Measurement method to determine the magnetic hysteresis effects of reluctance actuators by evaluation of the force and flux variation N.H. Vrijsen,a) J.W. Jansen, J.C. Compter, and E.A. Lomonova Eindhoven University of Technology, Electromechanics and Power Electronics, Department of Electrical Engineering, Eindhoven, The Netherlands. (Dated: 10 July 2012)
A measuring method is presented, which identifies the magnetic hysteresis effects occurring in linear reluctance actuators. The measuring method is applicable for many reluctance actuators, however in this paper the method is evaluated on an E-core reluctance actuator. The goal is to measure the effect of magnetic hysteresis on the force of the actuator. The force measurements are conducted with a piezoelectric load cell (Kistler type 9272). The high-bandwidth force measurement instrument (FMI) is calibrated with various voice-coil actuators (VCAs). The phase delay obtained from the calibration step is validated by the comparison of the measured force and flux variation of the E-core actuator. The measured flux variation in combination with the measured current is used to distinguish the phase delay between the current and the force of the E-core actuator, which occurs due to magnetic hysteresis in the soft-magnetic material. Finally, an open loop steady state AC model is presented that predicts the magnetic hysteresis effects in the force of the E-core actuator. I.
INTRODUCTION
Accurate force prediction of electro-mechanic actuators is a major issue for high-precision applications. For example in the semiconductor lithography the high accuracies are required to operate in the nanometer range. Simultaneously, the desired bandwidth of high-precision actuators is increasing. Because of the accuracy, usually highly linear VCAs1–4 are applied instead of nonlinear reluctance actuators5–15 . While, reluctance actuators are able to achieve a more than 10 times higher force density than VCAs16 . Up to now, the major restriction to apply reluctance actuators for high-precision applications is the nonlinear magnetic hysteresis introduced by the soft-magnetic material. Usually, a linear current-force relation is desired. However, ferromagnetic hysteresis results in a nonlinear, history dependent and rate-dependent relation between the current and the magnetic flux. This flux is directly related to the force and hence, the current-force relation is highly nonlinear. Research on ferromagnetic hysteresis and modeling of these effects has been done for years17–19 . These contributions are usually focused on the mathematical models predicting magnetic hysteresis in soft-magnetic toroids or strips. Some experiments are performed on reluctance actuators13,20 . However, these results are obtained for relatively low-frequent excitations and without prebiasing permanent magnet (PM). In this paper a laminated E-core reluctance actuator with and without pre-biasing PM is evaluated on the magnetic hysteresis effects present in the force. The proposed measurement method determines the phase delay between the current and the force, by analysis of the change of the magnetic flux21,22 . The combination of
a) Electronic
mail:
[email protected].
Four VCAs
E-core actuator
Force sensor
Currentforce relation
FMI response
Force sensor
Currentforce relation
FMI+ E-core response
Flux sensor
Currentflux relation
FMI response
E-core response
FIG. 1. Block diagram to determine the proposed measurement method.
these measurements shows the feasibility to distinguish magnetic hysteresis effects in the force of an E-core reluctance actuator. The steady state AC measurements are performed over a frequency range between [40-480] Hz. The steps to determine the measurement method as described in this paper are summarized by the block diagram in Fig. 1. First, the piezoelectric FMI is calibrated with four VCAs. Thereafter, the E-core reluctance actuator is identified and a non-hysteretic analytical model - based on a magnetic equivalent circuit (MEC)28 - is presented. This model describes the E-core actuator, with and without PM at the middle tooth. The force and flux variation are measured, the results are examined and the magnetic hysteresis effects in the force are identified. Finally, an open-loop hysteresis model is proposed that predicts the magnetic hysteresis effects of the E-core actuator for a steady state AC excitation.
