Evaluation of demodulation algorithms for robust self-sensing active magnetic bearings

Sensors and Actuators A 189 (2013) 441–450

Contents lists available at SciVerse ScienceDirect

Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Evaluation of demodulation algorithms for robust self-sensing active magnetic bearings G. van Schoor, A.C. Niemann, C.P. du Rand ∗ Department of Electrical, Electronic and Computer Engineering, North-West University, Potchefstroom, South Africa

a r t i c l e

i n f o

Article history: Received 17 August 2012 Received in revised form 24 October 2012 Accepted 24 October 2012 Available online 5 November 2012 Keywords: Self-sensing Active magnetic bearing (AMB) Amplitude modulation Demodulation Position estimation, Direct current measurement

a b s t r a c t Active magnetic bearings (AMBs) play a key role in various industrial applications. In the ongoing challenge to reduce the number of external sensing devices and manufacturing costs of AMBs, self-sensing techniques have positioned themselves in a dominant role to provide sensorless estimation of rotor displacement. A self-sensing arrangement employs an estimation algorithm that uses the modulated coil voltage and current signals to determine the air gap information. However, filters in the demodulation path of the estimator introduce additional phase-shift that results in lower achievable stability margins. Furthermore, a disadvantage of modulation self-sensing approaches is that the position estimates are nonlinearly dependent on the power amplifier voltage duty cycle. This paper firstly evaluates the static and dynamic performance of different demodulation techniques via an experimentally verified transient simulation model. The direct current measurement (DCM) approach, which comprises a minimum number of filters, is proposed for position estimation of self-sensing AMBs. The DCM algorithm incorporates a novel PA switching method that only uses the bearing coil currents as input. The estimator facilitates duty-cycle invariant position estimates with minimal additional phase-shift. According to simulated as well as experimental results, the sensitivity level of this estimator is the lowest compared to the other examined techniques. A practical implementation of the DCM approach shows that robust estimation can be realized for a 10 A magnetically coupled AMB that lends itself to industrial application. © 2012 Elsevier B.V. All rights reserved.

1. Introduction In modern industry, electromagnetic actuators provide contactless suspension of an object, rendering them key technology in applications such as active magnetic bearings (AMBs), turbomolecular vacuum pumps, turbo machines, high-speed flywheel energy storage systems, and magnetically levitated vehicles (MAGLEV) [1]. AMBs most frequently employ dedicated noncontact displacement sensors to measure rotor displacement. In the ongoing challenge to reduce the number of external sensing devices and associated wiring and interfacing, researchers and manufacturers of AMBs aim to produce compact integrated systems that are reliable and economical. Self-sensing techniques have positioned themselves in a dominant role to provide sensorless estimation of rotor displacement in AMBs. The self-sensing arrangement employs an estimation algorithm that uses the bearing coil currents and voltages to determine the air gap information. The general agreement in the literature is that self-sensing research can be grouped into two main categories [1–3]. In the first approach a classical linear time invariant

∗ Corresponding author. Tel.: +27 18 299 1978. E-mail address: [email protected] (C.P. du Rand). 0924-4247/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2012.10.033

state-observer estimates the rotor displacement by using available measurements of coil voltage and current [1]. The drawbacks of this method are low robustness, difficulty to realize feedback stabilization, and high sensitivity to parameter variations [1–4]. The second approach considers a nonlinear or linear time varying process model. Applying a periodic perturbation to the plant will result in a linear time periodic (LTP) system exhibiting improved selfsensing performance and robustness [5,6]. A large part of current research focuses on this solution for self-sensing AMBs [1–3,5–11]. Energy efficient AMBs use the switching signal of the pulsewidth modulation (PWM) power amplifier (PA) as source of periodic perturbation to reduce additional hardware requirements and manufacturing costs. The resultant amplitude modulated (AM) ripple components are then demodulated to estimate rotor displacement. Demodulation techniques inherently involve the use of different filters to isolate and manipulate the fundamental components of the bearing coil signals. These filters, however, introduce additional phase-shift that results in lower achievable stability margins [12]. Furthermore, a disadvantage of AM self-sensing is that the position estimate is a nonlinear function of the PA voltage duty cycle [13]. This paper extends the work presented in Ref. [3] and aims to address the aforementioned problem in two ways. Firstly, the performance of different demodulation techniques is evaluated, and

442

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

Fig. 1. Amplitude modulation self-sensing.

