Thermal analysis method of electromagnetic linear actuator based on multiphysics coupling model

Abstract

Electromagnetic linear actuators (ELAs) may be confronted with unsatisfactory performance when subjected to overheating. Therefore, it is significant to clarify its thermal characteristics and design the thermal performance requirements. A thermal analysis method based on multiphysics coupling model was presented, which uses the non-simplified loss distribution as the heat source to calculate the temperature field, adjusts the material properties by temperature, and considers the interaction between motion (including impact) and loss. More importantly, an improved universal equivalent winding to satisfy the condition of real compact concentrated winding was developed. Finally, the validity of this approach was verified through the experiment, and the regularity of temperature was summarized. The results show that the error of simulation and experiment is less than 6% and the permissible continuous operation frequency is no more than 30 Hz. The approach proposed in this paper can be employed not only to the ELA, but also to the design and analysis a wide range of electromagnetic machines.

Keywords

Electromagnetic linear actuator equivalent winding multiphysics coupling model thermal analysis method

1. Introduction

Electromagnetic linear actuator (ELA) as a kind of energy converter has been widely applied in vehicles [1,2] and fluid control systems [3]. Because of severe volumetric and thermal constraints, the ELA is at high risk of overheating, which can influence the reliability, efficiency and working capacity [4]. Therefore, clarifying ELA’s temperature rise and temperature distribution is necessary.

Finite element analysis (FEA) that based on the coupling model of multiphysics fields is commonly used in thermal calculation [4–6]. An ideal multiphysics coupling model involves the inclusion of detailed physical models that iteratively exchange the information [7–9] of energy consumption [10,11] and heat to accurately predict the electromagnetic and thermal properties. In general, the commonly used heat source loading method is to apply an average heat source to the machine parts [7–11]. It is considered that the loss density is uniform throughout those parts, and the loss is proportional to the volume [8,10–13]. Despite high operability, this method fails to fully consider the uneven distribution of the iron loss. The accuracy of the multiphysics coupling model can be improved by using non-simplified losses as the heat source, considering the effect of motion, and adjusting material properties according to temperature. Thus, an accurate multiphysics coupling model is proposed for thermal analysis and quantitative loss analysis furtherly in this paper.

Besides, the main loss within the ELA is concentrated on the winding region [14]. Meanwhile, there is a strong interaction between the electromagnetic and thermal designs as the temperature of windings is critically dependent upon the power density of ELA and vice versa. Numerous researches show that the insulation life is an exponential function of peak temperature [15,16]. Temperature rise with a few degrees at the peak operating point of winding would result in a sharp reduction of lifetime and performance degradation of the actuators [17]. The composition of windings is complex, which makes it difficult to build the actual winding distribution model. Hence, it is essential to establish a computationally efficient equivalent winding model with high precision in thermal analysis.

The computational methods employed in thermal analysis adopt various techniques to represent the winding region, including lumped parameter approaches [18–21], equivalent material representation [22,23] and layered winding method [13,24,25]. Compared with other equivalent winding methods, the layered winding method has the advantages of fast solving-time and high precision [20], including the hot-spot identification of temperature prediction. In reference [25], the wires are only equivalent to copper layers, which is simplified and can not obtain the appropriate results of thermal property. Besides, although the improved layered winding proposed in reference [13] can better reflect the real wires and go well with the common industrial motor whose windings are relatively loose, it dose not apply to high power actuators. The high power density of ELA determines the maximum volume fill factor of metal conductors in the winding to obtain better performance and reduce volume [14,18,21,26–28]. So more suitable layered winding needs to be applied to the thermal model. As a result, an improved layered winding method is proposed in analysis of the compact winding of the ELA. Then, this universal method can reduce the complexity of calculation and guarantee accuracy.

In this paper, the multiphysics fields model for thermal analysis of ELA based on equivalent winding is investigated. The structure parameters and coupling relationship between multiphysics fields of ELA are introduced in Section 2. Much operations are done for the multiphysics fields model and the improved layered winding method in Section 3. In Section 4, the results are verified by the experiment. The distribution of loss and temperature rising are quantitatively analyzed in Section 5. Besides, the alternation tendency of temperature is compared and discussed. The work is concluded in Section 6.