II. FORCE MEASUREMENT INSTRUMENT (FMI) IDENTIFICATION
The FMI used to perform dynamic force measurements on the E-core reluctance actuator, consists of a piezoelectric load cell, a aluminum frame around it and a charge
2 amplifier. A schematic illustration of the FMI is shown in Fig. 2. The FMI is first calibrate with four VCAs, for which magnetic hysteresis is negligible. However, eddy currents are significantly present in some of the PMs. The high-frequent force measurements are conducted with a piezoelectric load cell (Kistler type 9272). This load cell is chosen, because it has a high rigidity and hence, a high natural frequency23,24 . Similar load cells are regularly applied for force and torque measurements of drilling and cutting machines due to the linearity over a large amplitude range25,26 . Other common used passive load cells are the strain gage27 and hydrostatic sensor. However, both have a lower strain sensitivity (factor 1000) and a lower rigidity23 , which results in a larger airgap variation, because of elasticity. Disadvantages of piezoelectric load cells are the inability to measure statically over a longer period of time and the hysteresis present in the piezoelectric cell. Therefore, only steady state AC measurements are evaluated in this paper.
A.
Voice-coil configurations
Four VCAs are examined for the calibration of the FMI. The difference between the four VCAs is the applied PM, while the same excitation coil is used. These PMs are investigated on the amount of eddy currents, which introduce an extra phase delay in the current-force relation. The eddy currents in the coil of the applied VCAs can be neglected, because the copper wire diameter is 0.2 mm and the skin depth is 2 mm at 1 kHz. The schematic cross-section view of the first VCA is shown in Fig. 3(a). It is an axisymmetric coil-PM configuration with the coil fixed to a plastic holder, which is mounted to the load cell. The PM segments are glued to the back-iron, which is mounted to the frame cover of the FMI. The coil is placed between the radial magnetized PM segments. This VCA is able to apply a bi-directional force of approximately ± 15 N. The other three PMs are shown in Fig. 4 and the specifications are given in Table I. Figure 3(b) shows the schematic cross-section view that represents both toroidal PMs as shown in Fig. 4(a) and (b), which are Vacodym 510 HR (NdFeB) and ferrite (FeO) magnets, respectively. Figure 3(c) represents the cross-section of the square PM as shown in Fig. 4(c), which is from plastic bonded Samarium-Cobalt (SmCo). All three PMs of Fig. 4 are magnetized in the z-direction as shown in Fig. 3(b) and (c).
Current U , I in in amplifier
Toroidal PM
Cilindrical coil Fz Signal generator
Kistler 5070 UF charge amplifier
Kistler 9272 load cell Frame
FIG. 2. Measurement setup with the FMI (load cell, frame and charge amplifier) and the VCA with the Ferrite magnet.
Axisymmetric Back Coil iron
d
Cartesian PM
PM w
z
PM segments
z
h
h
w Coil z
Coil
y r
r
(a)
x
(b)
(c)
FIG. 3. Cross-sections view of the three VCAs (a) radial magnetized axisymmetric VCA configuration, (b) axisymmetric VCA with toroidal PM magnetized in the z-direction and (c) VCA with rectangular PM magnetized in the z-direction.
(a)
(b)
(c)
FIG. 4. Three PMs (a) Vacodym 510 HR (NdFeB) toroid shape (Fig. 3(b)), (b) ferrite (FeO) toroid shape (Fig. 3(b)) and (c) plastic bonded Samarium-Cobalt(SmCo) square shape (Fig. 3(c)).
TABLE I. Specifications of the three PMs (Fig. 4) B.
Frequency response measurement
The FMI is calibrated by the evaluation of the measured frequency response between the current Iin - and the force Fz -. The measured bode diagram of the FMI is obtained for a frequency range between 0 - 45 kHz as shown in Fig. 5. Figure 5(a) shows the measured amplitude response of the four PM configurations. The ampli-
Parameter NdFeB Ferrite SmCo Unit Height (h) 16 11 14.85 mm Width / outer diameter (w) 47.9 54 46 mm Depth / inner diameter (d) 11.9 24 42.9 mm Dipole moment (mz ) 25.27 3.86 10.11 Am2 Magnetic polarization (J) 1.17 0.24 0.43 T Electric resistivity (ρ) 1.4·10−6 1-100 7·10−7 Ωm
60
180
40
90 Phase [°]
20 0 -20
-60
0 -90
-180
-40 1
2
3
4
-270
10 10 10 10 Frequency (Hz)
1
2
3
4
10 10 10 10 Frequency (Hz)
(a)
(b) Radial magnetized NdFeB magnet SmCo magnet
10
Ferrite magnet Instrument resp.
