Collaborative optimization of multi-physical fields for composite electromagnetic linear actuators

Abstract

The composite electromagnetic linear actuator features a co-axial integrated design that merges the moving coil and moving iron actuator types. This design leads to significant interactions among electrical, magnetic, and thermal fields during operation, which in turn affect energy consumption and temperature increases. To further improve the material utilization rate of the actuator and optimize its comprehensive performance, such as force density and temperature rise, a multi-physical field collaborative optimization method was proposed. First, the energy consumption and temperature rise mechanisms of the actuator were analyzed. The bidirectional coupling relationships of multi-physical fields during its operation are studied. Simulations and experiments were conducted to establish the rules governing the spatiotemporal evolution of energy consumption and temperature fluctuations under standard operational conditions. Based on this, a system-level collaborative optimization model for the actuator was established. Sensitivity analysis helped segregate the design variables into system-level and independent categories. Subsequently, using a multi-objective genetic algorithm, the optimal parameter settings were determined. The optimization results revealed that the force constant of the main driving components increased by 5.4%, and the average temperature of the coil decreased by 6.3%; additionally, the starting force of auxiliary components increased by 2.3%, and the temperature of the yoke decreased by 0.8%.

Keywords

Composite electromagnetic linear actuator multi-physical field collaborative optimization temperature rise characteristics

1 Introduction

With the development of rare earth permanent magnet materials and modern control theories, new topologies [1] and control methods for linear motors are continuously emerging. These motors find widespread application in sectors such as automotive [2], maritime, robotics, and renewable energy generation. Electromagnetic linear actuators [3] are known for their high force-to-power ratio, quick response, and compact structure. They are integral to systems like fully variable valve trains, electronically controlled fuel injectors, as well as other high-precision linear servo [4] motion systems where space is at a premium.

The broadening range of application scenarios has led to enhanced requirements in terms of precision, force density, and reliability. As a mechano-electro-magnetic coupled system [5], the energy consumed by the actuator during its operation is irreversibly transformed into heat, causing temperature increases that adversely affect performance and durability. Traditional approaches such as core slotting [6] and widening the air gap to reduce energy loss and heat can compromise output performance and increase the system’s size. Thus, it is essential to maintain a balanced management of electromagnetic and thermal fields to enhance material efficiency and optimize overall performance.

Numerous researchers, both domestically and internationally, have conducted extensive studies on the multi-physics performance analysis and optimization of actuators or linear motors. Dai Jianguo [7] analyzed the intricate heat exchange process within electromagnetic linear actuators based on heat transfer theory, and developed a heat transfer theoretical model for these exchanges. Tan Cao [8] developed an electromagnetic-mechanical-thermal coupled model for a double-acting electromagnetic linear actuator, validating the model through experimental trials. The intricate multi-physics coupling relationship has greatly increased the complexity of optimizing actuator designs. A.K. Dhingra [9] proposed a multi-objective optimization method based on cooperative fuzzy game theory, effectively tackling the limitations of single-objective and multi-objective optimization designs. Hua Wei [10] reviewed modern motor optimization methods, highlighting prevalent models and algorithms, such as surrogate models, genetic algorithms, particle swarm optimization algorithms, and shared insights on multi-physics field optimization and robust optimization strategies.

The novel composite electromagnetic linear actuator integrates two types of actuators in a coaxial design [11], leading to a more intricate and compact structure that presents challenges with temperature rise and energy consumption. This paper explores the distribution rules of multi-physics field coupling in the composite actuator. In response to the limitations of traditional optimization methods in balancing efficiency and precision, we proposed a coordinated optimization method for the multi-physics field of the composite electromagnetic linear actuator, ultimately enhancing the actuator’s multi-physics comprehensive performance.

2 Structure and principle of the composite electromagnetic linear actuator

The structure of the composite electromagnetic linear [12] actuator (CELA), depicted in Figure 1, incorporates the Moving Coil Electromagnetic Linear Actuator (MCELA), Moving Iron Electromagnetic Linear Actuator (MIELA), spring, connectors, and valves [13].

Figure 1.

Structure diagram of the CELA.

The MCELA functions as the principal driving component, consisting of a moving coil skeleton, coils, permanent magnets, inner magnetic yoke, and outer magnetic yoke. The lower end of the moving coil skeleton is directly connected to the valve, facilitating valve motion control by adjusting the coil current. The internal magnetic field employs a Halbach array, offering linear output force and high control accuracy, albeit with a force density that is lower than that of other electromagnetic linear actuator types [14].