secondly, the DCM approach, which comprises a minimum number of filters, is proposed for position estimation of self-sensing AMBs. The DCM algorithm incorporates a novel PA switching method that facilitates estimates using only the bearing coil currents as input. The nonlinear dependency of the output on the voltage duty cycle is therefore removed. In Ref. [3], an estimator that includes an analogue demodulation circuit is evaluated on a low-current (1.2 A) magnetic bearing where the respective magnetic pole pairs are magnetically decoupled. The challenge is to improve estimator performance for higher current (10 A) magnetically coupled bearings, thereby increasing confidence in its suitability for industrial application. The contents of the paper are organized as follows. Section 2 gives an overview of the AM self-sensing approach, and describes the reference position estimator used in this study. Some general limitations of the analogue filter demodulation method are listed. The underlying modelling principles of different demodulation algorithms are outlined in Section 3. Section 4 reports the software models as well as simulation results for each of the algorithms listed in Section 3. The 10 A experimental AMB is presented in Section 5, together with the static and dynamic performance of the proposed direct current measurement (DCM) position estimator. Section 6 summarizes the concluding remarks and gives an outlook on future research. 2. Amplitude modulation self-sensing approach 2.1. Basic operating principle The amplitude modulation (AM) self-sensing approach extracts the rotor position information from the signals of the magnetic bearing coils. This operating principle is possible since the electrical inductance of the bearing coil, at the switching frequency, is inversely proportional to the rotor displacement [1]. The switching coil current waveform contains a high-frequency ripple component (id ) which is modulated onto the low-frequency control current (iL ). Employing demodulation techniques, the amplitude of the ripple component can be extracted to give an estimate of the rotor displacement (xe ). The modulated ripple component is obtained by either injecting a high-frequency signal, or by using the PWM switching voltage of the PA as a high-frequency source. This is the preferred configuration for industrial AMBs employing PWM amplifiers since less additional hardware is required. Fig. 1 shows a basic block diagram of the modulation self-sensing approach. One drawback of AM self-sensing is that the amplitude of the ripple component is not only a function of rotor displacement, but also depends on the switching voltage duty-cycle of the PA [13]. To compensate for the nonlinear effect of duty-cycle variation, a nonlinear observer or parameter estimator that considers the bearing

coil model is implemented [8–10], or the demodulated current is divided by the demodulated voltage [3]. 2.2. Reference approach The self-sensing estimator devised by Schammass et al. [3] is used as reference in this study. The estimation algorithm considers a simplified bearing inductor model that only uses the fundamental components (first harmonic) of the voltage and current waveforms. The model is shifted in frequency to reduce the processing bandwidth of the system. A static permeability curve compensates for the behaviour of material magnetization. Schammass et al. realized a practical implementation of the estimator using a low-current magnetically decoupled bearing test rig. The estimator showed improved robustness when compared to the state estimation selfsensing approach [3]. A simplified inductor model of a magnetic bearing coil is derived based on Fig. 2. Assuming constant permeability and neglecting nonlinear magnetic effects (i.e. hysteresis, saturation, eddy currents, fringing) as well as flux leakage, Faraday’s law of inductance states that:

v=N

d + Ri dt

(1)

where  denotes the coil voltage, N the number of coil turns,  the flux, R the coil resistance, and i the coil current. Given the magnetic circuit in Fig. 2, the total magnetic reluctance  can be expressed as:  = g + m =

2(g0 ± x) + lm /r 0 A

(2)

with g and m the air gap and magnetic material flux path reluctances respectively, g0 the nominal air gap length, x the positive or negative rotor displacement, lm the effective magnetic material path length, r the magnetic material relative permeability, 0 the permeability of free space, and A the pole face area. Assuming a

Fig. 2. Simplified magnetic bearing coil model [1].

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

443

Fig. 4. Analogue filter demodulation process.

Fig. 3. AM position estimator [3].

uniform distribution of flux throughout the magnetic circuit while applying Eq. (2), the magnetic flux can be written as: =

Ni 0 NA i. =  2(g0 ± x) + lm /r

(3)

Neglecting coil resistance and assuming that the movement of the AMB rotor is slow compared to the high frequency coil current, the air gap distance is described by substituting Eq. (3) in Eq. (1)



2(g0 ± x) =

0 N 2 A



v

di lm . − r dt

(4) 3. Demodulation algorithms

Eq. (4) indicates that the control current can be regarded as a carrier signal which is amplitude modulated by (g0 ± x). It is assumed that only the first harmonic of the voltage and current are selected by an ideal bandpass filter (BPF). Given the bandwidths of the rotor displacement and voltage duty-cycle (vd ) are lower than the amplifier switching frequency (ωs ), the low frequency model after demodulation is given by: id =





−1 lm 2(g0 ± x) + v Kωs r d

(5)

with K = 0 N2 A and id and vd the demodulated current and voltage respectively. The nonlinearity of the magnetic material is taken into account by considering an approximation of the anhysteretic magnetization curve via a quadratic equation 1/r ≈ a2 B2 + a1 B + a0 with B the magnetic flux density. Eq. (5) then translates to: id = [kx x + k2 B2 + k1 B + k0 ]vd

(6)

where kx = −2/Kωs , k2 = lm a2 /Kωs , k1 = lm a1 /Kωs , and k0 = lm a0 /Kωs , are coefficients that are empirically determined via simple experiments as described by Ref. [3]. It should be noted that depending on the nature of the nonlinearity of the magnetic material, the order of the approximating equation can be increased. An estimate of the magnetic flux density B is determined using the low-frequency control current iL and the leakage inductance Ls B=