0 Phase [°]
tude of the radial magnetized configuration is higher than the others, because the magnetic flux density in the coil cross-section is higher in this configuration. The measured natural resonance frequency of the four topologies is similar for each topology. The natural frequency is the response of the FMI including the load cell, the frame and the VCA. In addition to the resonance peak in the amplitude response, the natural resonance of the FMI results in a phase delay in the phase response. The phase response of the various PM configurations is shown in Fig. 5(b) and (c). Figure 5(c) shows the zoomed-in phase response for frequencies between 100-3000 Hz. This shows that an extra phase delay occurs for the cylindrical coil with radial magnetization, for the SmCo PM and for the NdFeB PM. This extra phase delay is a result of induced eddy currents in the PMs and for the radial magnetized configuration eddy currents are also induced in the back-iron. In a ferrite PM no eddy currents are present, because of the relatively high electrical resistivity of the material, 1-100 Ωm29 . Hence, no extra phase shift is measured for the ferrite PM. Therefore, the measured phase response with the ferrite PM is selected as a benchmark for the frequency response of the FMI.
Amplitude [dB]
3
−10 −20
C.
Frequency response model of the FMI
−30
The measured frequency response of the FMI is modeled for the prediction of the phase delay in further force measurements. The model of the measured response is split into two contributions, which are represented by a cascade of two second-order low-pass filters (LPFs) as shown in Fig. 6. The first contribution is from the mechanical system including the load cell, the frame and the actuator, for which the natural resonance is used as identification. The second contribution is the LPF of the charge amplifier, which introduces an extra phase delay for the evaluated frequency range. The input of the two filters is U (s) and the outputs of the mechanical system and the charge amplifier are Y1 (s) and Y2 (s), respectively. The phase response resulting from both LPFs is shown in Fig. 7, over the frequencies between 100-3000 Hz. The frequency response of the mechanical system in the Laplace domain is given by Hmech (s) =
Y1 (s) ωn2 = 2ζ U (s) s2 + ωnn s + ωn2
in which
√ ωn = 2πfn = c ζn = √ 2 km
(1)
500 1000 1500 2000 2500 3000 Frequency (Hz)
(c) FIG. 5. Bode diagram of the FMI for the four VCA configurations and the modeled instrument response. (a) The amplitude response, (b) the phase response and (c) the magnified (zoomed) phase response.
With m- the moving mass, k- the spring constant and cthe damping coefficient. The second-order frequency response of the charge amplifier is modeled by a cascade of two first order LPF’s, represented in the Laplace domain as HLP F (s) =
Y2 (s) ω02 = 0 2 Y1 (s) s2 + 2ζ ω0 s + ω0
(3)
where 1 R1 R2 C1 C2 R1 C1 + R2 C2 ζ0 = √ 2 R1 R2 C1 C2
ω0 = 2πf0 = √
k m (2)
From the measurements with the ferrite PM, the undamped natural resonance frequency is obtained as, fn = 4.45 kHz and the damping ratio ζn = 0.012.
(4)
with the cut-off frequency, f0 = 31 kHz determined by the phase response measurement of the VCA with the ferrite magnet. R1 , R2 , C1 and C2 are the filter resistances and capacitances as shown in Fig. 6. Satisfying, R1 C1 = R2 C2 the filter is designed such that the damping ratio, ζ0 = 1. The modeled phase response of the
4 Y1 (s)
U (s) Mechanical system
m k
Y2 (s)
Mover
Low-pass filter
c
R1
R2
C1
C2
Magnet
Uc
Coil E-core
FIG. 6. The low-pass filter model describing the frequency response of the FMI.