2. Structure and coupling relationship of ELA

2.1. Structure and operation principle

An ELA with a cylindrically symmetrical configuration is shown in Fig. 1. It consists of a compact concentrated winding (coil and framework), a sleeve, an armature, an outer stator (yoke, pole shoe and cover), and two permanent magnets magnetized along the axis inversely. The sleeve, armature and outer stator are made of steel-1008. The NdFeB PM is chosen with remnant flux density of 1.30 T, coercive force of 990 kAm⁻¹ and relative permeability of 1.06. A polytetrafluoroethylene is applied as coil framework to support the enamelled copper coil.

The hybrid working electromagnetic field is generated by winding and two PMs, which mainly includes three magnetic flux loops (𝜓₁, 𝜓₂, 𝜓₃) shown in Fig. 1. The magnetic flux loops 𝜓₁ and 𝜓₂ acting only by the PMs without winding excitation current can make the armature (black solid line shown in Fig. 1) remain stable at the end of the stroke. The winding is then excited with a release current, which generates a magnetic flux loop 𝜓₃ to weaken the flux 𝜓₁ but enhance the flux 𝜓₂. This causes an unbalanced force acting on the armature, the armature’s displacement (red dotted line shown in Fig. 1), and magnetomotive force are exported as a result. Similarly, the armature will move in the opposite direction by changing the polarity of the current applied to the winding, so the actuator can work stably and continuously. The main parameters of ELA are given in Table 1.

Fig. 1.

The schematic diagram of ELA.

Table 1

Main parameters of ELA

Parameter	Value/mm
Prototype height/mm	27
Prototype diameter/mm	54
Stroke/mm	3(±1.5)
Winding region height h/mm	18.7
Winding region width w/mm	8.4
Armature height/mm	8
Air gap/mm	0.3
PM height/mm	2.8
Coil turns	119

2.2. Analysis of coupling relationship

As a mechanical and electrical energy conversion device, the results of ELA’s input energy conversion directly determine its performance. But ELA’s dynamic process is a complex physical process, in which multiphysics coupling is performed, and the energy conversion of ELA is different when armature is in motion and stationary state. Figure 2 illustrated the coupling physical fields interaction in trigger phase and moving phase.

The trigger time is controlled by increased current, loading and eddy current, while the moving time is dominated by the change of current, back EMF and eddy current. The current in winding and the eddy current in soft magnetic material will generate a large amount of heat causing the temperature rise of ELA, and the temperature rise of ELA simultaneously changes the electromagnetic performance of the material. Hence, the eddy current is a strong factor in response and power consumption. So it is an important tool to study the performance of ELA by establishing a multi-physics coupling model that conforms to the actual working conditions and analyzing the energy consumption and temperature rise of ELA.

Fig. 2.

Interactions in coupling physical fields of ELA.

3. Multiphysics coupling calculation model

3.1. Electromagnetic and mechanical model

Electromagnetic model and mechanical model are inextricably linked in the whole energy conversion system. Using magnetic vector potential and electric scalar potential, the 3D transient mathematical model of the eddy current field is given as follow. $\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle {\nabla}\times \left(\frac{1}{{\mu}}{\nabla}\times \boldsymbol{A}\right)-{\nabla}\times \left(\frac{1}{{\mu}}{\nabla}\cdot \boldsymbol{A}\right)+{\sigma}\frac{\partial \boldsymbol{A}}{\partial t}+{\sigma}{\nabla}{\varphi}=0\\ \displaystyle {\nabla}\cdot \left(-{\sigma}\frac{\partial \boldsymbol{A}}{\partial t}-{\sigma}{\nabla}{\varphi}\right)=0\end{array}\end{eqnarray}$ (1) where A is the magnetic vector potential, φ is the electric scalar potential, σ is the conductivity, ∇ is the curl operator, μ is the material permeability.

The magnetic virtual displacement approach is adopted to calculate the magnetic virtual work force (F _m) produced by the polarzing and control the magnetomotive force. The F _m is calculated for all air elements with the magnetic virtual displacement specification adjacent to the armature body. $\begin{eqnarray}\displaystyle F_{m}=\lim _{{\Delta}x\rightarrow 0}\frac{{\Delta}W^{\prime }}{{\Delta}x}=\frac{dW^{\prime }}{dx}=-\left[\left(\frac{E_{x+{\Delta}x}+E_{x}}{{\Delta}x}\right)\right] & & \displaystyle\end{eqnarray}$ (2) where W is the incidental energy change for the small displacement Δ_x. E _x is the co-energy of the system at zero displacement point, and E _x+Δx is the co-energy at the assumed armature position. The mechanical model can be represented by Newton’s second law as $\begin{eqnarray}m\frac{d^{2}x}{dt^{2}}=F_{m}-c_{d}\frac{dx}{dt}-F_{l}\end{eqnarray}$ (3) where m is the quality of armature, c _d is the damping coefficient (20 N m⁻¹ s⁻¹ in this paper), F _l is loading force. The characters of ELA can be obtained by solving the model used finite element method.