  N 0

2(g0 ± x)

+



Ls iL . NA

(7)

Through model inversion, Eq. (6) translates to the following: xe = xg − xm =

1 kx

i  d

vd

−

[k2 B2 + k1 B + k0 ] kx

the PA, and the second filter employing approximately the same gain and phase response as the demodulation process. The second filter is required to ensure that the phase-shifts of xm and xg are the same. Compensation of magnetic permeability and duty-cycle change, as well as the magnetic flux estimation, are implemented via a digital signal processor (DSP). The ensemble in Fig. 3 shows that the performance of the estimator is highly dependent on the demodulation method. Analogue filters are temperature and noise sensitive which can influence system characteristics, limit the self-sensing algorithm bandwidth, and degrade self-sensing performance by introducing additional phase-shift that results in lower achievable stability margins. The following section presents different demodulation algorithms as well as new techniques to improve position estimation accuracy of self-sensing magnetic bearings. Compensation of the nonlinear magnetic material is realized by employing the technique discussed above.

(8)

where xe denotes the estimated position, xg the uncompensated estimated position, and xm the nonlinear magnetic material compensation term. Fig. 3 shows the configuration of the total position estimator. The estimator is composed of analogue BPFs, low-pass filters (LPFs), and demodulators (employs LPFs). The control current LPF comprises two analogue filters; filter one corresponding to the response of

3.1. Analogue filtering demodulation (Schammass’s approach) Analogue demodulation is most commonly used in self-sensing applications since it greatly simplifies implementation of the estimation algorithm on the controller. In this work, the bearing coil current and voltage are sensed using a basic current sensor and voltage divider. The analogue demodulation process is shown in Fig. 4. First, an analogue BPF removes the low frequency control current component. Next, envelope detection determines the ideal absolute value function of the fundamental component and shifts the position information to low frequencies. A LPF then selects only the low frequency baseband signal of interest. 3.2. Digital filtering demodulation The advantages of using digital filters are the ease of software programmability (i.e. changes are easy to implement), and insensitivity to temperature and noise variations. However, similar to the analogue demodulation method, the filters introduce additional phase-shift in the demodulated current envelope. This technique is a digital replica of the analogue filtering demodulation method. Fig. 5 shows a block diagram of the digital filter demodulation process. The analogue signal is passed through an anti-aliasing LPF before being digitized via a high-speed analogue-to-digital (A/D) converter. After digitization the digital filtering procedure implemented in the DSP is the same as the analogue demodulation method. Since the digital envelope detector determines the ideal absolute value function without analogue diodes, non-idealities such as distortion and additional voltage drops are eliminated.

Fig. 5. Digital filter demodulation process.

444

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

Fig. 6. Bandpass sampling demodulation method.

3.3. Bandpass sampling demodulation With the bandpass sampling demodulation method the number of filters in the demodulation path is reduced and the sampling requirements of the A/D converter are relaxed compared to conventional sampling. The self-sensing configuration is realized by sampling the bandpass filtered signal in synchronization with the carrier signal. The technique was proposed to reduce hardware complexity [14], but not fully implemented or evaluated in a magnetic bearing application. A block diagram of the bandpass sampling demodulation method is shown in Fig. 6. The modulated current and voltage signals are passed through a BPF to extract the fundamental components. Therefore, consider the filtered complex signal r(t) = rc (t) cos(ωs t) + rs (t) sin(ωs t) + r0 (t)

(10)

with T the sampling time, and k the sampling number. This is equivalent to a high pass filter (HPF). The absolute value function completes the demodulation process leaving only the carrier envelope component as required for self-sensing. The absolute value function can be mathematically included with −1k as multiplier for rk as follows: rd = −1k [rc (kT ) − rc [(k − 1)T ] + r0 (kT ) − r0 [(k − 1)T ]].

(11)

3.4. Fast Fourier transform (FFT) demodulation In the FFT demodulation method the filters in the demodulation path are removed. The fundamental components are extracted via an FFT analysis as shown in Fig. 7. The position information can then be obtained at a specific frequency. However, parameters such as sampling rate, frequency resolution, sampling length, trend removal, and biasing degrade self-sensing performance, and must therefore be chosen with care.

Fig. 7. FFT demodulation method.

The effect of under sampling leads to aliasing, and therefore, the sampling frequency must be at least double that of the maximum frequency component. Frequency resolution, which is defined as the frequency space between harmonic amplitudes, must be high enough to allow detection of the fundamental components. Detrending involves the process of obtaining and removing a linear estimation of a hidden trend in the bearing coil signals. In the selfsensing estimator, the trend of the low-frequency control current must be removed to obtain the high-frequency ripple information.

(9)

where rc (t) and rs (t) are the complex envelope components, and r0 (t) is the low frequency component with bandwidth equal to that of the BPF. The signal r0 (t) is considered an interference component since the BPF does not completely remove the low frequency component. The signal r(t) is sampled at the positive () and negative () peaks to obtain r(k), eliminating rs (t). To extract the carrier component rk , the time delayed signal r(k − 1) is subtracted from r(k), given by: rk = rc (kT ) − rc [(k − 1)T ] + r0 (kT ) − r0 [(k − 1)T ]

Fig. 8. Gradient demodulation self-sensing estimator.