FIG. 8. A schematic illustration of the E-core reluctance actuator.
10 F = Ni
5
PLeak
PLeak
Phase [°]
0 −5
P4 P3 Pair P1 P2
Pair
P1
P2
P3
−10 −15
Mechanical resp. − Hmech(s)
−20
Low−pass filter − HLPF(s)
−25 −30
FIG. 9. Airgap and leakage permeances as applied in the MEC model.
Instrument resp. 500
1000 1500 2000 Frequency (Hz)
2500
with L- the inductance and z- the length of the airgap. The inductance is dependent on the magnetic circuit, which is described by the flux linkage λ as
3000
FIG. 7. The modeled phase response of the FMI.
L(i, z) = FMI is shown in Fig. 7 together with the phase response of each LPF. This figure shows that the phase delay resulting from the mechanical natural resonance is much less than the phase delay from the LPF of the charge amplifier. However, for frequencies larger than 4.3 kHz the phase delay from the mechanical system becomes dominant. The FMI response is also compared to the measurements in Fig. 5.
III.
PT m P4
E-CORE RELUCTANCE ACTUATOR MODEL
A non-hysteretic actuator model is presented, which is used to evaluate the performed measurements on the E-core actuator in section V and VI. A schematic crosssection view of the E-core reluctance actuator is shown in Fig. 8. The E-core is manufactured from laminated SiFe sheets. The PM is placed on the middle tooth, which is shorter than the outer teeth. The tooth area of the middle tooth is twice the area of each outer teeth. The implemented non-hysteretic analytical actuator model is based on magnetic equivalent circuits (MEC). The airgap of the actuator is dominant considering a MEC model and hence, the iron is assumed infinitely permeable. In this case, the force produced by a reluctance actuator can be described as F (z) =
1 2 dL i 2 dz
(5)
dλ(i, z) N1 dϕ(i, z) = di di
(6)
and for infinitely permeable iron the inductance is obtained as L(z) =
λ(z) N1 ϕ(z) = = N12 Ptot (z) i i
(7)
with N1 - the number of turns of the excitation coil and 1 P = R is the permeance of the total magnetic circuit. The airgap permeances for every tooth are obtained including fringing effects28 as visualized in Fig. 9. With this representation the permeance of the left tooth PT l and right tooth PT r - is given by PT l = PT r = Pair + P1 + P2 + P3 + P4
(8)
The permeance of the middle tooth is similarly obtained. The force without the PM is calculated with Eq. (5). Including the PM the total force is given by F (z) =
dϕpm dϕpm 1 2 dL 1 i + Hc lpm + Ni 2 dz 2 dz dz
(9)
with lpm the PM length, Hc the coercive field strength of the PM and the flux generated by the PM described as ϕpm =
Hc lpm lpm µr
+g µ0 2·tw·ad
+
(10)
1 PT l +PT r
where µr - is the relative permeability of the iron, g- the airgap length, tw- and ad- are the width of the tooth and
5 Iin , Current Uin amplifier
E-core actuator
Integrator Uc
Uflux
400
Fz
Fz (N)
300
Signal generator
200
Kistler 9272 load cell Frame
100 0 0 −2
0.5 1 Aigrap (mm) 1.5
2
0 Current (A)
FIG. 10. Calculated force with variable current and variable airgap.
the depth of the actuator, respectively. This completes the non-hysteretic MEC model of the E-core reluctance actuator, which is applied to obtain the force error due to magnetic hysteresis in the following sections. The calculated force related to the airgap and current is shown in Fig. 10.
