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Edge effect of multi-degree-of-freedom oscillatory actuator driven by vector control

  • Masayuki Kato EMAIL logo , Katsuhiro Hirata , Tomoaki Mototsuji and Akira Heya
Published/Copyright: July 27, 2020
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Open Physics
From the journal Open Physics Volume 18 Issue 1

Abstract

This article proposes a thrust equation model of two-degree-of-freedom oscillatory actuator considering its edge effect. The proposed thrust equation model clarifies that asymmetric permanent magnet flux linkage and inductance characteristics cause undesirable magnetic thrust and reluctance thrust, respectively.

Keywords: linear oscillatory actuator; vector control; edge effect; multi-degree-of-freedom; electromagnetic actuators

1 Introduction

Linear oscillatory actuators (LOAs) have been used in a wide range of applications such as an electric shaver and an air compressor [1,2]. Various kinds of multi-degree-of-freedom (DOF) resonant actuators have been developed for further improvement of LOAs [3]. The authors have developed a two-DOF oscillatory actuator [4] driven by three-phase current and verified its effectiveness through 3D finite element analysis (FEA) and measurement.

This actuator has linearly expanded structure of four-pole three-slot surface permanent magnet synchronous motors (SPMSMs). Therefore, this actuator exhibits a specific property called the edge effect because its stator and mover have finite lengths. It is well-known that linear motors also show the edge effect [5]. For example, in case of linear synchronous motors, the air-gap flux distribution becomes unbalanced due to the edge effect [6]. To solve this problem, shape optimization or using auxiliary teeth has already been proposed [7,8]. On the other hand, we have not investigated how the edge effect influences thrust characteristics of the two-DOF actuator. Moreover, it has only three slots, which is much fewer than that of conventional linear motors. Therefore, the edge effect distorts sinusoidal distribution of the air-gap flux density from the four-pole permanent magnet. This fact indicates that the linearity of the thrust and the controllability of the two-DOF actuator decrease due to its edge effect.

In this article, we propose a thrust equation model of the actuator considering its edge effect. We clarify the thrust characteristics of the actuator under various conditions by using the proposed thrust equation model.

2 Two-DOF oscillatory actuator

2.1 Basic structure and operating principle

The basic structure of the two-DOF oscillatory actuator is shown in Figure 1. This actuator consists of a mover, a stator, and springs that support the mover. As mentioned previously, this actuator is similar to four-pole three-slot SPMSMs. Therefore, conventional vector control can be applied to operate this actuator. q-Axis current i q and d-axis currents i d correspond to the thrust in the x- and z-directions, respectively, as follows:

(1) F z F x = F d F q = ϕ i d i q

Three-phase current is determined by inverse d-q transformation as follows:

(2) i u i v i w = 2 3 cos θ − sin θ cos θ − 2 3 π − sin θ − 2 3 π cos θ + 2 3 π − sin θ + 2 3 π i d i q

(3) ϕ = s x l π

Here s x is the displacement in the x-direction and l is the pole pitch of the magnet mounted on the mover. This actuator is able to oscillate independently in x- and z-directions by AC inputs of i d and i q .

Figure 1 Basic structure of two-DOF resonant actuator.
Figure 1

Basic structure of two-DOF resonant actuator.

Table 1

Actuator’s parameter

Symbol Quantity (unit) Value
Φ f Magnetic flux ( μ W b ) 270
k Coefficient for flux amplitude 0.115
j Coefficient for flux slope 0.167
L a Effective inductance ( μ H ) 34.1
l a Leakage inductance ( μ H ) 12.2
L c Compensated self-inductance ( μ H) 4.6
M c Compensated mutual-inductance ( μ H ) 7.8
L 0 α axis inductance ( μ H ) 63.7
L 1 β axis inductance ( μ H ) 55.4
L p Summed inductance ( μ H ) 59.5
L m Subtracted inductance ( μ H ) 4.17
P n Pole pairs 2
R a Coil resistance ( Ω ) 0.16

2.2 Influence of edge effect

The vector control described in the previous section assumes that flux density distribution in air gap is sinusoidal. Moreover, it assumes that three phase coils (U, V, and W) have the same magnetic flux properties. The two-DOF resonant actuator has linearly expanded structure of SPMSMs. Therefore, this actuator exhibits the edge effect because its stator and mover have finite lengths. In this section, we investigate the influence of the edge effect on the PM flux linkage and phase inductance characteristics. These characteristics are computed through 3D FEA. Figure 2 shows the 3D finite element mesh of our actuator, excluding the air region.

Figure 2 3D finite element mesh without air region.
Figure 2

3D finite element mesh without air region.