3.5. Gradient demodulation Gradient demodulation offers the following advantages: firstly, no filters are employed in the demodulation path, and secondly, no voltage demodulation is required to compensate for the nonlinear effect of duty-cycle change. The proposed technique extracts the position information directly from the gradient of the current ripple component using Eq. (4). For this case, the voltage v applied to the coils is assumed to be a constant value, and therefore, the model described by Eq. (4) only requires the coil current as input. The gradient demodulation self-sensing algorithm is presented in Fig. 8. The current is sampled between 75% and 100% of the switching period to minimize the effects of eddy currents in the PA transients. The nonlinear effect of r is accounted for using the compensation method discussed in Section 2.2.

3.6. Direct current measurement demodulation The DCM approach presents the same advantages as the previous method, but is less sensitive to parameter variations (reduces estimation accuracy). Simulation as well as experimental results show that the proposed method can provide accurate and robust position estimates. DCM demodulation employs the simplified inductor bearing model documented in Section 2.2, and a novel PA switching method to compensate for the nonlinear effect of duty-cycle change on the demodulation output. This dependency is rejected by constraining the PA switching cycle to be the same each time the current ripple is measured. Controllability of the magnetic bearing, which requires a varying duty-cycle to generate dynamic forces, is met through compromise. The PA switching cycle is divided into alternative constant measurement and varying control cycles. Fig. 9 shows a graphical illustration of the proposed switching method. The optimal duty-cycle for measurement is 50% since the amplitude of the resulting triangular current waveform denotes rotor displacement [13]. This permits direct measurement of the maximum current ripple amplitude during a single PA switching cycle. The high-frequency current ripple is isolated by subtracting the average coil current (i.e. the current dc component) from the

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

445

Fig. 9. Raw coil voltage and current illustrating measurement and control cycles.

measured current during a single measurement cycle (TMC ). The maximum amplitude of the current ripple is given by: id(max) = max[i(t) − avg(i(t))] TMC

(12)

Section 4 discusses the practical implementation of Eq. (12) which presents unique challenges in terms of signal resolution. An estimate of the position can then be expressed as xe = xg − xm =

id(max) kx

−

1 [k2 B2 + k1 B + k0 ]. kx

(13)

Fig. 10 presents a diagram of the proposed estimator to highlight the fundamental differences if compared with the reference system. Since alternate switching cycles are fixed, the bearing’s maximum force slew rate is reduced. However, AM self-sensing, regardless of the estimation algorithm, benefit from limiting the voltage duty cycle to ensure sufficient excitation, thereby increasing system robustness [8,12,13]. The estimator stability is analyzed in Ref. [12] and verified on the experimental AMB. 4. Evaluation of demodulation algorithms in simulation 4.1. Reference AMB system The simulation platform is established in MATLAB® . This study considers an 8-pole heteropolar magnetically coupled radial AMB with referencing geometry shown in Fig. 11. Adjacent poles are paired in complementing polarity to form a coil, where two coils permit actuation in one degree-of-freedom (DOF). The magnetic bearing is emulated using an experimentally verified transient

Fig. 10. DCM self-sensing estimator.

Fig. 11. Geometry of an 8-pole heteropolar AMB.

simulation model (TSM). The model proposed by Noh et al. [15] is adopted which includes nonlinear effects such as magnetic hysteresis, material saturation, eddy currents, and cross-coupling. A flow diagram of the TSM is shown in Fig. 12. Details regarding the TSM modules and operating procedure are presented in Ref. [16]. Table 1 summarizes important bearing parameters that are representative of the experimental setup. 4.2. Self-sensing position estimator The block diagrams of the self-sensing estimators comprising different demodulation algorithms are shown in Fig. 13. In simulation, the position estimator permits actuation in one axis of the bearing TSM. The output of the TSM is assumed to be the real position, while the estimated position is given by the output of the self-sensing scheme. A complete description regarding each demodulator is presented in Ref. [12].

Fig. 12. Flow diagram of the TSM.

446

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

Fig. 13. Self-sensing estimator employing different demodulation algorithms. (a) analogue; (b) digital; (c) bandpass sampling; (d) FFT; (e) gradient; (f) DCM.