IV. FORCE MEASUREMENT METHOD TO IDENTIFY MAGNETIC HYSTERESIS EFFECTS
A measuring method is proposed, which identifies the phase delay in the force due to magnetic hysteresis. In section II the phase delay of the FMI was identified with VCAs. With reluctance actuators a similar phase delay is measured (between the current and the force). However, now this phase delay consists of two components. Firstly, from the FMI as earlier identified with the VCAs and secondly due to magnetic hysteresis and eddy currents in the soft-magnetic E-core actuator. Therefore, an extra verification step is introduced to check if the phase delay measured for the E-core actuator corresponds with the phase delay of the VCAs. The E-core actuator is excited with a sinusoidal current, while the force is measured. To verify the phase delay of the FMI, the variation of the magnetic flux through one tooth is measured, as shown in the diagram of Fig. 1. From the Maxwell stress tensor method12 can be obtained that the force is directly related to the squared magnetic flux density. Hence, no magnetic hysteresis is present between the magnetic flux density and the force. Therefore, the measured phase delay between the force and the variation of the flux is the delay resulting from the FMI. The variation of the magnetic flux density in the actuator tooth is measured with a sensing coil around one of the outer teeth of the actuator as illustrated in Fig. 8.
Kistler 5070 UF charge amplifier
FIG. 11. Measurement setup with the FMI (load cell, frame and charge amplifier) and the E-core reluctance actuator mounted on it.
The measured flux is assumed to be linearly related to the total flux that contributes to the force. The voltage on the sensing coil, with N2 = 10 windings, is integrated with an integrator circuit, as shown in Fig. 11. A lownoise high-bandwidth operational amplifier (NE5534) is used to integrate the measured voltage to obtain the magnetic flux Φ- through the tooth. The sensing voltage can be expressed as Uc = −N2
∆Φ ∆t
(11)
Only the time delay of the measured magnetic flux density with respect to the force is considered to obtain the phase delay of the FMI. The amplitude of the measured flux variation and is not used to predict the force amplitude. The evaluation of the phase delay is obtained by the maximum of the covariance function σXY of the force Fz - and the squared flux Φ2 -. The general covariance function is σXY = cov(X, Y ) =
N 1 ∑ (xi − x)(yi − y) N i=1
(12)
where X and Y are substituted by the force Fz - and one period of the flux Φ-, respectively. For the actuator without PM this becomes σF,Φ2 and for the actuator with PM this results in σF,Φ . Due to the offset of the flux for the E-core with PM there is a unidirectional flux and hence, the flux variation is directly related to the force. While for the actuator without PM where the flux is bi-directional and the force is unidirectional. From the maximum of the covariance the lag in time samples is obtained. Then the phase delay is calculated with the sample time and the excitation frequency. In Fig. 12 the results of all the force and flux variation measurements are compared to the frequency response of the voice-coil measurements and its model. The measured phase delay of the E-core actuator is shown for the frequencies (40,80,160,320,480,640,1280) Hz for various current amplitudes. The measurement results shown in next sections are for a peak-to-peak current of approximately 1.3 A-pp and for a frequency range of 40-320 Hz.
6 Ferrite magnet
40 Hz, meas.
40 Hz, meas.
Instrument resp.
80 Hz, meas.
80 Hz, meas.
Phase angle between F−Φ
160 Hz, meas.
160 Hz, meas.
5
320 Hz, meas.
200
4
320 Hz, meas. 0.2
Analyt. model
−100
0
0.1
3
−5
∆ F (N)
0
Force (N)
Phase [°]
Phase [°]
100
2 1
−200 1 10
2
3
10 10 Frequency (Hz)
4
10
−10 1 10
2
0 −1
(b)
FIG. 12. (a) The measured and modeled phase delay of the FMI (Fig. 5 and 7) and ◦- the measured phase delay between the force and flux, (b) magnified (zoomed) between 10-2000 Hz.
−0.5 0 0.5 Current (A)
1
−0.2 −1
(a)
−0.5 0 0.5 Current (A)
1
(b)
FIG. 13. E-core actuator without PM. Measured current (a) versus the measured and modeled force Fz -, (b) versus the force error ∆F - as defined by Eq. (13).