The MIELA serves as the auxiliary driving component, composed of an armature, a permanent magnet, a coil skeleton, a coil, and a magnetic conductive ring. The armature is tightly fixed to the moving coil skeleton through connectors, synchronizing with its movements. MIELA operates based on the principle of minimal magnetic reluctance and adjusts the output force by controlling the coil current [15]. MIELA possesses an outstanding end-holding capability when unpowered, thereby conserving energy while maintaining a high force density. However, the MIELA’s output force has a high degree of nonlinearity, posing challenges to maintaining precise control [16].

The composite electromagnetic linear actuator operates in two modes. In single-drive mode, only the MCELA primary drive component is powered, allowing MIELA to follow its motion. This mode is ideal for low-load scenarios. Conversely, in coordinated drive mode, both primary and auxiliary components are powered, making it optimal for high-load conditions. This dual-mode functionality allows the composite electromagnetic linear actuator to achieve a high force-to-power ratio, exceptional efficiency, and rapid responsiveness.

3 Establishment of the multi-physics field coupling model for the composite electromagnetic linear actuator

3.1 Mathematical model

3.1.1 Electromagnetic field model

Maxwell’s equations, consisting of Gauss’s law, Gauss’s magnetic law, Faraday’s law of induction, and Ampère’s law, provide the theoretical framework for studying electromagnetic fields. In engineering applications, the differential form of Maxwell’s equations is widely used, as shown in Eq. (1).

{\begin{matrix} \nabla \times H = J \\ \nabla \times E = - \frac{\partial B}{\partial t} \\ \nabla \cdot D = 0 \\ \nabla \cdot B = 0. \end{matrix}

(1)

Where, H is the magnetic field intensity, A/m; J is the current density, A/m²; E is the electric field intensity, V/m; B is the magnetic flux density, T; D is the electric displacement, C/m².

The relationships among H, J, E, B, D are shown in Eq. (2).

{\begin{matrix} D = ε_{0} ε_{r} E \\ J = σ E \\ B = μ_{0} μ_{r} H . \end{matrix}

(2)

Equation (3) is generally used to calculate the electromagnetic force acting on the electromagnetic coil in the air gap magnetic field:

F = \int_{l} I d l \times B .

(3)

For the calculation of the electromagnetic force in nonlinear magnetic fields, the boundary issue associated with finite elements can be converted into an equivalent variational problem. This problem is then broken down using interpolation functions to derive the finite element equation for the electromagnetic force, as presenteed in Eq. (4):

{\begin{matrix} J (A_{θ}) = 2 π \int_{S} (\int_{0}^{B} \frac{1}{μ} B d B - J_{θ} A_{θ}) ρ d ρ d z = min \\ ρ A_{θ} |_{l 1} = constant \\ F = \frac{d w_{e}}{d x} |_{i = c o n s t} = \frac{\partial}{\partial x} [\int_{V} (\int_{0}^{H} B \cdot d H) d V] \end{matrix}

(4)

Where, S denotes the boundary of the field domain; A_θ represents the magnetic potential; w_e represents the co-energy of the magnet, and the electromagnetic force is the derivative of the co-energy with respect to x.

3.1.2 Energy consumption field model

The form of energy loss in the composite electromagnetic linear actuator can also be determined according to the form of motor energy loss [17,18], which includes copper loss, iron loss, and mechanical energy loss, as shown in Eq. (5).

W_{t o t a l} = W_{c u} + W_{f e} + W_{m e c h} = \int | U I | d t

(5)

Where, W_total represents the input energy, W_cu indicates the copper loss, W_fe is the iron loss, W_mech is the mechanical energy loss, U₁ and U₂ are the input voltages for the MCELA and MIELA, respectively, while I₁ is and I₂ are their respective input excitation currents.

Copper loss is influenced by the winding coil resistance R₁ and R₂, and the excitation current I₁ and I₂ for the MCELA and MIELA, respectively. This can be calculated according to Eq. (6).

W_{c u} = \int I^{2} R d t .

(6)

The iron loss W_fe of the composite electromagnetic linear actuator is produced in a changing magnetic field. Based on the concept of loss separation, iron loss can mainly be divided into eddy current loss, hysteresis loss, and stray loss. The iron losses of the MCELA and MIELA are represented as W_e1, W_h1, W_ex1, W_e2, W_h2 and W_ex2, respectively. They can be represented by the mechanism in Eq. (7).

W_{f e} = W_{e 1} + W_{h 1} + W_{e x 1} + W_{e 2} + W_{h 2} + W_{e x 2} .