3.2. Loss model

The temperature rise of ELA is closely related to the loading mode of heat source. The difference in the location of the heat source will result in a different temperature distribution. Therefore, it is necessary to pay attention not only to the size of the loss but also to the distribution of the loss.

Copper loss and iron loss are the main losses of ELA. Copper loss P _copper defined as ohmic loss mainly depends on the resistance R of the winding and the current excited I by the winding, which can be computed by Eq. (4). $\begin{eqnarray}P_{copper}=\int _{0}^{t}I^{2}R\text{d}t.\end{eqnarray}$ (4)

Iron loss P _Iron caused by the soft magnetic material in changing magnetic field, can be mainly divided into eddy current loss P _e, hysteresis loss P _h and stray loss P _ex. In general, the calculation of ELA’s iron loss is based on the Steinmetz equation by loss separation method in frequency domain or time domain: $\begin{eqnarray}P_{iron}=P_{h}+P_{e}+P_{ex}.\end{eqnarray}$ (5)

However, due to the heavy workload and sophisticated calculation of ELA’s iron loss, it is necessary to model and simulate with the help of computer software to improve efficiency and precision. The calculation equations are as follows: $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}\displaystyle P_{h}=\frac{1}{T}\int _{0}^{T}k_{h}(f)B^{a(f)}∼\text{d}t=\frac{1}{T}\mathop{\sum }_{i=1}^{N}\left(\frac{1}{T}\right)B_{i}^{a(\frac{1}{T})}{\Delta}t\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}\displaystyle P_{e}=\frac{1}{T}\int _{0}^{T}k_{e}(B)\left(\frac{dB}{dt}\right)\text{d}t=\frac{1}{T}\mathop{\sum }_{i=1}^{N}k_{e}(B_{i})\left(\frac{B_{i+1}-B_{i}}{{\Delta}t}\right)^{2}{\Delta}t\end{eqnarray}$ (7) where f is the excitation frequency, k _h and k _e are iron loss coefficients. a is the empirical coefficient material related, N is the number of discrete points and the calculation step Δt is 0.1 ms in this paper. The coefficients k _h, k _e and a are related to the material, which are provided by soft magnetic material manufacturers. With the variations of excitation frequency and material’s magnetic flux density, those coefficients are taken into account in the calculation.

3.3. Thermal model

The coupling model of ELA is finally expressed in the form of thermal. The thermal calculation model is established to verify the accuracy of the coupling model, and then the temperature rise of ELA can be obtained.

3.3.1. Thermal field calculation model

The 3-D transient heat transfer equation of ELA is: $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \frac{\partial }{\partial x}\left(K_{x}\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(K_{y}\frac{\partial T}{\partial y}\right)+\frac{\partial }{\partial z}\left(K_{z}\frac{\partial T}{\partial z}\right)+q={\rho}c\frac{\partial T}{\partial t}\\[12.0pt] \displaystyle -{\lambda}\left.\frac{\partial T}{\partial n}\right|s_{i}=0\\[12.0pt] \displaystyle -{\lambda}\left.\frac{\partial T}{\partial n}\right|s_{j}={\alpha}(T_{w}-T_{f})\end{array}\right.\end{eqnarray}$ (8) where 𝜆 is the thermal conductivity, 𝛼 is the heat convection coefficient, q is the calorific value per unit mass per time, 𝜌 is the density of material, c is the specific heat capacity of material, S _i is the surface of heat insulation boundary, S _j is the surface of heat dissipation boundary, K _x, K _y and K _z are the coefficients of thermal conductivity in the direction of x, y, z of each material, T, T _w and T _f are expressed as the initial temperature, the surface temperature and the ambient temperature, respectively.