MEASUREMENT RESULTS E-CORE WITHOUT PM
∆F = Fmeasured − Fanalytical
(13)
The loops are a result of two phase delays, namely due to the FMI and due to magnetic hysteresis effects in the actuator. To distinguish both phase delays, the measured squared magnetic flux density is plotted against the measured force in Fig. 14(a). Figure 14(b) shows the modeled linear relation subtracted from the measured force. The resulting loops are solely due to the delay of the FMI, because no magnetic hysteresis is present between the squared magnetic flux density B 2 - and the force Fz -. To distinguish magnetic hysteresis effects from the measurements, the phase delay of the FMI must be compensated. Figure 15(a) shows the (B 2 − F ) relation with and without a phase correction of 1.52◦ on the force for a frequency of 320 Hz. The force error ∆F - present after the phase correction of the force is depicted in Fig. 15(b). This phase correction is obtained with the covariance as given in Eq. (12). The error for larger magnetic flux densities is a result of a DC-error of the force measurement,
40 Hz, meas.
40 Hz, meas.
80 Hz, meas.
80 Hz, meas.
160 Hz, meas. 4
160 Hz, meas. 0.2
320 Hz, meas.
2 1 0
320 Hz, meas.
0.1
3 ∆ F (N)
In spite that the E-core is designed to be used with PM, the proposed measurement method is first evaluated for the E-core without PM. In this case is the middle tooth of the actuator shorter than the outer tooth, which is not optimal due to a large effective airgap. The E-core actuator is examined with a constant airgap and excited with a sinusoidal current with a peak-to-peak value of 1.27 A. Figure 13(a) shows the measured current force relation compared to the non-hysteretic analytical actuator model presented in section III. The force error ∆F - between the analytical actuator model and the measurement is shown in Fig. 13(b) and is defined as
Force (N)
V.
−0.1
3
10 10 Frequency (Hz)
(a)
0
0
−0.1
0
0.01 B2 (T2)
(a)
0.02
−0.2
0
0.01
0.02
B2 (T2)
(b)
FIG. 14. Measured squared magnetic flux density (B 2 ) (a) versus the force Fz -, (b) versus the force error ∆F -.
which is also shown in Fig. 13(b). The same phase correction as applied in Fig. 15 is applied for the current-force relation, as shown in Fig. 16(a). The error that remains after the phase compensation of the force sensor is shown in Fig. 16(b). Now can be stated that the loop after compensation is a result of the magnetic hysteresis and eddy currents in the iron actuator core. The largest remaining force error ∆F - for an excitation of 320 Hz, is 1.23 % of the total force. The force error is smaller for excitations with lower frequencies.
7 320 Hz, meas.
40 Hz, meas.
40 Hz, meas.
320 Hz, sens. comp.
320 Hz, sens. comp.
80 Hz, meas.
80 Hz, meas.
160 Hz, meas.
160 Hz, meas.
0.2 0.1
2
300
0
1
−0.1
0
−0.2
250
320 Hz, meas.
480 Hz, meas. Analyt. model
200 Force (N)
∆ F (N)
3 Force (N)
320 Hz, meas.
∆ F (N)
4
320 Hz, meas.
150
0.01
0.02
0
B2 (T2)
0.01
0.02
100
B2 (T2)
(a)
50 −1
(b)
FIG. 15. Measurement with and without phase shift compensation at 320 Hz of the squared magnetic flux density B 2 - (a) versus the force Fz - and (b) versus the force error ∆F -.
320 Hz, meas. 0.2
Analyt. model
∆ F (N)
−0.1
0 −1
−0.2 −1
300
Analyt. model
250 200 150
−0.5 0 0.5 Current (A)
(a)
1
−0.5 0 0.5 Current (A)
1
(b)
FIG. 16. Measurement with and without phase shift compensation at 320 Hz of the current (a) versus the force Fz - and (b) versus th force error ∆F -.
VI.
