of-freedom zφ-module
Citation for published version (APA):Overboom, T. T., Jansen, J. W., Lomonova, E., & Tacken, F. J. F. (2010). Design and optimization of a rotary actuator for a two degree-of-freedom zφ-module. IEEE Transactions on Industry Applications, 46(6), 2401-2409. https://doi.org/10.1109/TIA.2010.2073430
DOI:
10.1109/TIA.2010.2073430
Document status and date: Published: 01/01/2010 Document Version:
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Design and Optimization of a Rotary Actuator for a
Two Degree-of-Freedom
zφ
-Module
T. T. Overboom, J. W. Jansen, E. A. Lomonova and F. J. F Tacken
Abstract—The paper concerns the design and optimization of
a rotary actuator of which the rotor is attached to a linear actuator inside a two degree-of-freedom zφ-module, which is part of a pick-and-place robot. The rotary actuator provides ±180° rotation, while the linear actuator offers a z-motion of ±5 mm. In the paper, the optimal combinations of magnet poles and coils are determined for this slotless actuator with concentrated windings. Based on this analysis, the rotary actuator is optimized using a multi-physical framework, which contains a coupled electromagnetic, mechanical and thermal model. Because the rotation angle is limited, both a moving-coil design with a double mechanical clearance and a moving-magnet design with a single mechanical clearance have been investigated and compared. Additionally, the influence of the edge effects of the magnets on the performance of the rotary actuator has been investigated with both 3D FEM simulations and measurements.
I. INTRODUCTION
Pick-and-place (P&P) robots consist of a long-stroke robot arm, which is responsible for moving electrical components (such as Surface-Mounted-Devices or Ball-Grid-Array com-ponents) from a feeder over the Printed-Circuit-Board (PCB). Attached to this arm, a placement module picks up the components, orientates and places them on the PCB. This high-precision, short-stroke, two degree-of-freedom (2 DoF) actuator, therefore, needs to enable rotational and translational motion. Moreover, they require a compact and light weighted design, due to higher accelerations and operational speeds. Such a module will be referred to as a zφ-module.
Several types for zφ-modules can be distinguished. In the first category and classic approach, the linear stroke actuator drives a trolley which holds a rotary actuator. As an alternative, in the second category the movers of the rotary and linear actuator are attached to the same shaft above each other. For example, in [1] a zφ-module from the second category is discussed which uses two separate magnet arrays along the axial length of the rotor; one array for the rotary actuator and one array for the linear actuator. Instead of stacking two magnetization patterns, they can also be integrated in a single magnet array, as is done in the third category. Using a checkerboard magnet array and by appropriately commutating the current inside the coils, the design in [2] offers both degrees of freedom.
Based on the second category, a zφ-module is designed, which has to pick-and-place 10.000 BGA-components in one
T. T. Overboom, J. W. Jansen and E. A. Lomonova are with the Electromechanics and Power Electronics group, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands. Email: T.T.Overboom@tue.nl
F. J. F Tacken is with Wijdeven B.V., De Scheper 303, 5688 HP, Oirschot, The Netherlands.
Fig. 1. 3D overview of the zφ-module containing a non-commutated short-stroke linear actuator (bottom) and three-phase slotless permanent magnet rotary actuator (top).
hour. Figure 1 shows a 3D overview of this design, where the rotary actuator is placed at the top and the linear actuator is placed at the bottom of the module. The specifications of the zφ-module are given in Table I. The rotary actuator is a slotless permanent magnet actuator and provides ±180° degrees rotation. Because the total stroke of the linear actuator is only 10 mm, a non-commutated short-stroke linear actuator is selected instead of a three phase linear actuator, like is used in [1].
This paper concerns the design and optimization of the rotary actuator as part of the zφ-module, while the design of the non-commutated short-stroke linear actuator is fixed and will be considered as load. First, the basic design regarding the rotary actuator is addressed and the test results of a non-optimized pre-prototype are discussed. Prior to the optimiza-tion step, an optimum combinaoptimiza-tion of the number of magnet poles and coils is selected. A multi-physical framework of the rotary actuator is presented and used to optimize the design, which is performed for both a configuration with moving magnets and a configuration with moving coils. Finally, the optimized designs are analyzed for the magnet edge-effects.
