + \frac{d^2 \tilde{B}_{y,n}^{III}}{dy^2} = j k_n \mu_0 \mu_r \sigma V_x \tilde{B}_{y,n}^{III} \end{array} \right. \quad (20) $$

By again exploiting (15), the coefficients of the general solutions can be related to each other such that:

$$ \left\{ \begin{array}{l} \tilde{B}_{x,n}^I(y) = C_{1,n}^I e^{k_n y} + C_{2,n}^I e^{-k_n y} \\ \tilde{B}_{y,n}^I(y) = -j C_{1,n}^I e^{k_n y} + j C_{2,n}^I e^{-k_n y} \end{array} \right. \quad (21) $$

$$ \left\{ \begin{array}{l} \tilde{B}_{x,n}^{II}(y) = C_{1,n}^{II} e^{k_n y} + C_{2,n}^{II} e^{-k_n y} \\ \tilde{B}_{y,n}^{II}(y) = -j C_{1,n}^{II} e^{k_n y} + j C_{2,n}^{II} e^{-k_n y} \end{array} \right. \quad (22) $$

$$ \left\{ \begin{array}{l} \tilde{B}_{x,n}^{III}(y) = C_{1,n}^{III} e^{l_n y} + C_{2,n}^{III} e^{-l_n y} \\ \tilde{B}_{y,n}^{III}(y) = -j C_{1,n}^{III} e^{l_n y} + j C_{2,n}^{III} e^{-l_n y} \end{array} \right. \quad (23) $$

These 6 unknown coefficients for each periodic component are found by expressing the boundary conditions. Since the magnetic field should be finite at infinity, $C_{1,n}^I$ and $C_{2,n}^{III}$ are zero. The other four coefficients are found by expressing the boundary conditions at the material interfaces:

$$ \left\{ \begin{array}{l} \tilde{H}_{x,n}^I(\delta_0) - \tilde{H}_{x,n}^{II}(\delta_0) = \tilde{I}_{b,n} \\ \tilde{B}_{y,n}^I(\delta_0) = \tilde{B}_{y,n}^{II}(\delta_0) \end{array} \right. \quad (24) $$

$$ \left\{ \begin{array}{l} \tilde{H}_{x,n}^{II}(0) - \tilde{H}_{x,n}^{III}(0) = 0 \\ \tilde{B}_{y,n}^{II}(0) = \tilde{B}_{y,n}^{III}(0) \end{array} \right. \quad (25) $$

where $\tilde{I}_{b,n}$ represents the n'th periodic component of the current sheet.

## B. Fourier decomposition of the current sheet

Given the adoption of a smooth stator model, the equivalent current sheet that produces the same magnetic field strength distribution in a smooth stator as the magnetic field strength distribution generated by the bias
current in the presence of pole shoes, and with the rotor at standstill, is derived. The equivalent current sheet is derived for the nominal bias current $I_b$ of 3 A. To obtain this magnetic field distribution, the field distribution is calculated using Altair Flux. This distribution is then approximated analytically as shown in Fig. 5. Where the transition from positive field strength to negative field strength between the pole legs is approximated linearly and the leakage between two consecutive pole shoes is approximated quadratically.

<figure><img src="image_5.png" /><figcaption>Fig. 5: Comparison between FEM and analytical approximated magnetic field strength distribution</figcaption></figure>

The periodic components from this analytical obtained field distribution can be calculated, with the n'th periodic component expressed as:

$$H_{y,n}(x) = c_n e^{j \frac{2\pi x}{T}} \qquad (26)$$

Where the Fourier coefficients are calculated as:

$$c_n = \frac{1}{T} \int_T H_y(x) e^{-j \frac{2\pi x}{T}} \qquad (27)$$

Having established the Fourier decomposition of the magnetic field, the equivalent current sheet producing the magnetic field can be found. By considering that the current harmonics lag the harmonics of the mmf drop over the air gap by 90°, given the positive current direction defined in Fig. 4, the following is obtained:

$$I_{b,n}(x) = -c_n j k_n e^{jk_n x} \delta_0 = \tilde{I}_{b,n} e^{jk_n x} \qquad (28)$$

Since the expressions for the magnetic flux density in the AMB and the periodic components of the equivalent current sheet are obtained, the resulting magnetic field distribution for a given rotational speed is determined by summing all solutions.