Referring to Fig. 13, the finite impulse response (FIR) LPF ahead of the position controller facilitates filtering of unwanted PA noise. An FIR filter is considered since a linear phase-shift for the frequency response is possible. The order of the filter is 20, with

Table 1 Bearing parameters. Symbol

Quantity

Value

fS Vp iL i0 ir(max) g0 N R L0 0 A r(max) lax rr rj rp rc rs w

PWM switching frequency Switching voltage Maximum control current Bias current Maximum current ripple Nominal air gap length Coil turns Coil resistance Nominal coil inductance Permeability of free space Pole face area Relative magnetic permeability Axial bearing length Journal inner radius Journal outer radius Stator pole radius Stator back-iron inner radius Stator outer radius Pole width

20 kHz 50 V 10 A 3A 400 mA 0.676e−3 m 50 0.2  5.2 mH 4␲ × 10−7 H/m 0.616e−3 m2 4000 44.358e−3 m 15.875e−3 m 34.95e−3 m 35.626e−3 m 60e−3 m 75e−3 m 13.89e−3 m

a cut-off frequency of 300 Hz, and a stop-band of 700 Hz. Reducing the order of the filter minimizes phase-shift, but reduces the robustness of the estimator. Each demodulation technique’s analogue module is shown in Fig. 13. The second order BPFs in Fig. 13(a)–(c) are simulated using infinite impulse response (IIR) filters with 1 MHz sampling frequency (corresponds with the TSM and experimental AMB), 20 kHz centre frequency, and bandwidth of 4 kHz. The demodulator constitutes an ideal absolute value function and a Butterworth LPF with a cutoff frequency of 1 kHz. The estimators in Fig. 13(e) and (f) implement analogue current ripple extraction circuits. Since ir represents only a small percentage of the total current range, sampling resolution will be poor if the measured current is digitized directly. Therefore, a high-speed ZOH isolates ir before digital sampling. The ripple component is then amplified by kr to the full range of the A/D converter. Table 2 presents a summary of the simulation and self-sensing parameters for stable suspension of the AMB rotor. The parameters of the inductor model are obtained using the procedure presented in Ref. [3]. Using Eq. (8), xm is approximated using a fourth order function to minimize the estimation error. An integrated method of deriving the compensation function is used where the position error due to material nonlinearity is separately measured for each demodulation technique by clamping the rotor in the zero position and varying the AMB magnetization level.

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

447

Table 2 Simulation and self-sensing parameters. Demodulation

PA bandwidth (Hz)

Position Kp

(a) Analogue (b) Digital (c) Bandpass sampling (d) FFT (e) Gradient (f) DCM

1300 1300 760 760 2500 2500

10,000 10,000 10,000 10,000 20,000 20,000

PA Kd

Kp

Ki

25 20 22 22 30 38

0.35 0.35 0.20 0.20 0.7 0.7

0.01 0.01 0.01 0.01 0.01 0.01

5. Simulation results 5.1. Static performance The static performance of the estimator is judged in terms of sensor linearity for static position disturbances. The rotor position is linearly varied from −300 ␮m to 300 ␮m under open loop conditions with a constant bias current. A comparison between the estimated position and the output of the TSM is presented in Fig. 14. Ideally, the resultant graph should be linear with a gradient of one. The bottom graph plots the error between the estimated and real position for the different demodulation algorithms (not discernible in top graph). The results presented in Fig. 14 compare favourably. A minimum error is observed for the gradient and DCM demodulation

400

desired

kx

k4

k3

k2

k1

k0

−1.50 −1.55 −1.58 −1.50 −1.02 −1.15

5.9e−4 7.9e−4 9.2e−4 9.1e−4 3.5e−4 5.6e−4

−4.8e−4 −6.7e−4 −6.7e−4 −6.1e−4 −2.3e−4 −4.7e−4

2.0e−4 2.7e−4 2.0e−4 2.2e−4 6.9e−5 1.8e−4

−3.1e−5 −3.8e−5 −3.7e−5 −1.4e−5 −2.0e−5 −9.4e−6

1.5e−3 1.5e−3 1.5e−3 1.6e−3 1.8e−4 1.0e−3

algorithms in the ±200 ␮m range. For maximum displacements, the estimated and real position differs the most since xm is determined whilst the rotor is clamped at the centre position. The proposed demodulation techniques show excellent linearity considering the nonlinear effects of the magnetic material as well as duty cycle variations. The variation in the simulated response is attributed to the variance introduced by the A/D conversion as well as the cyclic computation of filters and FFTs that are based on a varying window of data points. 5.2. Dynamic performance The dynamic performance of the estimator is evaluated through frequency response and sensitivity analyses. The frequency response ratio of the estimated (X(ω)) and real position (XTSM (ω)) is described by G(ω) = 20log[X(ω)/XTSM (ω)]. The frequency response must ideally have a magnitude of one and a phase of zero [3]. The AMB is perturbed with a 10 ␮m peak-to-peak

Magnitude [dB]

200 100 0 -100

(b) (a)

40 20 0

(c) 2

3

10

-200

10

0

-300 -400 -400

-300

-200

-100

0

100

200

300

400

Real position [µm]

(c) (a)

-100

(b) -200

2

3

10

25

(f)

10

Frequency [Hz]

(b)

20

Magnitude [dB]

(a) 15

Error [µm]

60

-20

Phase [deg]

Estimated position [µm]

300

10 5

60

20 0 -20

(d)

(d)

40

(e)

(f)

2

3

10

0

10

0

-10

(c)

(e)

-15 -400

Phase [deg]

-5

-300

-200

-100

0

100

200

300

400

Position [µm]

(f)

(d)

-100

(e) -200

2

10

3

10

Frequency [Hz] Fig. 14. Top: Estimated position linearity compared to the real position; Bottom: Static position errors for (a) analogue, (b) digital, (c) bandpass sampling, (d) FFT, (e) gradient, (f) DCM demodulation.