(7)

The experimental results show that the magnetic field of the moving-iron linear electromagnetic actuator is relatively variable when the actuator is moving, while the magnetic field is relatively stable when the actuator is moving, so the calculation formulas of eddy current loss are different. All iron losses can be calculated in the time domain using the following formula, where B_m represents the peak magnetic flux density, K_e1, K_h1, K_ex1, K_e2, K_h2 and K_ex2 are iron loss coefficients [19,20], as depicted in Eqs (8)–(11).

W_{e 1} (t) = \frac{1}{2 π^{2}} k_{e 1} {(\frac{d B}{d t})}^{2}

(8)

W_{e 2} (t) = \frac{1}{T} \int_{0}^{T} k_{e 2} (B) \frac{d B}{d t} d t = \frac{1}{T} \sum_{i = 1}^{N} k_{e 2} (B_{i}) (\frac{B_{i + 1} - B_{i}}{Δ t}) Δ t

(9)

W_{h 1} (t) = \pm \frac{1}{C_{β}} k_{h 1} | B_{m} \cos (θ) |^{β - 1} \frac{d B}{d t} .

(10)

Among,

\begin{aligned} C_{β} = 4 \int_{0}^{\frac{π}{2}} \cos^{β} θ d θ \\ W_{h 2} (t) = \frac{1}{T} \int_{0}^{T} k_{h 2} (f) B^{α (f)} d t = \frac{1}{T} \sum_{i = 1}^{N} k_{h 2} \frac{1}{T} B_{i}^{α (\frac{1}{T})} Δ t \end{aligned}

(11)

Where, f is the excitation frequency, β [21] is Coefficient of Steinmetz equation, typically between 1.6-2.2, α represents an empirical coefficient, in this paper N is the number of discrete points, and the computation step Δt is 0.1 ms. Consiering changes in excitation frequency and material magnetic induction strength, factors such as K_e1, K_h1, K_e2, K_h2 and α were taken into account. This is detailed in Eqs (12) and (13).

C_{e x} = {(2 π)}^{1.5} \frac{2}{π} \int_{0}^{\frac{π}{2}} \cos^{1.5} θ d θ .

(12)

By numerical integration: C_ex = 8.763,

{\begin{matrix} W_{e x 1} (t) = \frac{1}{C_{e x}} k_{e x 1} {| \frac{d B}{d t} |}^{1.5} \\ W_{e x 2} (t) = \frac{1}{C_{e x}} k_{e x 2} {| \frac{d B}{d t} |}^{1.5} . \end{matrix}

(13)

Mechanical loss refers to the energy lost due to the movement of mechanical components. At the beginning and end of a working cycle, the kinetic energy of the moving parts is zero. Given that the amount of spring compression remains consistent, the potential energy also stays unchanged. Hence, all energy exerted by the electromagnetic force is converted into mechanical loss. The calculation of mechanical loss for the composite electromagnetic linear actuator is outlined in Eq. (14):

W_{m e c h} = \int F_{m a g} \cdot v d t

(14)

Where F_mag is the electromagnetic force, and v is the average speed of the moving parts.

Calculating iron losses requires the loss coefficients K_e1, K_h1, K_ex1, K_e2, K_h2 and K_ex2. However, due to the complexity of calculating iron losses, finite element software is necessary.

3.1.3 Thermal field model

The heat generated by the composite electromagnetic linear actuator primarily consists of the copper loss from the coil winding and the iron loss from the magnetic material. This heat is radially transferred from the moving parts and the coils, eventually being emitted into the space between the outer yoke and the external air through convective heat exchange. Furthermore, as the moving parts reciprocate, heat is also axially transferred axially via complex mechanisms of conduction and convection. The three-dimensional transient heat conduction equation, along with initial and boundary conditions for the actuator, is presented Eq. (15):

{\begin{matrix} λ (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} + \frac{\partial^{2} T}{\partial z^{2}}) + q = ρ c \frac{\partial T}{\partial T} \\ T |_{t = 0} = T_{0} \\ - λ \frac{\partial T}{\partial n} = α (T_{w} - T_{e}) \end{matrix}

(15)

Where, λ is the thermal conductivity; q is the heat flux density per unit heat transfer area, W/m²; ρ is the density, kg/m³; c is the specific heat, J/(kg ⋅ °C); T_w is the temperature of the heat transfer surface, °C; T_e is the temperature of the environment near the heat transfer surface, °C; α is the conductivity heat of the heat transfer surface.

The main heat transfer methods between the external magnetic yoke and the external air are convective heat exchange and thermal radiation:

α_{V} = α_{V_{c}} + α_{V_{r}}

(16)

Where, α_Vc is the convective heat transfer coefficient, W/(m² ⋅ °C); α_Vr represents the radiative heat transfer coefficient, W/(m² ⋅ °C).