## IV. ADAPTIVE CONTROL

Motion-induced eddy currents weaken the magnetic field in the AMB. By weakening the magnetic field in

the air gap, the force exerted on the rotor is reduced. Therefore, the motion-induced eddy currents change the dynamic behaviour of the AMB. Since the bias magnetic field is determined by the bias current, reducing the bias current will lead to reduced air gap magnetic field. Therefore, the reduction of bias current or the motion-induced eddy currents counteracting the driving magnetic field have the same effect on the original magnetic field if the average magnetic field under a leg of a pole shoe is considered. The effect of the motion-induced eddy currents can thus be incorporated in the AMB model by reducing the bias current parameter $I_b$ in the model.

By the reduced bias current and thus magnetic field, the slope at the force-current characteristic at the operating point is reduced leading to a reduced current stiffness, using the definition of current stiffness as in [11]. This will also affect the displacement stiffness [12] of the bearing resulting in degraded disturbance rejection. To adaptively control the AMB to mitigate the effects of the motion-induced eddy currents, the parameter $I_b$ in the model is estimated and based on this estimate the adaptive control adjusts the real bias current $I_{b,r}$ through the phases to counter the effects.

### A. Augmented Kalman filter

An extended Kalman filter typically only estimates the system states which are the radial position and velocity of the rotor of the AMB. To be able to estimate the parameter $I_b$, the augmented Kalman filter is used. This Kalman filter operates as the extended Kalman filter but the system state is augmented with the parameter $I_b$:

$$x_{aug} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x \\ \dot{x} \\ I_b \end{bmatrix} \qquad (29)$$

With $x_{aug}$ the augmented state. By applying the augmented Kalman filter on the discretized augmented state-space model given by (30) the parameter $I_b$ can be estimated.

$$\begin{bmatrix} x_{1,t+1} \\ x_{2,t+1} \\ I_{b,t+1} \end{bmatrix} = \begin{bmatrix} x_{2,t} + \left( -\frac{mgl_c l}{\sqrt{2}I_y} + \frac{kl^2}{I_y} \frac{(I_b+i_{c,t})^2}{(\delta_0-x_{1,t})^2} - \frac{(I_b-i_{c,t})^2}{(\delta_0+x_{1,t})^2} \right) \Delta t \\ x_{1,t} + \frac{x_{2,t} + x_{2,t} \Delta t}{I_y} \\ I_{b,t} \end{bmatrix} \qquad (30)$$

With $\Delta t$ the time step at which the algorithm is executed.

### B. Adaptive compensation

The adaptive control loop for the x-axis of the AMB is shown in Fig. 6. The currents through the phases are an input to the real dynamics of the AMB. The currents
<figure><img src="image_6.png" /><figcaption>Fig. 6: Adaptive control scheme</figcaption></figure>

in both phases are each regulated by their own current control loop towards their current reference $i_{1,ref}$ and $i_{4,ref}$. These reference values for the current comprise of the references of the real bias current $I_{b,r,ref}$ and the control current $i_{c,ref}$. The control current $i_c$ is regulated to stably levitate the rotor and the real bias current is regulated such that the AMB behaves as desired. This is achieved by using the estimated parameter of the bias current $I_b$, which is estimated by the augmented Kalman filter. When the augmented Kalman filter estimates the bias current parameter lower then the reference value of the bias current parameter $I_{b,ref}$, it indicates that eddy currents are weakening the magnetic field. To counteract the effect of the eddy currents, the reference value for the real bias current $I_{b,r,ref}$ is increased by a PI controller until the estimated bias current parameter $I_b$ again equals its reference value. Then, the dynamic behaviour of the AMB again matches the desired nominal dynamic behaviour.

## V. SIMULATIONS & EXPERIMENTAL RESULTS

This section begins by discussing the results of the analytical model obtained in Section III and comparing them with FEM simulations from Altair Flux. Next, the reduction in performance of disturbance rejection due to decreased current stiffness is verified through both simulations and experiments. Finally, the performance of the augmented Kalman filter and the adaptive control loop is simulated and experimentally validated.