Fig. 2. Overview of the rotary actuator for (a) the moving-magnet configuration and (b) the moving-coil configuration. TABLE I
SPECIFICATIONS OF THEzφ-MODULE. Parameter Value Description
zstroke ±5 mm Linear stroke
φstroke ±180° degrees Angular stroke
zerr 5 µm Linear accuracy
φerr 3 mrad Angular accuracy
αz 150 m s−2 Linear acceleration
αφ 7700 rad s−2 Angular acceleration
vmax 3 m s−1 Maximum linear velocity
ωmax 135 rad s−1 Maximum angular velocity
dz 0.22 Duty cycle linear actuator
dφ 0.39 Duty cycle rotary actuator
Lzφ 105 mm Maximum length of zφ-module
ro,max 30 mm Maximum outer radius of zφ-module
ri,min 18 mm Minimum inner radius of zφ-module
II. BASIC DESIGN CONSIDERATIONS AND PRE-PROTOTYPE TESTING
The basic design of the three-phase AC brushless permanent magnet rotary actuator is shown in Fig. 2. It has a slotless structure in order to provide a low torque ripple. To increase the magnetic loading inside the airgap, a two segmented quasi-Halbach magnet array is used. Concentrated windings are selected for their ease of manufacturing. Because the rotary actuator also needs to accommodate the translational motion of the rotor, the coils are elongated in order to deliver constant torque for any given axial position. Because the rotational angle of the rotary actuator is limited to ±180°, both a moving-magnet and a moving-coil configuration will be optimized and compared. In the moving-magnet configuration the coils are attached to the back-iron leaving only a single mechanical clearance between the coils and magnets. For moving coils, however, a double mechanical clearance is required; one additional clearance between the coils and back-iron. Figure 2 shows an overview of the rotary actuator for the moving-magnet and moving-coil configuration. In both cases, the rotor is coupled to the rotor of the linear actuator.
A non-optimized pre-prototype of the rotary actuator has already been designed, built and tested. Also a prototype of the non-commutated short-stroke actuator is built, but as the linear actuator is out of the scope of this work, its specifications are
TABLE II
SPECIFICATIONS OF THE NON-COMMUTATED SHORT-STROKE LINEAR ACTUATOR.
Parameter Value Description
kt,lin 14 N A−1 Force constant
Rlin 8.2 Ω Coil resistance
Jlin 78 kg mm2 Inertia
Mlin 190 g Moving mass
given in Table II. The design of the rotary actuator is based on a configuration with moving coils and its dimensions are given in Table III. Figure 3 shows the pre-prototype. Each coil in the slotless PM actuator contains 92 turns and the magnets are anisotropic sintered NdFeB, with a remanent flux density of 1.33 T and a relative recoil permeability of 1.1. The core and back-iron are made of steel N398, which is a low-carbon steel (99.8% Fe).
Using the 2D semi-analytical magneto-static model, as described in Section IV, a torque constant of 0.93 Nm A−1
is estimated, however, tests on the pre-prototype showed a torque constant of 0.76 Nm A−1. To eliminate the influence
of friction, electro-motive force (EMF) measurements were performed, which showed a reduced amplitude of 11.7 % compared to the 2D semi-analytical model. This reduction is caused by the lower magnetic loading near the edges of the magnets, as will be discussed in Section V. The test also showed that the helical wound electrical wires (see Fig. 1) were insufficient to guarantee a long life operation.
III. SELECTION NUMBER OF MAGNET POLES AND COILS
As a first step in the design process of the rotary actuator, the appropriate combinations of the number of magnet poles (2p) and coils (Q) are selected. First of all, only combinations resulting in a balanced structure will be evaluated during the optimization step. Next, the selection is further reduced by choosing magnet pole and coil combinations based on the winding factor, kw. The winding factor is a measure for the
flux linkage of a certain winding layout and ranges from 0 (no linkage) to 1 (optimal linkage). The torque produced by a three-phase AC brushless PM actuator is related to the flux
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TABLE III
DIMENSIONS OF INITIAL DESIGN ROTARY ACTUATOR AND PERFORMANCE OF THEzφ-MODULE.
