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Simulation results of flux density in the back-iron regions must be analyzed and the radial length adjusted in order to avoid saturation. Considering a position-dependent synchronous electric drive, e. By means of these equations, the current density applied to a FEM produces the same magnetomotive force that would be observed in the experimental case. Considering a study in the quasi-static domain, it is possible to study the behavior of the electromagnetic force depending on the relative position between the stator and the translator, p z , which can be defined in terms of the electrical angle according to.
Therefore, the generated fields are in quadrature, producing the maximum force in relation to the applied current. Using equations presented in this subsection, it is possible to build parametric models of the geometry in 2D or 3D spaces. The geometry is axisymmetric, so 2D models satisfactorily represent the behavior of the spatial distribution of the fields, even if the technique of segmenting the rings into a minimum of eight sections with parallel magnetization is applied [ 10 ].
However, if the number of segments is lower than eight, and the asymmetry in the circumferential direction becomes relevant, the 2D model must be replaced by a 3D model. Thus, the model would be more complex and simulation would become time consuming, but the main structure of the proposed methodology would still be applicable. Thus, the spatial distribution of the field in these regions behaves as if they had infinite neighboring poles.
This approach neglects end effects, however; at this stage, the number of poles of the device to be designed is not known, so adverse effects should be avoided. The number of poles needed to meet the force requirement in continuous operation is computed in Section 3. The magnetic end poles are set as radial magnetized PMs with half the axial length, i. Table 1 presents the design variables, and Figure 3 shows the influence of the four parametric variables with variation between its lower and upper limits, according to the evaluated values.
The variable n BI was also kept constant and equals 0. In Figure 3 , the geometric dimensions affected by the parametric variables highlighted in bold below its respective figure are indicated.
The radial lengths of the back-irons are also a function of n form. The electromagnetic material properties should correspond to the properties of materials employed in the construction of the actuator, if such data are available. The initial electrical loading should be an estimation of what effective current density could be applicable in continuous operation mode. For conventional rotating machines, this is available in the references, e.
In this case, it was assumed that the effective current density is similar to the one used in electrical machines with natural convection, i. The geometrical constraints applied to this specific case study are summarized in Table 2. By assigning a constant value to these variables, the active region to be analyzed is limited radially. The limitation applied to the R oPMo can be justified due to restrictions of physical space where the actuator must be installed or due to magnetizer restriction. In this work, it was set considering the limitation imposed by the magnetizer fixture available, i.
The limitation imposed on R iPMi is justified by the need for radial magnetization and to enable air flow for cooling of the actuator. The radial magnetization on ring-shaped PMs would be physically impractical on a cylinder if R iPMi is zero; therefore, a limitation imposed by the magnetizer fixture available, if ideal radial magnetization is desired, may apply. On the other hand, to enable air flow passing through a hollow shaft in the inner back-iron, discussed in Section 3. In this case study, it is considered that radially magnetized PMs are active with segmentation and parallel magnetization [ 10 ], so that no restriction by magnetizer fixture is directly applicable.
Thus, considering the R oPMo dimension, and the needed air flow, R iPMi was initially set as 18 mm; however, if necessary, R iPMi could be reconsidered during the project. It should be noted that even if segmentation and parallel magnetization are also applied to the PMs of the outer array, restriction of R oPMo imposed by the magnetizer is still applicable in order to manufacture the axially magnetized PMs in a ring shape without segmentation.
The mechanical gaps depend on many factors, such as bearing backlash, manufacturing tolerances, thermal expansion of materials, etc. In this paper, inner and outer gaps were set as constant with a value that is typically found in the literature and in datasheets of manufactures and suppliers of linear actuators; however, such value might be minimized, if an in-depth study about the factors that affects mechanical gaps were performed. The radial length of the reel should be as short as possible, once its presence increases the magnetic gap.
In this topology, the reel is necessary to provide more stiffness to the moving coils, and its radial length was defined based on simulation of mechanical stress.
The constraints applied to the radial lengths of the PMs are discussed in detail in Section 3. If desired, after defining all other variables, n BI could be optimized in order to possibly reduce some of ferromagnetic material of the inner and outer back-iron pieces. Observing the flowchart presented in Figure 2 , it is possible to see that if some conditions, addressed in Section 3. These new geometrical constraints may apply especially to R oPMo or to R iPMi , depending on how flexible each one of those is.
Analysis of simulation results using 2D or 3D graphics of a space defined by the four parametric variables presented in Table 1 , plus force density and force ripple, can be a difficult and extensive task. An approach that was adopted in this work to analyze results in five dimensions is illustrated in Figure 4. This figure condenses five variables four parametric variables, which are independent variables, and one objective function variable, which is a dependent variable on a single graph.
It represents a top-oriented view of a combination of 40 3D graphs, in a way that the color map indicates the dependent variable, which, in the case of Figure 4 , is force density. The horizontal axis also holds two independent variables, i. Therefore, each of the rectangles in Figure 4 represents a variation of the parametric variables n form , on the horizontal axis, and N CPMs on the vertical axis. The form of presentation of five variables in a single figure allows one to compare results for the entire domain, which represent the combination of all parametric variables according to Table 1 , in a comprehensive way.
The force density F d of Figure 4 and Figure 5 is given by. Yet, in this work it was decided to keep them to increase performance and for mechanical support. However, they were not taken into account for computation of force density, because n BI was not considered in the parametric analysis; therefore, there would be no guarantee that force density is maximized if it were computed considering the back-irons. It should be noted that the shape, or color map, as presented, of force density would not be affected by its initial electrical loading of the uncoupled model, because effective current density is constant; however, its absolute value would be linearly proportional.