Based on the theory of natural convection heat transfer and the Stefan-Boltzmann law, Eqs (17) and (18) can be derived:

α_{V_{c}} = \frac{N_{μ} λ_{c}}{l} = C {(G_{r} P_{r})}^{m} \frac{λ_{c}}{l}

(17)

α_{V_{r}} = \frac{q}{Δ T} = \frac{ε σ (T_{w}^{4} - T_{e}^{4})}{T_{w} - T_{e}} .

(18)

Given the different structures and working principles of MCELA and MIELA, the convective heat transfer coefficients vary both among components and between the components and air.

3.2 Analysis of coupling interaction and multi-physics field coupling modeling

Within the actuator, several physical field couplings occur, encompassing electromagnetic field, temperature field, and energy consumption field. Given that the energy consumption field contributes directly to the temperature field, simplifying the analysis of these coupled fields can significantly save time and reduce complexity. Hence, this article mainly studies the bidirectional coupling between the electromagnetic and temperature fields.

The primary inputs of the temperature field are the copper and iron losses from the electromagnetic field, which are the main sources of heat generation. Concurrently, as the temperature continuously rises, the electrical conductivity of the magnetic material will change, affecting the magnetization effect. Temperature fluctuations also cause the impedance of various components to either increase or decrease, as detailed in Eqs (19)–(21).

R = K_{F} ρ_{t} \frac{l}{A_{0}}

(19)

ρ_{t} = \frac{0.0175 \times 10^{- 6} [1 + β (t - 15)]}{A_{0}}

(20)

B_{r t 1} = B_{r t 0} (1 - \frac{I L}{100}) [1 - \frac{α_{B_{r}}}{100} (t_{1} - t_{0})]

(21)

Where, R represents coil resistance, K_F is resistance increase coefficient, and ρ_t resistivity at t₀ temperature; β is the temperature coefficient of the conductor resistance, t is the conductor temperature, and A₀ is the cross-sectional area of the conductor. B_rt0 is the remanent magnetic strength at the temperature of t₀, B_rt1 is the remanent magnetic strength at the temperature of t₁, α_Br is the reversible temperature coefficient of the remanent magnetic strength, IL is the irreversible loss rate of the remanent magnetic strength, t₀ is the initial operating temperature, t₁ represents the operating temperature.

Considering the coupling between the electromagnetic and temperature fields and the relationship between energy consumption and temperature rise, a bidirectional coupling model is established. The specific process is shown in Figure 2.

Figure 2.

The setup process of multi-physical field coupling model.

Inside the composite actuator, soft magnetic material used is No. 8 steel, which has excellent magnetic conductivity, durability, and cost-effectiveness. The permanent magnet is made of Neodymium Iron Boron (NdFeB), and the excitation coil uses high-temperature resistant pure copper enamel wire. Nylon PA6 materials are used to make the coils skeleton.

3.3 Simulation and experimental results

The simulation process is set for typical conditions, featuring an 8 mm stroke, 20 ms, and a frequency of 12.5 Hz. In a collaborative working scenario, the composite electromagnetic linear actuator is simulated, and the results are verified through experimentation.

Given that our research team has previously conducted comprehensive simulations on the electromagnetic force characteristics [22] and dynamic behavior, achieving extensive experimental validation, this paper will not delve into those details. For more information, please refer to reference. The final results indicate that the force constant of the active drive component is approximately 24 N/A, the auxiliary driving component has a maximum starting force of about 110 N, and the holding force is 307 N. Verification confirms that, in collaborative mode, the experimental and simulation errors of the maximum stroke are controlled within a range of 0.02 mm for typical conditions.

3.3.1 Energy consumption and temperature distribution

The energy consumption distribution of the composite electromagnetic linear actuator varies based on its operating principles. In collaborative drive, both the MCELA and MIELA are energized simultaneously, influencing the overall energy distribution.

Figure 3.

Energy consumption and Temperature distribution of cooperative driven MCELA and MIELA. (a) Energy consumption distribution of cooperative driven MCELA. (b) Energy consumption distribution of cooperative driven MIELA. (c) Temperature distribution of cooperative driven MCELA and MIELA.

In a single cycle of collaborative drive mode, the total energy consumption of the composite electromagnetic linear actuator is 10.825 J, with the MCELA contributing 1.065 J and the MIELA accounting for 9.76 J. The distribution of energy losses for both actuator types is illustrated in Figures 3(a) and 3(b). As shown in the Figure 3, the copper-to-iron loss ratio in the moving coil electromagnetic linear actuator is 5:4, primarily resulting from eddy current losses in the end caps and both inner and outer magnetic yokes. However, in MIELA, the ratio is 3:4, with iron losses predominantly occurring in the armature and shell, exceeding those in the MCELA.