Description Parameter Value
Dimensions rotary actuator
Number of pole pairs p 8
Number of coils Q 12
Inner radius ri[mm] 18.0
Outer radius ro [mm] 30.0
Radial length core lc[mm] 1.5
Radial length magnets lm[mm] 5.7
Radial length coils lw[mm] 1.9
Radial length back-iron lb[mm] 2.5
Radial length mechanical clearance lcl[mm] 0.2
Active length actuator Lact[mm] 15.0
Magnet arc to pole arc ratio α 0.67 Coil opening angle βo [degree] 1.70
Volume V [cm3] 104.6
Peak airgap flux density Bgap[T] 0.87
Coil temperature Tw[°C] 42.9
Magnet temperature Tm[°C] 39.7
Performance zφ-module
Total copper losses Pcu,tot[W] 16.44
Rotary copper losses Pcu,rot[W] 10.90
Linear copper losses Pcu,lin[W] 5.55
Total inertia Jtot[kg.mm2] 142
Total moving mass Mtot[g] 282
Fig. 3. Pre-prototype of the rotary actuator with (a) the test set-up and (b) the coil array.
linkage according to T = 3 2ωI ˆE = 3 2ωIkw dΛmax dt , (1)
where I is the peak phase current, ˆE is the peak phase
back-EMF, ω is the angular velocity and Λmax is the maximum
possible flux linkage of a single phase. As the expression shows, the torque is directly linked to the winding factor, and it would be desirable to select a combination of magnet poles and coils resulting in the highest possible winding factor.
The winding factor can be split into
kw= kp· kd· kskew, (2)
where kp is the pitch factor, kd is the distribution factor and
kskew is the skewing factor. Since no skewing is applied, the
skewing factor is assumed to be unity. The winding factors can only be found for slotted structures, though [3], [4], [5], [6]. Therefore, for slotless machines with concentrated windings, the fundamental winding factor is calculated for different combinations of the number of magnet poles and coils. Below,
Fig. 4. Illustration of a slotless actuator for predicting the pitch factor.
both the pitch and distribution factor will be determined, when only the fundamental harmonic of the magnetic flux density due to the magnets is considered.
A. Pitch factor
The pitch factor is a measure for the flux linkage of a single coil. Ideally, the total flux through one magnet pole is linked by all turns in a coil, which is achieved in slotted machines having one coil per magnet pole and per phase. Whereas the teeth in a slotted machine provides a low reluctance path for the flux through the coil and all turns in the coil link the same amount of flux, in slotless machines however, this low reluctance path is not provided and, hence, not all turns show the same flux linkage. To determine the pitch factor in a slotless machine as is shown in Fig. 4, the average flux linkage of a single turn is calculated by varying the angular span, 2αt, of a turn at
radius r Ψav = 1 αc− βo Z αc βo Ψt(αt)dαt = 1 αc− βo Z αc βo Z αt −αt rLactBˆ1(r) cos(pφ)dφdαt = 2rLactBˆ1(r) αc− βo 1 p2[cos(pβo) − cos(pαc)] , (3)
where 2βo is the opening angle of a coil, 2αc is the angular
span of a coil, Lact is the length in axial direction and ˆB1 is
the amplitude of the fundamental harmonic of the magnetic flux density. With the maximum flux linkage of a single turn being equal to the flux through one magnet pole, Ψmax =
2
prLactBˆ1(r), the pitch factor, kp, for a slotless machine with
concentrated windings becomes
kp= Ψav
Ψmax =
1
pαc− pβo[cos(pβo) − cos(pαc)] . (4)
B. Distribution factor
The distribution factor is a measure of the electrical align-ment of all coils in a single phase and can be calculated by
Fig. 5. EMF phasor representation of a three-phase system with p=7 and
Q=12.
writing the back-EMF of a single coil in phasor representation (in per unit)
− →
Ei,pu= ej(γi),
where γi= 2πpQ · i is the electrical angle offset of coil i.