Fig. 15. Frequency response of estimator. (a) analogue, (b) digital, (c) bandpass sampling, (d) FFT, (e) gradient, (f) DCM.

448

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

Sensitivity [dB]

10

(b) (a) (c)

(d) (e) (f)

5

0

-5

-10

2

10

3

10

Frequency [Hz] Fig. 16. Input sensitivity plots of estimator. (a) analogue, (b) digital, (c) bandpass sampling, (d) FFT, (e) gradient, (f) DCM.

sinusoidal position reference at different frequencies. Fig. 15 shows the frequency response of the estimated position with respect to the reference output of the TSM. The results indicate that the frequency response of the estimators has zero gain and minimal phase-shift at low frequencies. The gain in magnitude is less than 20 dB at higher frequencies for the gradient and DCM algorithms. At frequencies above 100 Hz, a large phase-shift is observed for estimators (a)–(e). However, the phase remains small for the DCM algorithm at maximum frequency. This phase-shift is mainly attributed to the dynamics of the analogue ZOH circuit. Magnetic bearings are inherently unstable and require feedback control to operate in a stable equilibrium [17]. The sensitivity function evaluates the robustness of the AMB control for parameter variations and disturbance forces. However, established robustness indicators do not yet exist for self-sensing AMBs [8,11]. The analysis is performed according to ISO 14839-3 which documents the sensitivity analysis for AMBs with standard displacement sensors. The input sensitivity function Gs (ω) = E(ω)/Xref (ω) is evaluated, where E(ω) and Xref (ω) denote the frequency responses of the error and position reference respectively. The rotor is suspended with the estimated position, after which a 10 ␮m peak-to-peak sinusoidal position reference with varying frequency is applied. Fig. 16 shows the sensitivity functions when the rotor is suspended with the estimated position. The figure shows that the estimators yield peak sensitivities at low frequencies. A robustness comparison between the different demodulation algorithms reveals that DCM self-sensing produces the most robust estimates for a magnetically coupled AMB. According to the peak sensitivity zone limits for AMB machines [17], (a) and (b) are categorized in Zone C, rendering it unsatisfactory for long-term operation, (c)–(e) are acceptable for unrestricted long-term operation, and (f) is suitable for a newly commissioned industrial system. Table 3 presents a summary of the frequency response results, showing the corner frequencies where deviations in gain and phase occur, as well as the peak sensitivity for each technique.

Fig. 17. Experimental double heteropolar AMB.

frequency ripple of adequate amplitude to increase self-sensing robustness [1]. The amplifiers accommodate the estimators, controllers, as well as measurement and PA electronics. The power electronics implement two full H-bridge configurations, thereby realizing suspension of the rotor in one DOF via a single PA module. The integrated module, which is suitable for industrial application, is shown in Fig. 18. A complete description regarding the integrated electronic circuits is presented in Ref. [12]. 6.2. Experimental results The simulation results presented in Section 4 showed that DCM self-sensing are the least sensitive to disturbance forces. Therefore, this section reports the experimental evaluation of the DCM demodulation technique compared to a digital implementation of the estimator proposed by Schammass. Table 4 summarizes the simulation and self-sensing parameters for stable suspension of the AMB rotor. In the experimental setup, xm is realized using a third order function to decrease the computation complexity of the estimator. Note that although the self-sensing techniques are implemented in one DOF, the experimental AMB is fully suspended during dynamic evaluation. The results therefore include the coupling of flux between the axes of the magnetic bearing. 6.3. Static performance The static performance of the estimators is determined by varying the desired position linearly from −250 ␮m to 250 ␮m under open loop conditions with the bias current constant. The errors between the experimental estimated positions (digital and DCM) and the reference displacement sensors are plotted in Fig. 19.

6. Experimental evaluation of demodulation algorithms 6.1. Experimental system Fig. 17 represents the experimental radial AMB. The system comprises a 7.7 kg, 0.5 m flexible rotor, two magnetically coupled heteropolar bearings, and eddy-current displacement sensors to facilitate reference measurements. Compact 10 A switch-mode PAs are configured in two state (+Vp , −Vp ) in order to ensure high

Fig. 18. Integrated DCM self-sensing power amplifier module.