Figure 4.

Temperature rise test platform simulation and test changing curve of CELA. (a) Temperature rise test platform of CELA. (b) Simulation and test changing curve of CELA.

Figure 5.

Multi-physical field collaborative optimization process of CELA.

The temperature distribution cloud map from 5 to 25 min of the actuator is shown in Figure 3(c). It can be seen that from time 5 to 15 min, the temperature rises rapidly, with the armature temperature being the highest and reaching its peak in the shortest time. From 15 to 25 min, the temperature rises rate slows down. The peak temperature of coils of MCELA is slightly higher than that of armature of MIELA.

3.3.2 Temperature rise simulation and testing

To verify the temperature rise characteristics of the composite electromagnetic linear actuator, a specialized test platform was established, comprising the actuator, an integrated controller, a DC power supply, a host computer, and sensors for voltage, current, displacement, and temperature. To prevent irreversible damage to the actuator, only the temperature changes at five measurement points over a 10-minute period were monitored to confirm the accuracy of the simulation. The positions of the measurement points and the test platform are shown in Figure 4(a).

Among them, MI1 and MC3 measure the end cap surface temperatures of the MIELA and MCELA, respectively. MC1 and MC2 monitor temperature measurement points for the shell of the MCELA, while MI2 measures the temperature on the side of the shell of MIELA. The change curve of temperature is illustrated in Figure 4(a).

From the Figure 4(b), it can be seen that the simulation values and measured values are generally consisten within a 2% error range. This consistency verifies that the simulation and experimental results are in substantial agreement.

4 Multi-physics field collaborative optimization design

4.1 Optimization method and process

Cooperative optimization method [23], initially proposed by Ilan M. Kroo and colleagues at Stanford University, is widely applied in aerospace, power transmission, and other fields. This method, distinct from multi-objective optimization, hinges on its capability to decompose complex multi-objective, multi-field coupling challenges into a system-level optimization problem and optimization problems of several subsystems through parallel optimization. This approach addresses the variable coupling problem of optimization.

According to the collaborative optimization method, the optimization results and constraints obtained by the subsystems are fed back to the system level, and all the constraints of subsystems participate in the optimization are regarded as penalty functions [24], so as to avoid the optimization result being local optimal. To maintain consistency in constraints across subsystems during iterative optimization, the sum of squared differences of all system level variables and subsystem independent variables is kept below a specified positive real value.

Given the multitude of optimization parameters for the composite electromagnetic linear actuator and the need to simplify the optimization process, a sensitivity layered approach is used to analyze different physical parameters. Eventually, highly sensitive design variables are selected as independent optimization variables for each physical field’s subsystem [25], while those with lower sensitivity are treated as system-level design variables. Independent variables for subsystems are optimized within their specific context, whereas system-level design variables are integrated from system-level optimization into the subsystems. The optimization process is outlined in Figure 5.

4.2 Establishment of the system optimization model

System-level optimization models for both moving coil and moving iron electromagnetic linear actuators are established as shown in Eqs (22) and (23):

{\begin{matrix} min M_{M C E L A} = f (B_{i}, B_{o}, B_{p}, H_{s}, H_{l}, r_{i}) \\ s . t . \sum_{1}^{n_{X} e l e c} {(X_{i} - X_{i}^{e l e c})}^{2} + \sum_{1}^{n_{Y} e l e c} {(Y_{i} - Y^{e l e c})}^{2} \leq ε \\ \sum_{1}^{X^{t e m p}} {(X_{i} - X_{i}^{t e m p})}^{2} + \sum_{1}^{n_{Y} t e m p} {(Y_{i} - Y_{i}^{t e m p})}^{2} \leq ε \\ {(X, Y)}_{min}^{T} \leq {(X, Y)}^{T} \leq {(X, Y)}_{max}^{T} \end{matrix}

(22)

{\begin{matrix} min M_{M I E L A} = f (h_{p}, b_{p}, h_{p s}, h_{g}, h_{a s}, h_{a}, r_{a}) \\ s . t . \sum_{1}^{n_{X} e l e c} {(X_{i} - X_{i}^{e l e c})}^{2} + \sum_{1}^{n_{Y} e l e c} {(Y_{i} - Y^{e l e c})}^{2} \leq ε \\ \sum_{1}^{X^{t e m p}} {(X_{i} - X_{i}^{t e m p})}^{2} + \sum_{1}^{n_{Y} t e m p} {(Y_{i} - Y_{i}^{t e m p})}^{2} \leq ε \\ {(X, Y)}_{min}^{T} \leq {(X, Y)}^{T} \leq {(X, Y)}_{max}^{T} \end{matrix}