320 Hz, meas.
320 Hz, sens. comp.
0
1
1
320 Hz, sens. comp.
0.1
2
−0.5 0 0.5 Current (A)
(b)
320 Hz, meas.
3 Force (N)
(a)
Force (N)
4
1
480 Hz, meas.
FIG. 17. E-core actuator with PM. Measured current versus the (a) force Fz - and (b) force error ∆F - as defined by Eq. (13).
320 Hz, meas.
320 Hz, sens. comp.
−0.5 0 0.5 Current (A)
∆ F (N)
0
5 4 3 2 1 0 −1 −2 −3 −4 −5 −1
MEASUREMENT RESULTS E-CORE WITH PM
The measurement method is now evaluated for the Ecore with a PM placed on the middle tooth. Pre-biasing the magnetic flux density with the PM, increases the force range from (0-3.3) N up to a force range of (125220) N, with a similar current excitation. Besides the higher force range with the PM, a disadvantage could be an increase of eddy currents, because the PM is not laminated. For this actuator, sinusoidal excitations of (40,80,160,320,480) Hz are investigated with a peak-topeak current of 0.25, 0.5, 1 and 1.4 A-pp. Only the measurements with a peak-to-peak current of 1.4 A are shown in this section, because the magnetic hysteresis and eddy current effects are better distinguishable for larger excitations. Figure 17(a) shows the nonlinear current-force relation. Fig. 17(b) shows the error between the measured force
100 50 −1
−0.5 0 0.5 Current (A)
(a)
1
5 4 3 2 1 0 −1 −2 −3 −4 −5 −1
320 Hz, sens. comp.
−0.5 0 0.5 Current (A)
1
(b)
FIG. 18. Measured current versus the (a) force Fz - and (b) force error ∆F -, with and without phase shift compensation at 320 Hz.
and the non-hysteretic analytic actuator model. Similar as in previous section, the elliptic shapes are a result of two phase delays. The compensation for the phase delay of the FMI for a frequency of 320 Hz, is shown in Fig. 18. The compensated phase shift at 320 Hz is 1.52◦ , as obtained from the covariance of the measured force and flux variation. The largest remaining force error between the analytic and measured force, ∆F , for an excitation of 320 Hz, is 0.63 % of the total force. The force error due to magnetic hysteresis and eddy currents - after compensation for the FMI - for all evaluated frequencies is shown in Fig. 19. The maximum force error occurs approximately for a current of 0 A and hence, the maximum force error is defined as the height of the loop-eye for a current of 0 A. Figure 20 shows the percentages of force error divided by the force range for
8
1 40 Hz, sens. comp. 0.5
80 Hz, sens. comp. 160 Hz, sens. comp. 320 Hz, sens. comp. 480 Hz, sens. comp.
40 Hz B (T)
∆ F (N)
1.5 5 4 3 2 1 0 −1 −2 −3 −4 −5 −1
80 Hz
0
160 Hz
−0.5
320 Hz 480 Hz
−0.5 0 0.5 Current (A)
−1
1
−1.5 −500
FIG. 19. The force error due to hysteresis after compensation for the phase shift of the FMI.
0 H (A/m)
500
∆ Fmax/Frange (%)
FIG. 21. Magnetic hysteresis of a toroid, for Bmax = 0.4 T (solid), Bmax = 0.8 T (dotted) and Bmax = 1.2 T (dashed). 5 4 3
0.25 A−pp, sens. comp.
2
0.5 A−pp, sens. comp. 1 A−pp, sens. comp.
1 0
1.4 A−pp, sens. comp. 0
200 400 Frequency (Hz)
FIG. 20. The maximum force error due to magnetic hysteresis in the E-core actuator with PM, in percentages of the force range.