Figure 5 shows an example of a phasor representation of the back-EMF for the individual coils and also the resulting phase back-EMF phasors, which are symmetrically shifted by 120° electrical degrees from each other. The minus sign for a phasors means that the windings of the coil are connected in the opposite direction. The phase back-EMF phasors (in per unit) can be calculated according
−→ Eph,pu= X Q/3 − → Ei,pu.
The distribution factor is found by dividing the magnitude of the resulting phase EMF phasor by the number of coils per phase, as given by kd= | − → Eph,pu| Q/3 . (5)
The method presented in [5] is used to obtain the winding layout which results in the highest distribution factor for a given number of magnets poles and coils.
C. Analysis of the winding factor
The winding factor inside a slotless machine with concen-trated windings is calculated for different combinations of the number of magnets poles and coils using (2), (4) and (5). Combinations resulting in the same number of coils per magnet pole and per phases, q, have the same winding factor and, therefore, the winding factor as function of q is shown in Fig. 6. As the winding factor also depends on the opening angle of the coil, it is calculated for the case when the opening angle is zero and half the total coil span (βo=0 and βo=0.5αc,
respectively). For both cases it can be noticed that machines having q=1/4 give the highest winding factor, kw=0.716 and
kw=0.955, respectively. These types of machines have kd=1
and, therefore, create the peaks in Fig. 6.
Fig. 6. Winding factors for slotless machines with concentrated windings versus the number of coils per magnet pole per phase.
In the further analysis of the rotary actuator, only magnet poles and coils combinations resulting in q=1/4 are considered because they provide a high winding factor and can be ob-tained with every number of magnet poles which is a multiple of four. During optimization the number of magnet poles is varied and the number of coils is determined according to
Q=6qp.
IV. MULTI-PHYSICAL FRAMEWORK
A multi-physical framework, containing a 2D semi-analytical magneto-static, thermal and mechanical model, is derived to obtain an optimized design of the rotary actuator. In the following subsections these models are discussed.
A. 2D semi-analytical magneto-static model
The magnetic field distribution inside the slotless permanent magnet actuator, as depicted in Fig. 7, is obtained by using the method as described in [7]. This method provides an accurate and fast means of determining the torque capabilities, while accounting for the relative recoil permeability, µr, of
the magnets. In this subsection the models obtained from this technique are further extended to account for the two-segmented Halbach array with straight magnetization, which is shown in Fig. 8.
The method assumes two-dimensional fields in cylindrical coordinates, infinite permeable iron and linear demagnetization of the magnets in the second-quadrant of the B-H curve. Due to the infinite permeability of the iron parts, only the magnetic fields inside the airgap (rm≤ r ≤ rb, indicated by subscript
I) and the magnets (rc ≤ r ≤ rm, indicated by subscript
II) have to be determined. By introducing the vector potential according to
B = ∇ × A, (6) and assuming no current density inside the airgap, Amp`ere’s law for the airgap region can be rewritten into the Laplace equation
∇2A
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Fig. 7. Schematic of the slotless permanent magnet moving-coil actuator including the coils.
(a)
(b)
Fig. 8. Straight (parallel) magnetized Halbach array with (a) the magnet array and (b) the magnetization waveform.
Likewise, the Poisson equation for the magnet region can be obtained
∇2A
II = −µ0∇ × M0. (8)
For both regions the following constitutive relations are used BI = µ0HI, (9)
BII = µ0µrHII+ µ0M0. (10)
where M0 is Fourier series representation of the remanent
magnetization vector and is given by
M0=
∞
X
n=odd
[Mnrcos(npφ)er− Mnφsin(npφ)eφ] . (11)
Fig. 9. Comparison of the flux density inside the airgap calculated analyti-cally and simulated using 2D FLUX (rc=19.5mm, rm=25.7mm, rb=27.5mm,
α=2/3, Brem=1.33T, µr=1.1).