(23)

Where X^elec and X^temp are system variables for the electromagnetic sub-system and temperature-rise subsystem respectively, Y^elec and Y^temp are variables of the electromagnetic subsystem and temperature-rise subsystem respectively. The difference between the optimal solution of independent variables and system variables in each subsystem is less than the small value ε. The structural parameter variables are shown below respectively:

For MCELA: inner magnetic yoke width is represented as B_i, outer magnetic yoke width is denoted as B_o, permanent magnet width is represented as B_p, short permanent magnet height is denoted as H_s, long permanent magnet height is represented as H_l, inner diameter of inner magnetic yoke is denoted as r_i.

For MIELA: permanent magnet height is denoted as h_p, permanent magnet width is represented as b_p, seating height of the permanent magnet is expressed as h_ps, magnetic conductive ring height is denoted as h_g, armature seating height h_as, armature height is expressed as h_a, armature radius is represented as r_a.

The schematics of the structure of the design variables for the two actuators are shown in Figures 6(a) and (b).

Figure 6.

Schematic of the planar structure of the CELA design variables. (a) Schematic diagram of the design variables of MCELA. (b) Schematic diagram of the design variables of MIELA.

4.2.1 Electromagnetic field subsystem optimization model

In the steady-state electromagnetic field, the electromagnetic force significantly impacts the performance of the CELA. Due to the different working principles and variations in their internal structural parameters, electromagnetic subsystems are established separately for MCELA and MIELA.

For MCELA, the primary goal is to maximize output force. This ensures its effectiveness in scenarios requiring high-speed movement and quick responses. The electromagnetic force can be calculated using the formula F = K_mI [26,27]. Where I represents the excitation current, K_m is taken as the force constant of the actuator, serving as the standard for measuring the magnitude of the output force. The larger the force constant, the higher the output force.

K_{m} = \frac{F}{I} .

(24)

The fluctuation rate of force [28] is a standard of measuring the linearity of the output force. Its calculation formula is depicted in Eq. (25):

K_{f} = \frac{F_{m} - F_{t}}{F_{m}} \times 100 %

(25)

Where F_m is the electromagnetic force when the position of mover is located at the midpoint inside the actuator, and F_t is the electromagnetic force when the position of mover is at the end. The smaller the fluctuation rate of force, the more linear the output.

Ultimately, the electromagnetic subsystem optimization model for MCELA is shown in Eq. (26):

{\begin{matrix} f i n d X^{e l e c}, Y^{e l e c} \\ min \sum_{i = 1}^{n_{Y} e l e c} {(Y_{i} - Y_{i}^{e l e c})}^{2} \\ s . t . K_{m_{min}} \leq K_{m} \leq K_{m_{max}} \\ K_{f_{min}} \leq K_{f} \leq K_{f_{max}} \\ X_{min}^{e l e c} \leq X^{e l e c} \leq X_{min}^{e l e c} \\ Y_{min}^{e l e c} \leq Y^{e l e c} \leq Y_{min}^{e l e c} . \end{matrix}

(26)

For MIELA, no losses or temperature changes occur in single-drive mode. However, due to its internal structure, a passive “holding force” is generated without power input, which can significantly impact the response speed if excessive. This holding force F_h is not represented by a regular straight line but a smooth curve. To obtain an optimal force characteristic curve, the linear coefficient K_c of the passive force-displacement characteristic curve [29] for MIELA is employed, as depicted in Eq. (27).

K_{c} = \frac{4 F_{0.375}}{3 F_{h}}

(27)

Where F_0.375 is the electromagnetic force at a position of 0.375 times the midpoint of the travel, F_h is the holding force. The maximum value of K_c is 1, and the closer to 1, the higher the linearity of passive force-displacement characteristic curve of MIELA.

During collaborative work process, the response speed is influenced by the magnitude of the starting force. As the armature moves, the internal magnetic field of the actuator changes, causing a nonlinear relationship between electromagnetic force and current. Thus, the force constant does not adequately represent the electromagnetic force output performance of MIELA. Research indicates that when current value is 11 A, the starting force peaks, enabling quick actuator response. Therefore, optimizing the starting force has been prioritized. The electromagnetic subsystem optimization model for MIELA is detailed in Eq. (28):