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

449

Table 3 Dynamic performance of demodulation algorithms. Demodulation

0 dB (Hz)

20 dB (Hz)

0◦ (Hz)

180◦ phase peak (Hz)

Sensitivity (dB)

(a) Analogue (b) Digital (c) Bandpass sampling (d) FFT (e) Gradient (f) DCM

150 150 300 150 250 100

400 400 450 400 700 700

100 100 100 100 100 300

400 400 400 600 400 50◦ @ 1 kHz

13.3 13.7 11.6 9.7 10.3 6.2

Table 4 Simulation and self-sensing parameters for experimental AMB. Demodulation

PA bandwidth (Hz)

(b) Digital (f) DCM

1300 2500

Position

PA

Kp

Kd

Kp

Ki

10, 000 10, 000

15 25

0.2 0.7

0.01 0.01

kx

k3

k2

k1

k0

7.80e−7 1.56e−7

3.5e−1 1.7e−1

4.3e−1 −3.9e−2

−1.5e−1 −1.6e−2

3.2 5.9e−1

Table 5 Dynamic performance of experimental AMB. Demodulation

0 dB (Hz)

20 dB (Hz)

0◦ (Hz)

180◦ phase peak (Hz)

Sensitivity (dB)

(b) Digital (f) DCM

300 200

550 450

100 100

450 300

16.3 10.3

Magnitude [dB]

Simulated predictions and the corresponding experimental results show that for maximum displacement the estimated positions differ less than 20 ␮m from the desired signal. Although both experimental etimators display excellent linearity, improved results are observed for the digital estimator (b). 6.4. Dynamic performance

60

(b)

40

(f)

20 0 -20 1 10

2

10

10

3

10

3

The frequency response and sensitivity functions shown in Figs. 20 and 21 describe the dynamic performance of the experimental estimators. Important performance measurements are summarized in Table 5. The test values and procedures applied in the dynamic simulations are consistent with those used in the practical setup. The results presented show that the performance of the experimental and simulated digital estimator (b) compares relatively well. The higher practical sensitivity obtained is attributed to high frequency switching noise (degrades signal-to-noise ratio) and nonlinear magnetic coupling effects. As shown, the DCM technique (f) has zero gain up to 200 Hz and an average gain of 10 dB at high frequencies. The additional phase-shift observed in the practical results is mainly attributed to

Phase [deg]

200

(b) 0

(f)

-200 1 10

2

10

Frequency [Hz] Fig. 20. Frequency responses of experimental estimator for (b) digital and (f) DCM demodulation.

non-idealities in the ZOH circuit resulting in an amplitude disturbance during the negative slope of the current. Further investigations revealed that only a 28◦ shift is associated with the self-sensing technique, and the remainder is caused by 20

15 10

15

(b)

(f)

10

Sensitivity [dB]

5

Error [µm]

(b)

0 -5

(f) -10

5 0 -5

-15 -10 -20 -15 -25 -400

10 -300

-200

-100

0

100

200

300

400

2

10

3

Frequency [Hz]

Position [µm] Fig. 19. Experimental static position errors for (b) digital and (f) DCM demodulation.

Fig. 21. Input sensitivity plots of experimental estimator for (b) digital and (f) DCM demodulation.

450

G. van Schoor et al. / Sensors and Actuators A 189 (2013) 441–450

the PID input filter [12]. The experimental sensitivity curve yields a peak value of 10.3 dB, rendering it acceptable for unrestricted longterm operation [17]. 7. Conclusion This paper evaluates different demodulation algorithms for position estimation of self-sensing AMBs. To ultimately realize industrial application, estimator robustness and accuracy are the main concerns of this work. Amplitude demodulation, which is predominantly used in self-sensing literature, requires analogue filters to isolate and manipulate the fundamental components of the bearing coil signals for position estimation. However, these filters limit system bandwidth and introduce additional phase-shift that lowers the achievable stability margin. The challenge to reduce the number of demodulation filters and compensate for nonlinear voltage duty cycle variation is addressed with the proposed DCM approach. According to the simulated and experimental results, the sensitivity level of this estimator is the lowest compared to the other examined techniques. Furthermore, the DCM algorithm contributes minimal phase-shift to the position estimates at high frequencies, thereby increasing selfsensing robustness. Given the ISO peak sensitivity zone limits for AMB machines, control using the DCM approach is satisfactory for unrestricted long-term operation. A practical implementation comprising the proposed estimator shows that accurate and robust estimates can be realized for a 10 A magnetically coupled bearing. The developed system, which includes integrated PAs and mechanically connected stator coils, facilitates a reliable and economical solution for self-sensing AMBs. However, self-sensing dynamic performance is still limited compared to dedicated displacement sensors. A comparison of the achieved practical results with the theoretical limits as proposed by Ref. [5] warrants further investigation. The results presented herein support the argument that improvements in the practical estimator’s signal-to-noise ratio and ripple extraction methodology can further enhance AMB performance. References [1] G. Schweitzer, E.H. Maslen, Magnetic Bearings: Theory, Design, and Application to Rotating Machinery, Springer-Verlag, Berlin, 2009. [2] E.H. Maslen, Self-sensing for active magnetic bearings: overview and status, in: Proceedings of the 10th International Symposium on Magnetic Bearings, Martigny, Switzerland, 2006, pp. 13–19. [3] A. Schammass, R. Herzog, P. Bühler, H. Bleuler, New results for self-sensing active magnetic bearings using modulation approach, IEEE Transactions on Control Systems Technology 13 (4) (2005) 509–516. [4] N. Morse, R. Smith, B. Paden, J. Antaki, Position sensed and self-sensing magnetic bearing configurations and associated robustness limitations, in: Proceedings of the 37th IEEE Conference on Decision and Control, vol. 3, 1998, pp. 2599–2604. [5] E.H. Maslen, D.T. Montie, T. Iwasaki, Robustness limitations in self-sensing magnetic bearings, ASME Journal of Dynamic Systems, Measurement, and Control 128 (2) (2006) 197–203.