Consequently, the mean radius of the coil is shifted and so its volume is altered; however, this effect is more significantly observed with N CPMs.
Electromagnetic Linear Machines with Dual Halbach Array. Design and Analysis. Authors: Yan, L., Zhang, L., Peng, J., Zhang, L., Jiao, Z. Free Preview. Electromagnetic Linear Machines with Dual Halbach Array The design concept and analytical approaches can be implemented to other linear and rotary.
From Figure 4 , and with more detail in Figure 5 , it is possible to infer that the parametric variable N CPMs has a higher influence over force density than n form. This can be explained by a compromise between electrical and magnetic loading, which is directly related to N CPMs.
On the other hand, n form is related to the interpolar leakage flux, which is more significant for lower values of this parametric variable. A maximum force density of 2. The minimization of force ripple is a desirable feature with regard to the system linearity, reducing problems related to the positioning control and minimizing the inclusion of oscillations by the actuator when the speed is not zero. Depending on the electric drive technique, there are time-dependent induced harmonics that may generate force ripple during dynamic operation; however, these harmonics are not taken into account in this study, once force ripple is computed based on a magnetostatic model.
The force ripple can be estimated statically calculating the difference between the forces obtained with current in 2-phase and 3-phase, corresponding to, e. From Figure 6 , it is possible to infer that maximum and minimum values of force occur at two particular electrical angles as discussed in the previous paragraph; thus, a fair estimation of the variation of force can be carried out by evaluating F 2 and F 3 and computing the force ripple according to Equation Parametric simulation results of F 3 and F 2 applied to Equation 11 are shown in Figure 7 , where the results of absolute value of static force ripple are presented.
Based on the result obtained in Section 3. Results for the uncoupled model are summarized in Section 4. Dimensional results in axial direction are computed in the next subsection. The active volume of the actuator that is able to cope with effective force specifications can be obtained by dividing the rated force F r Section 3. The axial active length of the actuator L z is then calculated by relating it with its active volume, so that.
The number of poles P of the actuator can also be determined based on previous results. Therefore, P is given by. In order to obey the condition imposed by Equation 14 , L z or n form must be recalculated. In this case, it can be observed from Figure 5 that n form slightly affects F d over a range of 0. That means it could be adjusted without having a significant effect on the overall results. If the choice was to adjust L z , this could lead to an unnecessary increase in the active volume of the device.
If n form , calculated by Equation 15 , returns a value outside of the range 0. In this situation, P should be set as the closest even number to zero, if the dimensional design conditions verified in the next subsection are met, and the final n form calculated with Equation P was found to be 4 with a final n form of 1. This step in the designing process verifies whether the results obtained so far are consistent in the sense that the final shape of the actuator presents acceptable dimensions and parameters, which are defined by the following inequalities:.
The first inequality, which is the ratio between the axial active length and the windings axial length, is set so that at least half of the total length of the windings is active during operation. This ratio is related to the efficiency of the actuator, once there are Joule losses along the total length of the windings, but only the portion that is placed within the active length produces force. On the other hand, the second inequality in Equation 16 requires that the minimum number of poles of the actuator is four. This criterion is established in order to limit the way that end effect affects the overall performance of the actuator.
End effect is intrinsic to linear machines and is more significant if the device has a low number of poles. The third and fourth inequalities of Equation 16 limit the radial length of the permanent magnets to a minimum, which are imposed to prevent the PMs from breaking during assembly, which would easily happen if the radial length of the PMs is too small.
The shaded area in Figure 8 indicates a domain of the parametric variables N PMi and N CPMs in which radial length of both inner and outer PMs are bigger than 3 mm, whereas the areas that are not shaded indicate whether inner or outer magnets present a radial length smaller than specified. In this case whether R oPMo should be decreased or R iPMi increased, so that the active axial length of the actuator would become larger. The radial length of the inner and outer PMs is also higher than 3 mm, once in Section 3. If the former discussed conditions were not attained, it would be suggested to alter N PMi , which presents very little sensitivity in relation to design objective, instead of changing N CPMs.
The inequalities given by Equation 16 could be different from the ones that were set; they are basically criteria imposed by the designer. The heat sources associated with losses of an actuator can be of many kinds; however, in this particular actuator, the main source of losses is Joule losses at the conductors. Very low levels of induced current appear in the PMs, which can be neglected.
This specific topology presents an intrinsic low level of iron losses because there is no relative movement between back-irons and PMs and it presents a coreless armature. Still, there may be induced current in the back-irons by variation of the flux produced by the armature, but in lower levels than would be observed within actuators with cored armature and low compared to Joule losses, so it can be neglected. Friction losses at bearings are also not considered.
In order to achieve higher force and power density, the device must present an improved capacity of heat exchange. In the proposed methodology, thermal analysis plays an important role, since it allows for determining absolute force density and, thereafter, active volume for a given specification.
In this work, it was considered that a forced air flow was imposed to a hollow shaft, which insufflates the actuator with air at ambient temperature. In order to improve heat exchange in this topology, a forced inlet flow with a speed of 0.
This characterizes a forced convection at the hollow shaft and at the inner and outer gaps, which significantly improves heat transfer. It must be clear that the decision about whether forced or natural heat transfer is applied must be made in accordance with the mechanical characteristics of the actuator. Depending on the operation conditions, the actuator may be completely sealed, or, in a different situation, it might be open and its own movement produces an air flow that characterizes forced convection.
In any case, it must be defined at this step because this significantly affects the designing process. The forms of heat transfer considered in the simulation were by radiation, by conduction, and by forced and natural convection. The models for convection and radiation used were Boussinesq and Rosseland, respectively, which are appropriate for this topology [ 20 ].