The coefficients Mnr and Mnφare obtained from the Fourier
transformation of the magnetization waveform, which for the two-segmented Halbach array with straight magnetization is given by Mr = −Bµrem0 cos(φ + π/p) Mφ = Bµrem0 sin(φ + π/p) ) −πp <φ <−(2−α)π2p , Mr = Bµrem0 sin(φ + π/2p) Mφ = Bµrem0 cos(φ + π/2p) ) −(2−α)π2p <φ <−απ 2p, Mr = Bµrem0 cos(φ) Mφ = −Bµrem0 sin(φ) ) −απ 2p <φ <απ2p, Mr = −Bµrem0 sin(φ − π/2p) Mφ = −Bµrem0 cos(φ − π/2p) ) απ 2p <φ < (2−α)π 2p , Mr = −Bµrem0 cos(φ − π/p) Mφ = Bµrem0 sin(φ − π/p) ) (2−α)π 2p <φ <πp.
Here, Bremis the remanent flux density of the magnets and
α = φp
φp+φc is the radial magnet arc to the total magnet pole
arc ratio.
A description of the magnetic fields inside the the magnet and airgap region is obtained by finding solutions for the vector potential which are governed by the Laplace and Poisson equations and by applying the appropriate boundary conditions [7]. The semi-analytical field solutions show good agreement with results obtained from finite element modeling (FEM) in 2D FLUX (error <1%), as can be seen in Fig. 9 where the flux density inside the air gap is shown.
For a machine with 1/4 coil per magnet pole per phase the winding arrangement for a machine with moving coils is shown in Fig. 7. The torque is calculated by using the Lorentz’ force law T = Z V r × (J × BI)dv = Z V JzBrIrdvez, (12)
where V is the volume occupied by the coils, r is the vector to the point about which the torque is computed, J is the current
Fig. 10. Radial heat flow inside the actuator with moving coils.
density vector and BIis the airgap flux density vector, which is
assumed to be constant along the axial length of the magnets.
B. Thermal model
The amplitude of the current density inside the coils is constrained by the temperature distribution inside the actuator. A similar approach like is proposed in [8], is used and modified to predict the temperature distribution inside the rotary actuator. In this approach a thermal equivalent circuit (TEC) of the actuator is created, which offers a fast way to predict the thermal behavior. The actuator geometry is divided into lumped components and the thermal behavior of these components is presented in a network of thermal resistances, capacitances and heat sources.
While the transient thermal behavior of a machine can be simulated using TEC, in this case only the steady-state temperature distribution is of concern, and hence, only the thermal resistances and heat sources need to be determined. The thermal model is further simplified by considering one section of the actuator and assuming only radial heat flow. For the moving coil configuration, Fig. 10 shows the resulting radial heat flow by conduction, convection and radiation. Heat is produced inside the coils and is determined from the ohmic losses. The linear dependency of the electrical resistivity of copper on temperature is accounted for by modeling it as a negative thermal resistance. Iron losses inside the magnets are neglected because of the low magnetic field created by the coreless coils. In the moving coil concept, no changing magnetic field is induced inside the back-iron, because the magnets and back-iron are fixed relative to each other. In the moving magnet concept, the iron losses inside the back-iron, which are caused due to the rotating magnets, are approximately 12 W kg−1 at 1.5 T and 50 Hz [9]. Compared
to the ohmic losses, these iron losses are small and can be ignored in the thermal model.
The conductive heat flow across the actuator is modeled by thermal resistances which are determined for the different cylindrical components. To account for radiation in the TEC, this mode of heat flow is linearized and modeled like convec-tion with a heat transfer coefficient h² ≈ 6², where ² is the
emissivity. Convection at inner and outer radii of the actuator is modeled by a heat transfer coefficient, which is estimated to be 10 W m−2 K−1 [10]. Inside the airgap, forced air cooling
is applied, which is also modeled by convection with a heat transfer coefficient, hgap.