the various frequencies between 40 Hz and 480 Hz. This shows that the magnetic hysteresis and eddy currents are not dependent on the excitation amplitude, which can only be the case when the E-core is not saturated for those excitations. Measurements on a laminated toroid of the same material as the SiFe E-core are performed to confirm this assumption. A toroidal construction is considered, because the magnetic field is homogeneously distributed over the whole magnetic path. For this E-core actuator an inhomogeneous magnetic flux distribution occurs in the corners. The measured magnetic hysteresis in a laminated toroid is depicted in Fig. 21. According to the measurements on the laminated toroid in combination with the analytic actuator model, the iron of the E-core would approach the saturated region from a current amplitude of about 3 A-pp.
tance actuator. The effect of magnetic hysteresis in the force, appears to be similar to the elliptic shape of a regular phase delay. Therefore, a phase delay model is proposed to predict the effects of the magnetic hysteresis and eddy currents in this E-core actuator for steady state AC measurements, which is given as 1 dL i(t − θh )2 (14) 2 dz with θh - the phase delay of the current as a result of the magnetic hysteresis effects. With this method is the force error (due to magnetic hysteresis and eddy currents) reduced by a factor 5 and 20 for the actuator without PM and with PM, respectively. Figure 22(a) shows the remaining force error after hysteresis compensation for the actuator without PM, and Fig. 22(b) shows the remaining force error for the E-core with PM. The force error that is left after hysteresis compensation is the disturbance in the force measurement, which is due to the natural resonance of the FMI and due to amplifier noise. This phase compensation model reduces the error significantly, because the E-core actuator is not saturated and therefore only minor loops are observed. This is also illustrated by the measurements on the laminated toroid, as shown in Fig. 21. It can be seen that hysteresis in a toroid could be approximated by an ellipse for low current excitations. This presented hysteresis model is a steady state AC model, which is not applicable for arbitrary input signals, because no dynamic rate-dependencies are taken into account. F (t) =
VIII. VII. STEADY-STATE AC MAGNETIC HYSTERESIS MODEL
Previous force measurements showed the magnetic hysteresis, which is present in the force of this E-core reluc-
CONCLUSIONS
A measuring method has been presented, which identifies the magnetic hysteresis effects occurring in an E-core reluctance actuator. This method can be extended for other types of reluctance actuators, because it is based on the measurement of the rate of change of the flux through
9 40 Hz, hyst. comp.
40 Hz, hyst. comp.
80 Hz, hyst. comp.
80 Hz, hyst. comp. 160 Hz, hyst. comp.
160 Hz, hyst. comp. 0.2
5
320 Hz, hyst. comp.
320 Hz, hyst. comp. 480 Hz, hyst. comp.
2.5 ∆ F (N)
∆ F (N)
0.1 0
−0.1 −0.2 −1
0
−2.5
−0.5 0 0.5 Current (A)
(a)
1
−5 −1
−0.5 0 0.5 Current (A)
1
(b)
FIG. 22. Force error ∆F - after hysteresis compensation. (a) E-core without PM and (b) E-core with PM.
the soft-magnetic material. The high-bandwidth FMI is calibrated with various VCAs. This analysis showed that the force measurement introduces an additional phase lag. This phase delay is obtained for each frequency and used to compensate for the phase delay in the force measurements of the E-core reluctance actuator. This obtained phase delay is verified by measurements of the changing magnetic flux in the E-core actuator. After the phase compensation the actual magnetic hysteresis effects in the E-core actuator are obtained. For a frequency of 320 Hz, the non-hysteretic analytical model has an accuracy of 98.5 % and 98.8 % for the Ecore without and with PM, respectively. For the E-core with PM, hysteresis errors are measured in the range of 0.1 % to 2.4 % of the total force. These force errors depend on the excitation frequency, which is analyzed within a range of (40-480) Hz and the amplitude of the current is evaluated between (0.25-1.4) A-pp. Finally, a hysteresis model is presented, which is based on the prediction of the phase delay between the current and the force. With this model the force error is reduced with a factor 5 and 20 for the actuator without PM and the actuator with PM, respectively. The reduction is more significant for the actuator with PM, because in this case the change of the magnetic flux density B- is larger. 1 J.
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