{\begin{matrix} f i n d X^{e l e c}, Y^{e l e c} \\ min \sum_{i = 1}^{n_{Y} e l e c} {(Y_{i} - Y_{i}^{e l e c})}^{2} \\ s . t . K_{c_{min}} \leq K_{c} \leq K_{c_{max}} \\ F_{s_{min}} \leq F_{s} \leq F_{s_{max}} \\ X_{min}^{e l e c} \leq X^{e l e c} \leq X_{max}^{e l e c} \\ Y_{min}^{e l e c} \leq Y^{e l e c} \leq Y_{max}^{e l e c} . \end{matrix}

(28)

4.2.2 Thermal field subsystem optimization model

Based on the temperature distribution law from Chapter 2, it is known that the temperature of the actuator’s armature inside is the highest temperature the actuator can reach. To effectively optimize temperature rise, the average highest temperature of the armature is used as the objective function, for MCELA, the average highest temperature of coils is expressed as T_C, and the average highest temperature of the MIELA armature expressed as T_I. Considering the prolonged calculation time for the transient temperature field, a neural network model is employed to simulate and replace the thermal field finite element model.

The most widely used structure in neural networks are consisted of 3 layers: input layer, hidden layer, and output layer. Given the relatively simple nonlinear function mapping between actuator inputs and outputs, the layers are each designed as single layers. The number of neurons in the input and output layers corresponds to the number of system input and output parameters. Specifically, for MCELA, there are 6 input neurons and 1 output neuron, and for MIELA, there are 7 input neurons and 1 output neuron. The selection of neurons in the hidden layer [30] has a significant impact on the prediction error and training speed of the surrogate model. The method used in this paper is shown in formula (29).

m = \sqrt{n + 1} + a

(29)

Where m represents the number of hidden layer neurons, n is the number of input neurons, l is the number of output layer neurons, and a is a constant ranging from 1–10. The final number of hidden layer neurons selected is 10.

Neural networks are a type of multilayer feed-forward network model based on error backpropagation. The training function “trainlm” is chosen. The Latin hypercube method is used as the sampling method, uniformly sampling 80 sets of data, and using the first 20% of the data as training values.

Error analysis is critical for assessing model fit quality. The average absolute errors for MCELA and MIELA are 0.06 and 0.08, respectively. Maximum relative errors are respectively 0.35% and 0.31%; average relative errors are 0.06% and 0.1%; and mean square errors are severally 0.009% and 0.01%, respectively.

Ultimately, the final temperature-rise subsystem models for MCELA and MIELA, based on the conditions and requirements mentioned, are presented in Eqs (30) and (31), respectively.

For MCELA:

{\begin{matrix} f i n d X^{e l e c}, Y^{e l e c} \\ min \sum_{i = 1}^{n_{Y} e l e c} {(Y_{i} - Y_{i}^{e l e c})}^{2} \\ s . t . T_{C_{min}} \leq T_{C} \leq T_{C_{max}} \\ X_{min}^{e l e c} \leq X^{e l e c} \leq X_{max}^{e l e c} \\ Y_{min}^{e l e c} \leq Y^{e l e c} \leq Y_{max}^{e l e c} . \end{matrix}

(30)

For MIELA:

{\begin{matrix} f i n d X^{e l e c}, Y^{e l e c} \\ min \sum_{i = 1}^{n_{Y} e l e c} {(Y_{i} - Y_{i}^{e l e c})}^{2} \\ s . t . K_{I_{min}} \leq K_{I} \leq K_{I_{max}} \\ X_{min}^{e l e c} \leq X^{e l e c} \leq X_{max}^{e l e c} \\ Y_{min}^{e l e c} \leq Y^{e l e c} \leq Y_{max}^{e l e c} . \end{matrix}

(31)

4.3 Parameter sensitivity analysis of design variables

To more effectively identify the impact of each parameter on the subsystem, the sensitivity analysis method [31] is adopted. Taking the design variable parameters in the linear motor optimization process as an example, assuming the relationship between the j-th optimization goal and the input variable can be represented by a function G_j(x₁, x₂, …x_n), for x_i, its meaning is the output value of the i-th parameter variable in the j-th optimization goal, where i ∈ (1, n). The parameter sensitivity of the variable can be expressed as:

S_{i}^{G_{j}} (x_{i}) = \frac{G_{j} (x_{i} + Δ x_{i}) - G_{j} (x_{i})}{Δ x_{i}} \frac{x_{i}}{G_{j} (x_{i})} .

(32)

However, since G_j(x₁, x₂, …x_n) may not always be an explicit function of x_i, direct differentiation is not feasible. Instead, the finite difference method is used for computational analysis, ultimately obtaining the sensitivity of the two optimization objectives. The greater the sensitivity, the more pronounced the change in the output state variables relative to the alteration of the input design variables. When the sensitivity is greater than zero, the output increases as the design variables increases.