Fig. 11. Influence of the airgap heat transfer coefficient on the coil and magnet temperatures when 0.5 W of heat is produced in the coils at an ambient temperature of 20 °C. TABLE IV MATERIAL PROPERTIES. m k ρ [kg m−3] [W m−1K−1] [Ω m] NdFeB 7350 10 -N398 7850 73 -Copper 8900 1 1.678×10−8 Air 1.067 0.0285 -@T=25°C
The resulting TEC is solved as described in [8] and using the dimensions of the initial design of the slotless actuator and the material properties from Table IV, the temperatures inside the coils and magnets as function of the heat transfer function, hgap, for both the moving-coil and moving-magnet
configuration are shown in Fig. 11. As the figure shows, the temperature drops significantly when hgapis increased from 5
to 30 W m−2 K−1. Although high values for the heat transfer
function can be achieved by forced air cooling [11], hgap is
set to a value of 15 W m−2 K−1 during the optimization
procedure. This value was estimated from measurements on the pre-prototype with an air flow of 25 liters per minute through the airgap.
C. Mechanical model
The performance of the actuator strongly depends on the mechanical load on it. When no external load is considered, the required torque and force to achieve a certain acceleration are linked to the inertia and total mass of the moving parts of both the rotary and linear actuator. While the design of the linear actuator is fixed, the inertia and moving mass of the rotary actuator are determined for the moving-magnet and moving-coil configuration.
V. DESIGN OPTIMIZATION
The multi-physical framework is implemented in MATLAB and used in the optimization procedure, which is performed with sequential quadratic programming (SQP). In this section
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TABLE V
LIST OF ALL CONSTRAINTS FOR THE ROTARY ACTUATOR. Constraint Description
Bcore<1.4 T Saturation in core
Bback<1.4 T Saturation in back-iron
ri>18 mm Minimum inner radius
ro<30 mm Maximum outer radius
L<37 mm Maximum length actuator
lw≥0.9 mm Radial length coils (windings)
Tw<100 °C Maximum coil temperature
Tm<60 °C Maximum magnet temperature
the optimization objective, design constraints and results are discussed.
A. Optimization objective
To create an efficient design of the zφ-module the rotary actuator is optimized with the objective to minimize the copper losses inside the rotary and linear actuator combined. These losses are calculated for a third order motion profile. By taking the losses inside the linear actuator into account, the moving mass of the rotary actuator is limited. The objective function can be written as
f (α, βo, ri, lc, lm, lw, lb, Lact) = Pcu,rot+ Ilin2 Rlin, (13)
where α, βo, ri, lc, lm, lw, lb and Lact are the optimization
variables, which are described in Table III and indicated in Fig. 2 and 7. Ilin is the current and Rlin is the resistance of
the linear actuator, as is given in Table II. The relatively small iron losses in the back-iron are again ignored.
The initial design of the rotary actuator already showed that the specifications given in Table I are easily met within the volume constraints, making this design oversized for the application. Therefore, the design is also optimized with the objective to minimize the volume of the rotary actuator. The objective function can be expressed as
f (α, βo, ri, lc, lm, lw, lb, Lact) = πr2oL, (14)
where rois the outer radius of the actuator and L is the total
length of the coils.
B. Constraints
The design of the rotary actuator is subject to several constraints. First, the volume of the actuator is constrained by the available height inside the zφ-module, the radius of the shaft and the outside radius. Furthermore, a lower bound is set to the radial length of the coils. To guarantee a linear response of the actuator to the injected current, saturation of the steel is avoided. Finally, to prevent damage of the winding insulation, possible irreversible demagnetization of the magnets or substantial loss of performance due to a lower value of the intrinsic magnetization, the coil and magnet temperatures are constrained. Table V lists all the constraints.
C. Optimization results
The optimization is performed for the moving-magnet con-figuration and moving-coil concon-figuration and Fig. 12 shows
Fig. 12. Minimized copper losses for different magnet pole pairs.
Fig. 13. Minimized volume for different magnet pole pairs.
the minimized copper losses for different magnet counts. Both configurations have a minimum at p=14 and are constrained by the volume. In Fig. 13, the results of the optimization with the objective to minimize the volume of the rotary actuator, are shown. The figure shows a minimum for p=18 and
p=16 for the moving-magnet and moving-coil configuration,
respectively. In this case the minimization is limited by the magnet temperature.