[6] D.T. Montie, Performance limitations and self-sensing magnetic bearings, Ph.D. Thesis, University of Virginia, Virginia, USA, 2003. [7] J.-S. Yim, J.-H. Kim, S.-K. Sul, H.-J. Ahn, D.-C. Han, Sensorless position control of active magnetic bearings based on high frequency signal injection method, in: Proceedings of the 18th IEEE Applied Power Electronics Conference, vol. 1, 2003, pp. 83–88. [8] E.O. Ranft, G. van Schoor, C.P. du Rand, Self-sensing for electromagnetic actuators. Part II: Position estimation, Sensors and Actuators A 172 (2011) 410–419. [9] T. Gluck, W. Kemmetmuller, C. Tump, A. Kugi, A novel robust position estimator for self-sensing magnetic levitation systems based on least squares identification, Control Engineering Practice 19 (2011) 146–157. [10] T.M. Lim, S. Cheng, Magnetic levitation of a one DOF system using simultaneous actuation and displacement sensing technique, Mechatronics 21 (2011) 548–559. [11] R. Herzog, P. Blanc, A comparison of linear periodic and nonlinear control strategies for self-sensing magnetic bearings, in: International Symposium for Magnetic Bearings ISMB12, Wuhan, China, August, 2010. [12] A. Niemann, Self-sensing algorithms for active magnetic bearings, Ph.D. Thesis, North-West University, Potchefstroom, South Africa, 2008. [13] M.D. Noh, E.H. Maslen, Self-sensing magnetic bearings driven by a switching power amplifier, University of Virginia, Virginia, Charlottesville, VA, 1996. [14] A. Schammass, A self-sensing active magnetic bearing: modulation approach, Ph.D. dissertation, EPF, Lausanne, Switzerland, 2003. [15] M.D. Noh, D.T. Montie, E.H. Maslen, A simulation model for the analysis of transient magnetic bearing performance, in: Proceedings of the Seventh International Symposium on Magnetic bearings, 2000, pp. 177–181. [16] E.O. Ranft, An improved model for self-sensing heteropolar active magnetic bearings, Ph.D. Thesis, North-West University, Potchefstroom, South Africa, 2008. [17] ISO TC108/SC2/WG7 AMB, Mechanical vibration: vibration of rotating machinery equipped with active magnetic bearings – part 3: evaluation of stability margin, ISO Standard 14839-3, 2006, 2007.

Biographies G. van Schoor received the B.Eng., M.Eng., and D.Eng. degrees in electrical and electronic engineering from the Rand Afrikaans University (RAU), South Africa in 1985, 1987, and 2001, respectively. He worked in the power electronics industry for 9 years before joining the Program Group Engineering at Technikon SA (now UNISA), South Africa, where he managed the Electrical Engineering Program. In 2001, he was appointed professor of modelling and control at the NWU in the School of Electrical, Electronic, and Computer Engineering. He founded the Magnetic Bearing – Modelling and Control research group (McTronX) in 2005 at the Faculty of Engineering. In 2006, he became Director of the Unit for Energy Systems, NWU. In 2012, he is appointed in the position of research professor. Prof. van Schoor is a registered professional engineer with the Engineering Council of South Africa. In 2002 and 2011, he receives an award for best paper in the Transactions of the South African Institute of Electrical Engineers. His research interests are in the fields of modelling and control, magnetic bearings, and power electronics. A.C. Niemann received the B.Eng. and M.Eng. degrees in computer and electronic engineering from the North-West University (NWU), South Africa in 2003 and 2005, respectively, and his Ph.D. on the topic of self-sensing algorithms for magnetic bearings in 2008 from the NWU. From 2006, he worked as a research associate and a design engineer in the School of Electrical and Electronic Engineering at the NWU in the field of active magnetic bearings. In 2010, he was appointed to work at the South African Council for Scientific and Industrial Research (CSIR). Further interests include power electronics and control systems. C.P. du Rand received the B.Eng., M.Eng., and Ph.D. degrees in computer, electrical, and electronic engineering from the NWU, South Africa in 2002, 2004, and 2007, respectively. From 2005, he worked as a senior research associate in the McTonX research group in the fields of smart sensors and fault detection systems for dynamic processes. In 2009, he received a post-doctoral fellowship in the School of Electrical and Electronic Engineering, NWU. His research interests include smart sensor technologies, fault detection and isolation algorithms, and advanced control systems.