3.3.2. Heat convection coefficient calculation

The ELA discussed in this paper operates in the atmospheric environment without high-temperature, the influence of thermal radiation can be neglected. Heat convection coefficients can be gained directly from thermal coupling field analysis. Based on the natural-convection heat transfer theory and Stefan–Boltzman Law, the heat convection coefficient 𝛼_c can be obtained as: $\begin{eqnarray}{\alpha}_{c}=\frac{N_{u}{\lambda}_{a}}{l}=C(G_{r}\cdot P_{r})^{m}\frac{{\lambda}_{a}}{l}\end{eqnarray}$ (9) where N _u, G _r, P _r, C, m are the numbers of Nusselt, Grashof, Prandtl, constant and Nusselt coefficient respectively. 𝜆_a is the air thermal conductivity coefficient at the certain temperature, and the characteristic length l represents the height of ELA. A specific heat transfer model has been done in our previous work [27].

3.3.3. Equivalent winding conduction calculation model

Under severe volumetric constraint demands, ELA requires compact winding to ensure its performance. Due to the mutual force between the wires, the regular section layout of winding arrangement is shown in Fig. 3. This arrangement gives a maximum packing factor close to the actual winding, and the regular hexagonal unit in Fig. 4, which describes the common geometrical structure of coil, can be used as the basic model for the analysis.

Fig. 3.

Layout and basic model of winding.

Fig. 4.

Schematic representation process of equivalent thermal conductor layer.

It is unrealistic to build each wire in the thermal model. At the same time, the whole winding is composed of copper, coating film, framework, impregnating varnish and air, which increases the complexity of the calculation model. Therefore, an improved layered winding method is established on the premise of ensuring the accuracy of temperature prediction.

It’s assumed that the heat is transferred from a single wire’s copper to the coating film at steady state without circumferential heat flow. First, as shown in Fig. 4(a), the temperature T and heat flow Φ from the center of the conductor to the outer diameter of the wire can be deduced from the differential equation of heat conduction and the boundary conditions: $\begin{eqnarray}\left[\begin{array}{@{}c@{}}T_{0}\\ {\phi}_{1}\end{array}\right]=\left[\begin{array}{@{}l@{}}1\\ 0\end{array}\right]T_{1}+\left[\begin{array}{@{}c@{}}\displaystyle \frac{R_{1}^{2}}{4{\lambda}_{cu}}+\frac{R_{1}^{2}}{2{\lambda}_{\mathit{cf}}}\ln \frac{R_{2}}{R_{1}}\\[13.0pt] {\pi}R_{1}^{2}l\end{array}\right]Q_{1}\end{eqnarray}$ (10) where, R ₁ and R ₂ are the radius of copper layer and coating film of a single wire respectively, T ₀ is the temperature at the copper center, T ₁ is the temperature at outer diameter of the wire, 𝜆_cu is the thermal conductivity of copper, 𝜆_cf is the thermal conductivity of coating film, l is the length of wiring, and Q ₁ is the heat production rate in the copper.

Then, the wire is equivalent to the circular thermal conductor of uniform material as shown in Fig. 4(b), and the temperature and heat flow can be calculation by Eq. (11). $\begin{eqnarray}\left[\begin{array}{@{}c@{}}T_{0}\\ {\phi}_{1}\end{array}\right]=\left[\begin{array}{@{}l@{}}1\\ 0\end{array}\right]T_{1}+\left[\begin{array}{@{}c@{}}\displaystyle \frac{R_{2}^{2}}{4\bar{{\lambda}}_{1}}\\[12.0pt] {\pi}R_{2}^{2}l\end{array}\right]Q_{2}\end{eqnarray}$ (11) where, $\bar{{\lambda}}_{1}$ and Q ₂ are the thermal conductivity and the heat production rate of the equivalent circular thermal conductor, respectively. According to the equal heat production rate, the equivalent thermal conductivity can be calculated: $\begin{eqnarray}\bar{{\lambda}}_{1}=\frac{{\lambda}_{cu}{\lambda}_{\mathit{cf}}}{{\lambda}_{\mathit{cf}}+2{\lambda}_{cu}\ln {\displaystyle \frac{R_{2}}{R_{1}}}}.\end{eqnarray}$ (12)