Figure 3 shows the analyzed PM flux linkage of U, V, and W phase coils. Solid lines in Figure 3 are approximated to sine waves with the same amplitude. The results show that the edge effect makes a difference between ideal PM flux linkage (solid line) and the 3D FEA results. Figures 4 and 5 show the analyzed self-inductance and mutual inductance of U, V, and W phase coils. Because of this actuator’s structual symmetry, inductance characteristics of V and W phase coil are identical. Figure 4 reveals that U phase self-inductance L u is slightly higher than V (W) phase self-inductance L v ( L w ). Additionally, mutual inductance between U and V phase M u v is slightly higher than that between V and W phase M v w . In order to discuss the difference, we compare the following two models. Figure 6 shows the analyzed flux density distribution of an enhanced model. The flux passing through V phase teeth is equally divided into the two adjacent teeth (U and W). Figure 7 shows the analyzed flux density distribution of the normal model. The flux passing through V phase teeth is not equally divided into the two teeth (U and W). This is because W phase teeth is far from V phase teeth.

Figure 3 PM flux linkage characteristics.
Figure 3

PM flux linkage characteristics.

Figure 4 U-phase inductance characteristics.
Figure 4

U-phase inductance characteristics.

Figure 5 V(W)-phase inductance characteristics.
Figure 5

V(W)-phase inductance characteristics.

Figure 6 Flux density distribution of enhance d model.
Figure 6

Flux density distribution of enhance d model.

3 Voltage equation model considering edge effect

3.1 Modeling of flux linkage characteristic

In this section, we derive a flux linkage model considering the edge effect mentioned in the previous section. Two parameters i and k are used to compensate the difference between the ideal PM flux and 3D FEA results in Figure 3. The three-phase PM flux linkages Φ u , Φ v , and Φ w are divided into a symmetric flux vector Φ sym and an asymmetric flux vector Φ asym as the following equation:

(4) Φ u Φ v Φ w = Φ sym + Φ asym = Φ f cos θ cos θ − 2 3 π cos θ + 2 3 π + Φ f 0 k cos θ − 2 3 π − j θ k cos θ + 2 3 π + j θ

Here, Φ f is amplitude of PM flux linkage. Two parameters L c and M c are used to compensate the bias shown in Figures 4 and 5. The inductance models are obtained as follows:

(5) L u = l a + L a + L c

(6) L v = L w = l a + L a

(7) M u v = M v u = M w u = M u w = − 1 2 L a

(8) M v w = M w v = − 1 2 L a + M c

Here, l a and L a are the leakage and effective inductance, respectively.

3.2 Voltage equation on dq reference frame

The voltage equation on UVW reference frame is expressed as follows:

(9) v u v v v w = R a + p L u p M u v p M w u p M u v R a + p L v p M v w p M w u p M v w R a + p L w i u i v i w + p Φ u Φ v Φ w

Here, R a is the coil resistance of each phase and p is the differential operator. The Clarke transformation gives the voltage equation on α β γ reference frame.

(10) v α v β = R a + p L 0 0 0 R a + p L 1 i α i β + p Φ a 1 + k 3 cos θ ( 1 + k ) sin θ − 2 j 3 θ

(11) L 0 = l a + 3 2 L a + 2 3 L c + 1 3 M c

(12) L 1 = l a + 3 2 L a − M c

(13) Φ a = 3 2 Φ f

Unlike the conventional SPMSMs, the diagonal element in equation (10) does not become equal because L 0 and L 1 contain the parameters L c and M c . The Park transformation gives the voltage equation on dq reference frame.

(14) v d v q = A 11 A 12 A 21 A 22 i d i q + ω B 11 B 12 B 21 B 22 i d i q + p Φ a 1 + 2 3 k − k 3 cos 2 θ − 2 j 3 θ sin θ k 3 sin 2 θ − 2 j 3 θ cos θ

(15) A 11 = R a + p ( L p + L m cos 2 θ ) A 12 = p L m sin 2 θ A 21 = − p L m sin 2 θ A 22 = R a + p ( L p − L m cos 2 θ )

(16) B 11 = − L m sin 2 θ B 12 = − L p − L m cos 2 θ B 21 = L p − L m cos 2 θ B 22 = L m sin 2 θ

(17) L p = L 0 + L 1 2

(18) L m = L 0 − L 1 2 = 2 3 L c + 4 3 M c

We show the voltage equation model of conventional PMSMs to discuss equation (14).

(19) v d v q = R a + p L d 0 0 R a + p L q i d i q + ω 0 − L q + L d 0 i d i q + p Φ a 0

Here, ω is the rotational speed.