As is already mentioned in Section II, the reduced amplitude of the EMF measurements compared to the 2D semi-analytical magneto-static model, is caused by the magnet edge-effects. For the design of the pre-prototype, the peak airgap flux density along the axial length of the magnets is simulated with 3D FEM, and Fig. 14 shows a comparison to the flux density predicted with the 2D semi-analytical model. A reduction of 10.3 % of the flux linkage by the coils and, hence, a similar reduction of the EMF, is predicted from this analysis. Likewise, the magnet edge-effects are predicted for the optimized designs and the copper losses in the optimized designs are recalculated while the reduced magnetic loading is accounted for. The sizes and specifications of these designs
Fig. 14. 3D FEM to 2D semi-analytical magneto-static modeling comparison of the airgap flux density along the axial length of the permanent magnet.
are listed in Table VI for both minimized copper losses and minimized volume.
With the objective to minimize the copper losses, the optimized design with moving coils produces the lowest losses, which is 30 % less compared to the moving-magnet configuration. This is due to a lower inertia, a smaller moving mass and a higher magnetic loading. On the other hand, from a practical point of view, the electrical wiring and additional mechanical clearance with the moving-coil configuration may be of a concern. As Table VI shows, these two designs occupy the maximum available space inside the zφ-module. In the new designs with minimized volume, both the moving-coil and moving-magnet configuration occupy 50 % and 45 %, respectively, less volume than the first two designs. Because the magnet edge-effects were analyzed after the optimization step, the magnet temperature has slightly exceeded its thermal constraint. In this case the moving-magnet configuration is preferred because it has lower copper losses. In all four new designs, it can be noticed that the linear actuator is the dominating mechanical load and can account for 85 % and 87 % of the total inertia and moving mass, respectively. In both moving magnet designs, the iron losses are less than 2.2 % of the copper losses. Therefore, it is justified to ignore these iron losses in the thermal model and in (13).
VI. CONCLUSIONS
In this paper an optimized design of a rotary actuator, which is coupled to a linear actuator, has been obtained using a multi-physical framework. Only combinations of the number magnet poles and coils resulting in 1/4 coil per magnet pole per phase have been considered during optimization, because, for slotless machines, they have the highest winding factor. Within the available volume, a design with moving coils results in the lowest combined copper losses of the rotary and linear actuator. This design has 30 % less copper losses compared to a configuration with moving magnets. Although only a limited rotational stroke is required, electrical wiring with moving coils may still form a point of concern. It is also shown that
OPTIMIZED DESIGNS OF THE ROTARY ACTUATOR WITH MINIMIZED COPPER LOSSES OR MINIMIZED VOLUME(MC =MOVING COIL, MM =
MOVING MAGNET).
Objective Min. Copper losses Min. Volume Configuration MC MM MC MM p 14 14 18 16 Q 21 21 27 24 ri[mm] 21.2 24.7 18.0 18.0 ro[mm] 30.0 30.0 22.9 22.6 lc[mm] 0.5 0.5 0.5 0.5 lm[mm] 5.4 2.2 2.5 2.2 lw[mm] 0.9 1.2 0.9 0.9 lb[mm] 1.6 1.2 0.6 0.8 L [mm] 37 37 31.7 36.2 Lact[mm] 18.7 18.6 16.7 20.7 α 0.5 0.7 0.57 0.59 βo [degree] 3.4 1.5 1.27 1.60 V [cm3] 104.6 104.6 52.3 57.9 Bgap[T] 1.09 0.77 0.72 0.81 Tw[°C] 30.2 42.7 72.4 71.1 Tm[°C] 28.8 39.4 65.1 63.8 Pcu,tot[W] 7.74 11.08 20.83 16.17 Pcu,rot[W] 4.28 6.69 17.52 12.15 Pcu,lin[W] 3.46 4.39 3.32 4.02 Jtot[kg.mm2] 103 120 92 97 Mtot[g] 223 251 218 240
the same specifications for the zφ-module can be met within a 50 % smaller design of the rotary actuator.
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