Moreover, as shown in Fig. 4(c), the wire can be equivalent to a regular hexagon under this circumstance of the same area. Finally, considering the cross-section of winding, the regular hexagon of wire is homogenized into a rectangle, as shown in Figs 4(d) and 4(e). It’s regarded as the thermal conductor layer. The heat conduction process similar to the above formula can be expressed as: $\begin{eqnarray}\left[\begin{array}{@{}c@{}}T_{0}\\ {\phi}_{2}\end{array}\right]=\left[\begin{array}{@{}c@{}}1\\ 0\end{array}\right]T_{1}+\left[\begin{array}{@{}c@{}}\displaystyle {\alpha}\frac{\left(2\sqrt{3}-3\right){\pi}R_{2}^{2}}{2{\lambda}}+{\beta}\frac{\left(6\sqrt{3}-9\right){\pi}R_{2}^{2}}{8{\lambda}}\\[6.0pt] \displaystyle \frac{3\sqrt{3}}{2}L^{2}l\end{array}\right]Q_{3}.\end{eqnarray}$ (13)

Based on equal area and equal heat production rate, the equivalent thermal conductivity of thermal conductor layer with different heat transfer paths can be obtained by Eq. (14). In this condition, the integral interval is the side lengths of equivalent rectangle, as shown in Fig. 4(e). Where, the proportional coefficients 𝛼 and 𝛽 are determined by the cross-section dimension of the compact winding, and can be calculated by Eq. (15). Then, the wire can be regarded as a standardized equivalent square, as shown in Fig. 4(f). $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}\displaystyle {\lambda}={\alpha}[\left(4\sqrt{3}-6\right){\pi}]\bar{{\lambda}}_{1}+{\beta}\left[\left(3\sqrt{3}-\frac{9}{2}\right){\pi}\right]\bar{{\lambda}}_{1}\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle & \text{} & \displaystyle \text{}\displaystyle {\alpha}=\frac{h}{h+w};\quad {\beta}=\frac{w}{h+w}.\end{eqnarray}$ (15)

Due to the ratio of the inside diameter to the outside diameter of the wire can be obtained by measured before calculated, $\bar{{\lambda}}_{1}$ can be considered as a constant. The equivalent thermal conductivity 𝜆 is 3.72 W/(m ⋅ K). $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \bar{c}=\frac{c_{1}\cdot {\rho}_{1}\cdot v_{1}+c_{2}\cdot {\rho}_{2}\cdot v_{2}+c_{3}\cdot {\rho}_{3}\cdot v_{3}+\cdots }{{\rho}_{1}\cdot v_{1}+{\rho}_{2}\cdot v_{2}+{\rho}_{3}\cdot v_{3}+\cdots }=\frac{\displaystyle \mathop{\sum }_{i=1}^{n}c_{i}\cdot {\rho}_{i}\cdot v_{i}}{\displaystyle \mathop{\sum }_{i=1}^{n}{\rho}_{i}\cdot v_{i}}\\[12.0pt] \displaystyle \bar{{\rho}}=\frac{{\rho}_{1}\cdot v_{1}+{\rho}_{2}\cdot v_{2}+{\rho}_{3}\cdot v_{3}+\cdots }{v_{1}+v_{2}+v_{3}+\cdots }=\frac{\displaystyle \mathop{\sum }_{i=1}^{n}{\rho}_{i}\cdot v_{i}}{\displaystyle \mathop{\sum }_{i=1}^{n}v_{i}}.\end{array}\right.\end{eqnarray}$ (16)

It can be seen that the single wire is homogenized into a thermal conductor layer of the uniform material consisting of the copper and coating film. Then, the rest parts of the winding are referred as insulation layer. Based on the principle of equal area and equal heat generation rate, the thermal conductor layer and the insulation layer are arranged orderly. Then, the equivalent specific heat capacity $\bar{c}$ and equivalent density $\bar{{\rho}}$ of the thermal conductor layer can be estimated using the volume ratios of each material by Eq. (16). This calculation is also applicable to the insulation layer. This method is universal and can also be used for slotted motors.