### A. Simulation of magnetic field distribution

The results for the analytical model and FEM simulations are summarised in Table. II. The magnetic field distributions are given in Fig. 7 for a rotational speed of 0 RPM, 3000 RPM and 50000 RPM. The red curves represent the analytical solutions whereas the blue curves are the solutions obtained from the FEM analyses. Despite the slight underestimation of the field in the air gap by the analytical solution, the results from both

analyses are very similar. Particularly up to 3000 RPM, the results align very well. Most important, it is seen that the magnetic field inside the air gap is influenced considerably, even at a moderate rotational speed of 3000 RPM. The corresponding force on the rotor as function of the rotational speed is given in Table. III. It is seen that the force significantly reduces, such that an effect on the dynamic behaviour of the AMB is expected.

<figure><img src="image_7.png" /><figcaption>Fig. 7: Magnetic flux density distribution in the air gap for different rotational speeds</figcaption></figure>
<table><caption>TABLE II: Model parameters</caption><thead><tr><th>Parameters</th><th>Value</th></tr></thead><tbody><tr><td>δ₀[mm]</td><td>1</td></tr><tr><td>μ<sub>r</sub></td><td>1000</td></tr><tr><td>σ[1/Ω]</td><td>10<sup>7</sup></td></tr><tr><td>I<sub>b</sub>[A]</td><td>3</td></tr></tbody></table>

<table><caption>TABLE III: Force exerted by one pole for a bias current of 3A and different rotational speeds</caption><thead><tr><th></th><th>FEM force[N]</th><th>Analytical Force[N]</th></tr></thead><tbody><tr><td>0 RPM</td><td>24.3</td><td>25.8N</td></tr><tr><td>1000 RPM</td><td>18.1</td><td>17.5</td></tr><tr><td>3000 RPM</td><td>14.8</td><td>13.4</td></tr><tr><td>10000 RPM</td><td>10.8</td><td>8.7</td></tr><tr><td>25000 RPM</td><td>6.2</td><td>5.7</td></tr><tr><td>50000 RPM</td><td>4.0</td><td>3.9</td></tr></tbody></table>

### B. Impulse response under eddy current field weakening effect

The impulse response is simulated for various bias currents, representing the eddy current field weakening. This approach is used to validate the reduction in disturbance rejection performance caused by the motion-induced eddy currents. The simulation results are shown in Fig. 8a in which a bias current of 3 A represents a rotor at standstill and a bias current of 1 A a rotor rotating at 50000 RPM. It is seen that the maximum radial displacement increases with decreasing bias current and thus increasing rotational speed. The motion-induced eddy currents thus indeed decrease the disturbance rejecting performance of the AMB. The experimental measurements shown in Fig. 8b agree with the observations from the simulation.

### C. Augmented Kalman filter

This subsection simulates and experimentally verifies the performance of the augmented Kalman filter. The Kalman filter will be verified for a step in bias current from 3 A to 2 A. The simulation and experimental results are shown in Fig. 9. It is seen from the simulation result that the augmented Kalman filter is able to observe changes in the parameter $I_b$ and estimates the bias correctly to be 2 A after about 1 s. From the experimental result it is seen that initially the augmented Kalman filter accurately estimates the bias current to be 3 A. At the step, the augmented Kalman filter clearly observes a change in parameter and the estimate decreases to approximately 1.8 A. The augmented Kalman filter thus somewhat underestimated the bias current parameter at

<figure><img src="image_8.png" /><figcaption>Fig. 8: Simulated and experimental impulse response under eddy current field weakening effects</figcaption></figure>

2 A but has accurate estimation at the nominal current of 3 A.

### D. Adaptive control

The adaptive control loop is verified both by simulation and experimentally for a step in bias current from 3 A to 2 A. The results are shown in Fig. 10. It is seen that both in simulation and experimentally that the estimated bias current is effectively increased again to the nominal value of 3 A.

## VI. CONCLUSION

An analytical method to quantify the field weakening by motion-induced eddy currents in a solid-rotor AMB is proposed. This model shows good agreement with the results from FEM simulations. An adaptive control loop is proposed to mitigate the effects of these motion-induced eddy currents by means of An augmented Kalman filter which estimates the bias current parameter in the model. Based on this estimate, the real bias current is increased to strengthen the magnetic field. Simulation and experimental results demonstrate that the adaptive control loop effectively adjusts the bias current to achieve, resulting in an improved control performance.
<figure><img src="image_9.png" /><figcaption>Fig. 9: Simulation and experimental results of the augmented Kalman filter</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 10: Simulated and experimental response of the adaptive control for a step in bias current</figcaption></figure>

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