First, we focus on the first terms on the right side in equations (14) and (19). The diagonal elements in equation (14) depend on the electrical angle of the mover θ . On the other hand, the diagonal elements in equation (19) do not depend. This is because dq transformation means the conversion of AC components to DC ones. Moreover, the first matrix in equation (14) has nonzero elements, which correspond to interference between d- and q-axes.

Finally, we visualize the voltage equation model considering the edge effect. Figures 8 and 9 show the magnetic flux and voltage vector diagrams of the two-DOF resonant actuator. Vector diagram of conventional PMSMs is shown in Figure 10 for comparison. From Figure 8, PM flux linkage vector Φ a is dependent on the mover position θ , which indicates that the direction of Φ a does not always become the same as the direction of d-axis. The other flux vector except Φ a also depend on θ . Therefore, the total flux vector Φ o has complicated behavior when the actuators are operated to oscillate.

Figure 7 Flux density distribution of normal model.
Figure 7

Flux density distribution of normal model.

Figure 8 Magnetic flux vector diagram of two-DOF resonant actuator.
Figure 8

Magnetic flux vector diagram of two-DOF resonant actuator.

Figure 9 Voltage vector diagram of two-DOF resonant actuator.
Figure 9

Voltage vector diagram of two-DOF resonant actuator.

3.3 Thrust equation model considering edge effect

In this section, we derive thrust equation model from the voltage equation model considering the edge effect. The thrust of the actuator F x is calculated from the cross product of a current vector i a = ( i d , i q ) and the flux vector Φ d q , as follows:

(20) F x = − P n i a × Φ d q = F 1 + F 2 + F 3 + F 4 = P n Φ a i q + P n Φ a 2 k 3 + k 3 cos 2 θ + 2 j 3 θ sin θ i q + P n Φ a k 3 sin 2 θ − 2 j 3 θ cos θ i d + P n ( ( L m sin 2 θ ) ( i d 2 − i q 2 ) + 2 L m cos 2 θ i d i q )

Here, P n is the pole pairs. The first term on the right side corresponds to the magnetic thrust caused by the symmetric PM flux linkage Φ sym . The second and third terms correspond to the magnetic thrust caused by the asymmetric PM flux linkage Φ asym . Especially, the third term interferes the x-axis thrust Φ x by d-axis current i d . The fourth term is the reluctance thrust caused by asymmetric inductance characteristics. Surprisingly, this actuator can generate reluctance thrust though the mover does not have saliency.

This paper clarifies the thrust performances of the actuator on the basis of equation (20). The two parameters L c and M c are determined from the following equations:

(21) L c = L u − L v

(22) M c = M v w − M u v

Table 1 shows the actuator’s parameter. These values are determined from the 3D FEA results. As expected, L m is much smaller than L p because the edge effect is not dominant. Figures 11, 12, and 13 show each thrust component when the vector control is applied under various conditions. These results are normalized by the value of the first term Φ 1 . From Figure 12, the thrust cannot be kept constant when i q is comparatively large. It is because Φ 4 is proportional to the square of i q . From comparison between Figures 11 and 13, the d-axis current i d interferes with the thrust characteristic because of F 2 and F 3 . These results suggest that the thrust characteristics are successfully investigated on the basis of the proposed thrust equation model.

Figure 10 Vector diagram of conventional PMSMs.
Figure 10

Vector diagram of conventional PMSMs.

Figure 11 Thrust characteristics (i d = 0, i q = 1).
Figure 11

Thrust characteristics (i d = 0, i q = 1).

Figure 12 Thrust characteristics (i d = 0, i q = 10).
Figure 12

Thrust characteristics (i d = 0, i q = 10).

Figure 13 Thrust characteristics (i d = 1, i q = 1).
Figure 13

Thrust characteristics (i d = 1, i q = 1).

4 Conclusion

This paper proposed a thrust equation model of two-DOF oscillatory actuator considering its edge effect. 3D FEA revealed that PM flux linkage and phase inductance characteristics were different from the ideal one due to the edge effect. The proposed thrust equation model clarified that asymmetric PM flux linkage and inductance characteristics caused undesirable magnetic thrust and reluctance thrust, respectively.

From the opposite point of view, this actuator has a potential to be driven as a kind of linear reluctance actuator although the mover has no saliency. In the near future, we will design a new linear reluctance actuator without saliency of the mover, which utilize the reluctance thrust caused by the edge effect effectively.

References

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Received: 2019-12-15
Revised: 2020-05-27
Accepted: 2020-06-21
Published Online: 2020-07-27

© 2020 Masayuki Kato et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2020-0169
Keywords for this article
linear oscillatory actuator; vector control; edge effect; multi-degree-of-freedom; electromagnetic actuators
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