3.4. Two-way coupling model

It is noteworthy that these physical fields are not independent of each other. In order to ensure the accuracy of simulation calculation, data exchange between the multiphysics fields can be carried out by a two-way coupling model. In this study, FEM and analytical model are used to analyze the transient characteristic of ELA considering the interactions among electromagnetics, mechanics, loss and thermal, as shown in Fig. 5. The coupling model adopts the nonlinear B–H curve of the magnetic material to obtain the accurate electromagnetic field results and distribution of loss. Also, the distributions of copper loss and iron loss are taken as the heat source for the thermal field. The analysis results of the temperature will reversely be coupled to adjust the material properties in the electromagnetic field. Besides, the interaction between motion and loss as well as the impact between armature and yoke are also considered. This progress will be repeated until thermal stability is achieved. The primary factors affecting the ELA’s thermal analysis results and thermal distribution are mainly loss coefficients, heat conductivity and heat transfer coefficient of each area. Based on the analytical model above, the heat conductivity and heat transfer coefficient as well as the iron loss coefficients are all related to the material, which are set in JMAG-Designer’s material library.

Fig. 5.

Mechanical-electromagnetic-thermal multiphysics two-way coupling model.

Fig. 6.

Prototype and test bench.

4. Experiment and verification

It is difficult to measure the internal temperature, due to the compact cylindrical configuration of ELA. Therefore, a general dynamic performance test bench [28] is established and the temperature of the external surface monitoring point is used for testing, as shown in Fig. 6. The platinum thermistors (Pt100) are arranged outside ELA to verify the accuracy of the thermal model, and the conversion relationship between temperature and thermistor can be expressed as Eq. (17). In this equation, R _Pt is the resistance of platinum thermistor Pt100, which is measured in real-time by a resistance tester. Additionally, the fixed-point DSP TMS320F2812 is chosen as the digital controller. The DSP control board, interface circuit, power amplifier and current sensors are integrated into the control unit. Besides, the comparison of test and simulation temperature at different test points under 25 Hz is shown in Figs 7 and 8. $\begin{eqnarray}T=\frac{R_{Pt}-100}{0.385}.\end{eqnarray}$ (17)

The experiment is conducted without any cooling measures and the ambient temperature is 23 °C. The excitation amplitude is 13 A and the excitation frequency is 25 Hz. The end surface of the actuator is in contact with the fixed metal plate during the test, while the lateral surface is exposed to air. As a result, the temperature of surface at point A is always lower than that at points B and C during the test. The test temperature at point A shows an almost linear upward trend within 9 minutes, and it also increases within 9–34 minutes, but the rate of increase slowed down significantly. After 34 minutes, it begins to reach thermal stability. This trend is similar to the results at points B and C. The simulation results at points B and C coincided with each other due to the symmetrical structure of ELA. But the test temperature value at point C is always higher than that at point B because the movement of point B always preceded point C in the process of testing. The error between experimental temperature and simulation temperature is less than 6%, which can prove the correctness and precision of the thermal analysis method using the multiphysics coupling model. The multiphysics coupling model also lays a foundation for the follow-up research on quantitative analysis of loss, quantitative analysis of temperature, and temperature rising law.

Fig. 7.

Temperature comparison of test and simulation at point A.

Fig. 8.

(a) Temperature comparison of test and simulation at point B. (b) Temperature comparison of test and simulation at point C.

5. Results and discussions

The correctness and precision of the multiphysics coupling model have been validated in above section, and the current curve obtained from the test is the only input coupled to the multiphysics coupling model. So the quantitative analysis of the loss and temperature as well as research on temperature rising law of ELA is carried out without changing other conditions based on the multiphysics coupling model in simulation.

5.1. Quantitative analysis of loss

The circulating energy of ELA is shown in Fig. 9, which illuminates the change rules of energy consumption with working frequency. As can be seen from Fig. 9, the output circulating copper loss decreases by 23.3% with the increase of working frequency. This is because the free phase of the current shortened with the decrease of cycle time. Meanwhile, the intensified changing of the electromagnetic field makes the output circulating iron loss increased by 12.2%. As a result, the input circulating energy increases by 2.7%. The iron loss is greater than that of copper loss in all cases. Moreover, since the circulating energy of ELA remains basically unchanged, the loss distribution and peak density of ELA under the condition of static and moving armature are calculated and shown in Fig. 10.

When the armature is static, the area with high loss density concentrates near the end of the armature, and when the armature is moving, the area with high loss density concentrates on the armature face in contact with the sleeve. Moreover, the loss density of ELA when the armature is moving is much larger than when the armature is static, which can be proved that the movement has a significant influence on the loss distribution and density of ELA. Furthermore, the coupling relationship of physical fields is verified. It can be seen that although the loss distribution is not uniform and is greatly affected by movement, the overall trend is concentrated on the regions of armature and winding. Meanwhile, the iron loss distribution of each part is an important basis for temperature calculation. Figure 11 shows the percentage of iron loss from each component to the total iron loss at different operating frequencies.