Design variable parameters are classified into two levels based on different sensitivity values. Specifically, when S_va is greater than or equal to 0.2, the design variable is taken as an independent variable of the subsystem. When S_va is less than 0.2, the design variable is considered a system variable. The final system level division is shown in Table 1.

Table 1.

The hierarchical division of the MCELA and MIELA subsystem design variables.

	Subsystem	Independent design variables	System design variables
MCELA	Electromagnetic subsystem	B_i, B_o, H_l	B_p, H_s, r_i
	Temperature rise subsystem	B_p, H_l, H_s	B_i, B_o, r_i
MIELA	Electromagnetic subsystem	b_p, h_as, h_g, h_p, r_a	h_a, h_ps
	Temperature rise subsystem	h_as, h_p	b_p, h_a, h_g, h_ps, r_a

Figure 7.

Optimization process of MCELA collaborative optimization target.

4.4 Optimization results and simulation verification

Given that the optimization variables of the actual CELA are discrete variables and the objective function has discontinuities, the MOGA-II algorithm [32] for global optimization was adopted. This algorithm focuses more on the crossover and mutation effects within the population than traditional genetic algorithms, thereby enhancing optimization efficiency. For both MCELA and MIELA, the population size is set at 60, with 50 iterations, a crossover rate of 0.9, and a mutation rate of 0.1.

Based on the optimization process, the final optimization results for each objective and the iterative optimization process are shown in Figures 7 and 8.

Figure 8.

Optimization process of MIELA collaborative optimization target.

The electromagnetic field and temperature field before and after optimization were compared through simulation. The distribution cloud diagrams of these fields prior to optimization are shown in Figures 9(a) and 9(c), while post-optimization diagrams are in Figures 9(b) and 9(d). Given that both MCELA and MIELA have axial symmetric structures, only half of each actuator’s electromagnetic and temperature field cloud diagram is displayed.

Figure 9.

Simulation comparison of temperature field before and after optimization of CELA. (a) Simulation comparison of temperature field before optimization of MCELA. (b) Simulation comparison of temperature field after optimization of MCELA. (c) Simulation comparison of temperature field before optimization of MIELA. (d) Simulation comparison of temperature field after optimization of MIELA.

Post-optimization modifications include enlarging the diameter of MCELA’s inner magnetic yoke, which disperses the axial magnetic flux. The height adjustments of the long and short permanent magnets concentrate the magnetic flux closer to the coil side, enhancing the electromagnetic force. Furthermore, reducing the thickness of the inner magnetic yoke improves heat dissipation, thereby lowering the actuator’s temperature. Additionally, increasing the armature height and slightly decreasing the armature radius in MIELA enhances the starting force.

To verify that the performance of the optimized actuator, a prototype was constructed, and the force characteristics of the composite electromagnetic linear actuator before and after optimization were documented through experiments, as shown in Figure 10.

Figure 10.

Before and after force characteristics optimization of CELA.

To save energy, the power supply is turned off during the second half of the travel in the moving iron electromagnetic linear actuator, leading to a discontinuity in force characteristics. As shown in Figure 10, there is a notable increase in output following optimization.

5 Conclusion

This study focuses on the CELA, establishing an electromagnetic field-temperature field coupled model and analyzing its electromagnetic characteristics and temperature rise characteristics. A multi-physics field collaborative optimization method is proposed for optimization analysis, ultimately yielding the best design parameters. The specific conclusions are as follows:

(1) The multi-physics field coupling relationships during the operation of MIELA and MCELA were elucidated, and an electromagnetic-temperature field coupled simulation model was established. This model offered insights into the patterns of energy consumption and temperature rise under standard operating conditions.

(2) Combining the finite element method and the neural network surrogate model method, optimization models for both the electromagnetic and temperature field systems were developed. Sensitivity analysis was employed to hierarchically organize the design variables for each subsystem optimization model. Through the application of collaborative optimization methods and the MOGA-II genetic algorithm, the optimal design parameters for the actuator were obtained.

(3) Compared to the initial design, improvements were observed: the main driving component, MCELA, showed a 5.4% increase in force constant, a 6.3% decrease in the highest average temperature, and a slight increase in force fluctuation rate. The auxiliary driving component, MIELA, experienced a 2.3% increase in starting force, while its passive force linearity and armature temperature decreased by 3.8% and 0.8%, respectively. These changes significantly enhanced the overall performance of the CELA, validating the effectiveness of the optimization method.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant number 52205271] and the Innovation team project of “Qing-Chuang science and technology plan” of colleges and university in Shandong Province, China [2022KJ232].

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