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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Active magnetic bearings (AMBs) have become a key technology in various industrial applications. Self-sensing AMBs provide an integrated sensorless solution for position estimation, consolidating the sensing and actuating functions into a single electromagnetic transducer. The approach aims to reduce possible hardware failure points, production costs, and system complexity. Despite these advantages, self-sensing methods must address various technical challenges to maximize the performance thereof. This paper presents the direct current measurement (DCM) approach for self-sensing AMBs, denoting the direct measurement of the current ripple component. In AMB systems, switching power amplifiers (PAs) modulate the rotor position information onto the current waveform. Demodulation self-sensing techniques then use bandpass and lowpass filters to estimate the rotor position from the voltage and current signals. However, the additional phase-shift introduced by these filters results in lower stability margins. The DCM approach utilizes a novel PA switching method that directly measures the current ripple to obtain duty-cycle invariant position estimates. Demodulation filters are largely excluded to minimize additional phase-shift in the position estimates. Basic functionality and performance of the proposed self-sensing approach are demonstrated via a transient simulation model as well as a high current (10 A) experimental system. A digital implementation of amplitude modulation self-sensing serves as a comparative estimator.

Active Magnetic Bearings (AMBs) permit frictionless suspension of the rotor through magnetic forces, rendering them a key technology for various industrial applications [

Self-sensing facilitates rotor position estimation by consolidating the sensing and actuating functions into a single electromagnetic transducer. In magnetic bearings, the stator coil electrical inductance is influenced by the displacement of the rotor within the air gap [

The general agreement in the literature is that self-sensing research can be grouped into two main categories [

Amplitude demodulation techniques inherently involve the use of band-pass (BPF) and low-pass (LPF) filters to isolate and manipulate the high frequency fundamental components (voltage and current) for position estimation [

This paper extends the work presented in [

The contents of this paper are organized as follows: Section 2 presents the underlying modeling principles of the DCM self-sensing approach. The reference transient simulation model, the high current experimental heteropolar AMB, and the practical implementations of the DCM and digital demodulation algorithms are described in Section 3. Section 4 reports the static and dynamic performance of the self-sensing sensors. Finally, Section 5 summarizes the concluding remarks.

The DCM method exploits the fact that rotor displacement is directly related to the current ripple amplitude during a switching cycle [_{0} the permeability of free space, _{g}_{r}

The DCM self-sensing approach is based on this simplified inductor model, a novel PA switching method, and the least-square algorithm proposed by [

Alternatively, this work proposes a more simplistic approach by measuring the maximum amplitude of the current ripple directly during a constant 50% duty switching cycle (discussed in next section). Due to the constant 50% duty cycle each time the current ripple amplitude is measured, the voltage in _{r}_{ge}_{m}_{r_}_{max} the maximum current ripple amplitude, _{x}_{e}_{0,1,2} empirically determined coefficients for the bearing inductor model. In _{e}_{0,1,2} realize a 2nd order estimation function to compensate for magnetic material nonlinearities. The parameters of the inductor model _{x}_{0,1,2} are determined via simple experiments as described by [_{e}_{e-}_{1} the delayed estimated position (one sample), _{0} the nominal air gap length, _{s}_{L}_{s}

In [

Alternating switching cycles are therefore fixed, thus reducing the magnetic bearing's maximum force slew rate. Modulation self-sensing, regardless of the signal-processing algorithm, benefit from limiting the voltage duty cycle to ensure sufficient excitation (

In modulation self-sensing, the high frequency current ripple is isolated by passing the measured current through an analog BPF or high-pass filter (HPF) [_{MC}_{s}_{r}_{s}

In order to facilitate a stability analysis of the self-sensing algorithm, the position estimation loop must be linearized. The nonlinear compensation function _{m}_{2}^{2} + _{1}_{0} is linearized around the nominal low-pass filtered current (_{L}_{0}) and the nominal rotor position (_{0}), given by:
_{LC}_{L}_{0} (_{L}_{L}_{0} + _{LC}_{0}. _{mi}_{mx}

Since _{d}

Rewriting

Since the self-sensing algorithm is dependent on _{ge}_{s}_{s}_{ge}

In _{mx}_{x}_{mx}_{x}_{ge}_{d}_{s}_{mx}_{x}

The poles of a heteropolar AMB are coupled magnetically through the rotor and stator back iron, as well as leakage flux [

This section presents the effects of magnetic cross-coupling due to a 50% measurement cycle. Using Faraday's law, the current in coil 1 (top actuator) is determined by:
_{1} the coil resistance, number of coil turns, and the magnetic flux in coil 1 respectively. Rewriting

_{(1,}_{n}_{)} the mutual inductance between coil 1 and coils

In

The accuracy of the self-sensing simulations is dependent on the comprehensiveness of the AMB model. An experimentally verified transient simulation model (TSM), which includes nonlinear effects such as magnetic hysteresis, material saturation, eddy currents, and cross-coupling, is adopted to emulate the experimental system. The flow diagram of the TSM implemented in simulation is shown in

In

The DCM self-sensing approach is evaluated via an 8-pole heteropolar AMB with referencing geometry shown in _{p}_{p}

A compact integrated PA is designed in-house. The system accommodates the self-sensing scheme, position and current controllers, as well as the measurement and PA electronics. The power electronics implement two full H-bridge configurations, thereby realizing suspension of the AMB rotor in one DOF via a single PA module. The integrated system is shown in

In practice, PA switching noise degrades the signal-to-noise ratio, which makes direct application of _{r}_{s}_{r}

The ZOH isolates the working point current (_{w}_{w}_{s}_{r}_{r}_{r}_{s}_{r}i_{r}_{r}i_{r}_{r}_{_max}. The estimated position is subtracted from the reference position to produce a position error, which is fed to the position controller. The current controller then generates the appropriate correction signal for the amplifiers using the control error. A low order finite impulse response (FIR) filter is implemented after the A/D converter to reduce the high frequency switching noise. The filter does introduce some unwanted phase-shift, but the cut-off frequency is chosen well beyond the sensor bandwidth at 20 kHz. An FIR filter is considered since a linear phase-shift for the frequency response is possible. Furthermore, classical position and current controllers are used to achieve stable suspension of the experimental AMB rotor, thereby demonstrating its feasibility for industrial application.