Fig. 9.

Circulating loss versus working frequency.

Fig. 10.

Loss distribution and density cloud of ELA at peak value.

Fig. 11.

Loss distribution of each component.

The percentage of iron loss ratio of each component to total iron loss ratio remains stable with the change of working frequency. Among them, armature and cover accounted for the most proportion of 44.2% and 21.3% of total iron loss, respectively. While, armature iron loss and winding copper loss account for 24.7% and 42.8% of the total loss, but the mass and volume of the armature are far less than that of the winding and cover, indicating that the armature is the main heat source compared to cover and winding. This tendency provides a basis for thermal field analysis of heat source loading based on loss distribution.

5.2. Quantitative analysis of temperature

Figure 12 illustrates the temperature distribution of ELA applying 25 Hz current excitation at thermal stability, and the emphasis is laid on the layered windings. Figure 13 reflects the average temperature changing in the main parts of ELA with thermal time constants, while the temperature changing of windings is replaced by the temperature at the center of the winding.

Fig. 12.

Cloud chart of temperature distribution.

Fig. 13.

Average temperature changing of ELA.

Fig. 14.

Maximum balance temperature in working process.

As shown in Figs 12 and 13, the variation trend of average temperature is similar with each part. The balance temperature is listed from high to low: armature, winding, cover, sleeve and PMs. But the component with the fastest temperature rise and highest temperature under thermal stability is the armature, followed by the winding, due to both armature and winding are the main heat source of ELA and located in the close circumstance, but the small volume of armature makes it more intense. Though PMs and sleeve are installed inside of ELA, the movement of armature makes it have the effect of air-cooled heat dissipation. In addition, heat convection between cover and environment makes the temperature of cover lower than that of winding. The results are in good agreement with loss analysis.

5.3. Research on temperature rising law

Since the steady operation of ELA is limited by the permissible temperature rising of PMs and winding. The maximum permissible temperature rising of winding is 220 °C, which is higher than 160 °C of PMs. Due to the temperature of winding is higher than that of PMs during the working process, so it is necessary to compare the temperature of the two parts to secure its running.

Figure 14 shows the maximum balance temperature of winding and PMs in the working process with different work frequencies. As can be seen that the higher working frequency is, the higher the temperature is. As a result, the time for PMs to reach limiting temperature is more than 190 s. Without destroying the PMs characteristics, the ELA can work continuously under the condition that the working frequency is no more than 30 Hz. While the time of winding to reach limiting temperature is more than 280 s. Meanwhile, if the working frequency is less than 40 Hz, the winding allows ELA to operate continuously. Although the temperature rise of winding is faster than that of PMs, the PMs always reach the limiting temperature firstly. Therefore, the limiting temperature of ELA is deemed as the PMs’ temperature. Besides, the common operation condition in application of ELA is intermittent with variable frequency. It’s obvious that this law derived by the means of simulation with the model verified by the experiment shows a significant guidance for the design of high power density actuators. Meanwhile, it can lay a foundation for controller design of low-cost parameters.

6. Conclusion

In this article, a thermal analysis method considering the distribution of loss for ELA based on the multiphysics coupling model was established and verified. Particular focus was put on the equivalence method of the compact winding. The distribution of loss and the rules of temperature were analyzed. The main results are summarized as follows:

(1) The experimental data agree well with the simulation results, and the error is less than 6%, which validates the correctness and precision of the equivalent winding thermal analysis method.

(2) The armature iron loss and copper loss account for 24.7% and 42.8% of the total loss respectively, and are the main components of the heat source. Besides, the balance temperature is listed from high to low: armature, winding, cover, sleeve and PMs.

(3) The limit temperature of ELA is the limit temperature of PMs. The higher working frequency is, the higher the temperature is. Thus, the ELA can operate continuously at a working frequency of less than 30 Hz.

Footnotes

Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (Grant Nos. U1806215, 11572063, 51905319 and 51905319), the 111 Project (B14013) and the Fundamental Research Funds for the Central Universities of China (DUT18ZD103).

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