The static performance of the position estimators are judged in terms of sensor linearity for static position disturbances and currents. The desired position is linearly varied from −250 μm to 250 μm under open loop conditions with a constant bias current of 3 A.

The results presented in

Ideally, the frequency response must have a magnitude of one and a phase of zero [

Magnetic bearings are inherently unstable and require feedback control to operate in a stable equilibrium [

The experimental curve yields a peak sensitivity of 10.3 dB for DCM self-sensing. According to the peak sensitivity zone limits [

Magnetic cross-coupling has the potential to significantly degrade self-sensing performance [

This work presents the DCM approach for self-sensing AMBs. The proposed method is realized via a compact integrated PA that facilitates stable suspension of the experimental AMB rotor in one DOF. Position estimation is accomplished using only the measured current ripple of the sensing bearing coil. A novel switching method is proposed to reduce nonlinear modulation effects associated with voltage duty cycle change. The results indicate that phase-shift introduced by demodulation filters greatly influences self-sensing stability and bandwidth. The DCM approach employs minimal filtering in the demodulation path of the estimator, thereby minimizing additional phase-shift in the position estimates.

The DCM self-sensing AMB is evaluated in terms of static and dynamic performance. The linearity results show good agreement between the reference and estimated rotor displacement. In addition, the simulated and experimental gain of the DCM estimator compare favorably. However, some discrepancies are observed at high frequencies, which are mainly attributed to the unmodeled dynamics of the current ripple extraction circuit, as well as the high frequency switching noise in the experimental system. Although the improvements observed in the practical results are limited, the simulated results clearly highlight the performance advantages of the proposed method. Evaluation of the sensitivity function indicates that the robustness of AMB control using DCM self-sensing is satisfactory for unrestricted long-term operation. The proposed switching method minimizes the influence of magnetic cross-coupling on the position estimates without mechanical separation of the bearing coils, thereby reducing manufacturing costs.

The high current practical implementation of the DCM method for AMB control demonstrates feasibility for industrial application. However, self-sensing AMB dynamic performance is still limited compared to dedicated position sensors due to the duty cycle limitation imposed. Future directions of research will aim to improve the current ripple extraction methodology (eliminating the analog ZOH phase effect), as well as digital signal processing that enhance the signal-to-noise ratio of the practical estimator.

The authors declare no conflict of interest.

Simplified electromagnetic actuator.

Modulation self-sensing algorithm.

DCM self-sensing approach.

Raw coil voltage and current showing measurement and control cycles.

Linearized block diagram (

Flow diagram of the transient simulation model.

Geometry of an 8-pole heteropolar magnetic bearing.

Experimental double heteropolar AMB.

Integrated self-sensing power amplifier module.

Configuration of the practical DCM position estimator.

Configuration of the digital modulation position estimator.

Simulated and experimental static position errors. Simulated: (a) modulation, (b) DCM; Experimental: (c) modulation, (d) DCM.

Frequency response of the simulated and experimental position estimators. Simulated: (a) modulation, (b) DCM; Experimental: (c) modulation, (d) DCM.

Simulated and experimental input sensitivity functions. Simulated: (a) modulation, (b) DCM; Experimental: (c) modulation, (d) DCM.

FFT position plots illustrating magnetic cross-coupling effects.

Experimental magnetic bearing and self-sensing parameters.

_{S} |
PWM switching frequency | 20 kHz |

_{p} |
Switching voltage | 50 V |

_{L} |
Maximum control current | 10 A |

_{0} |
Bias current | 3 A |

_{r_}_{max} |
Maximum current ripple | 400 mA |

_{0} |
Nominal air gap length | 0.676e−3 m |

Coil turns | 50 | |

Coil resistance | 0.2 Ω | |

_{0} |
Nominal coil inductance | 5.2 mH |

_{0} |
Permeability of free space | 4π × 10^{−7} H/m |

Pole face area | 0.616e−3 m^{2} | |

_{r_}_{max} |
Relative magnetic permeability | 4,000 |

_{LPF} |
LPF cutoff frequency | 5 kHz |

_{ax} |
Axial bearing length | 44.358e−3 m |

_{r} |
Journal inner radius | 15.875e−3 m |

_{j} |
Journal outer radius | 34.95e−3 m |

_{p} |
Stator pole radius | 35.626e−3 m |

_{c} |
Stator back-iron inner radius | 60e−3 m |

_{s} |
Stator outer radius | 75e−3 m |

Pole width | 13.89e−3 m | |

_{P} |
Proportional constant (position controller) | 10,000 |

_{D} |
Derivative constant (position controller) | 25 |

_{P} |
Proportional constant (PA controller) | 0.7 |

_{I} |
Integral constant (PA controller) | 0.01 |

_{x} |
Conversion constant | 156.25e−9 A/m |

Summary of self-sensing dynamic performance.

Simulation | |||||

(a) Modulation | 150 | 400 | 50 | 400 | 13.7 |

(b) DCM | 100 | 750 | 400 | 50° @ 1 kHz | 6.2 |

Experimental | |||||

(c) Modulation | 300 | 550 | 100 | 450 | 16.3 |

(d) DCM | 200 | 450 | 100 | 400 | 10.3 |