Thermal Robustness Redesign of Electromagnet Under Multi-Physical Field Coupling

Abstract

Aiming at the durability problem of the proportional electromagnet used in the proportional valve of engineering machinery, in order to improve its thermal failure resistance under random load conditions, a parametric redesign model of the proportional electromagnet was proposed based on the multi-physics coupling theory and robust optimization theory. This article takes the proportional electromagnet with a basin-type suction structure as the research object. The parameter model was verified through steady-state proportional electromagnet tests and temperature distribution tests. On the premise of ensuring the accuracy of electromagnetic calculation force, the conductivity and heat transfer parameters with fuzzy magnitude in the system were calibrated. Taking the key structural parameters of the proportional electromagnet and coil as the control factors, and the enameled wire diameter of the coil caused by the uncertainty of the production process conditions as the noise factor, an orthogonal experiment was designed based on the Taguchi method, and the thermal robustness redesign evaluation function of the proportional electromagnet was defined. Multi-factor weighted form. The thermal load of the proportional electromagnet obtained from the excavator field test was used as the response to calculate the heat source. Under the constraint of allowable temperature rise that can not cause coil insulation failure, a redesign method for key structural parameters that minimizes changes in system response under noise interference is given. Studies have shown that coil length and number of turns are the main factors affecting the thermal robustness of proportional electromagnets. The window shape of the coil is determined by the winding process and determines the magnetic properties and heat transfer capabilities of the system. The thermal robustness redesign method of proportional electromagnets proposed in this article has engineering reference value for the customized design of electromechanical products under magnetothermal coupling conditions.

1 Introduction

With the rapid popularization and application of electro-hydraulic proportional pilot technology, the layout of hydraulic excavators is more flexible, the dynamic response is improved, the noise source is reduced, and the combination of control accuracy and position closed loop provides conditions for the automation of construction machinery, thereby enabling dangerous situations It is possible to realize remote control and wireless remote control for construction machinery operating in regional areas. The electro-hydraulic proportional multi-way valve mainly uses a cartridge proportional pressure valve as the pilot stage, and controls the system flow by controlling the displacement of the main spool through the pilot valve.

Pproportional solenoid electromagnet (PSE) is the main component of proportional solenoid valve (PSV). In decades, researchers have conducted lots of research on proportional electromagnets. The main research directions are optimization strategies, fault detection and control technology. Technical means are computer simulations and engineering tests. Liu [1] and others modeled finite element model of PSV based on FE theory. Thermal deformation and temperature distribution of PSV under different conditions were simulated, and compared with the experimental results to verify the reliability of model and provide a theoretical basis for the reliability design of the PSV. Wang [2] and others verified the numerical model of the PSV based on the finite element theory, then used genetic algorithms and multi-objective optimization methods to optimize the shape design parameters of the electromagnet. The influence of the optimized parameters on the dynamic and static performance of the electromagnet was verified through experiments. Fan [3] modeled a multi-physics performance model considering the influence of temperature field. The purpose is solving the problem of non-uniformity between multi-physics fields during model establishing process. The correctness of the model was verified through relevant experimental design, and an evaluation method of electromagnetic force was proposed, which provides theory basis for rapid and customized design and engineering processing of PSVs.

Because of large design parameters of the proportional electromagnet structure, it is hard to design the PSE by changing single variable, and most of the design parameters are nonlinear with the steady-state PSE force [4,5,6,7]. In actural practice, traditional design methods can no longer fully meet the requirements of reliability and robustness. From perspective of design quality, design process not only meet the requirements of functional reliability, but also satisfy the requirements of robustness and be able to fit complex conditions [8, 9]. This article takes the PSE as the research object. Parametric redesign model under magnetothermal coupled condition is proposed. Finite element theory, multi-physics coupling simulation and Taguchi test methods are applied to the robust redesign of the complex non-homogeneous proportional electromagnet magnetothermal coupling system, and thermal robustness redesign of PSE is defined in the form of multi-factor weighting. Designed evaluation function. The combination of redesign method, experiment and approximate model technology broadens the idea of research on the durability of proportional electromagnets.

2 Parametric Modeling of Proportional Electromagnet

2.1 Structural Characteristics of Proportional Electromagnet

PSE has direct-acting electromagnetic actuator and special magnetic isolation structure. As shown in Fig. 1, 1/2 cross-sectional structural diagram of an proportional electromagne, which consists of an PSE end cover, magnet steel, shell, bobbin, seal, armature, gasket, coil, and sleeve.

Fig. 1

Proportional electromagnet model 1/2 section structure. 1-end cover 2-magnetic steel 3-shell 4-frame 5-seal 6-armature 7-gasket 8-coil 9-sleeve

The coil is the key component to provide power input for the electromagnet. The coil properties are directly related to the electromagnetic performance, thermal reliability and life of the electromagnet and even the proportional solenoid valve. The parameter expression of proportional coil is necessary in optimization of proportional electromagne. Turns of the coil is N, excitation current in the coil is I, inner radius is r_xqinner, outer radius is r_xqouter, and winding length is 2L. The initial design range of each characteristic variable has been shown in Table 1.

Table 1 The definition and range of coil windings design parameters

2.2 Magnetic Field Response Based on Coil Optimal Shape Optimization

The coil model parameters have been normalized: $X = r_{{xqouter}} /r_{{xqinner}}$, $Y = {L \mathord{\left/ {\vphantom {L {r_{xqinner} }}} \right. \kern-0pt} {r_{xqinner} }}$.

According to the theory of electromagnetics, there are solenoid coil center field strength as shown in Eq. (1):

$$dH_{0} = \frac{2\pi }{{10}}j\lambda \frac{{r^{2} }}{{(r^{2} + z^{2} )^{3/2} }}drdz$$

(1)

In Eq. (1), λ is filling factor, j is current density.

Furthermore, power function of the PSE coil, as shown in Eq. (2), the field intensity at every point on central axis of PSE coil could be simplified by Eq. (3). Determinant G representing shape type of PSE is shown as Eq. (4).

$$P = \int {dp = \int {j^{2} } } \rho dv = \frac{{\pi \rho N^{2} I^{2} }}{{2\lambda r_{xqinner} }}\frac{X + 1}{{Y(X - 1)}}$$

(2)

$$H_{z} = C_{1} \sqrt {\frac{P}{{R_{1} }}} G(X,Y,\overline{z} )$$

(3)

$$\begin{gathered} G(X,Y,\overline{z} ) = \frac{1}{{2\sqrt {2\pi Y(X^{2} - 1)} }} \cdot \hfill \\ [(\overline{z} + Y)\ln \frac{{X + \sqrt {X^{2} + (\overline{z} + Y)^{2} } }}{{1 + \sqrt {1 + (\overline{z} + Y)^{2} } }} - \hfill \\ (\overline{z} - Y)\ln \frac{{X + \sqrt {X^{2} + (\overline{z} - Y)^{2} } }}{{1 + \sqrt {1 + (\overline{z} - Y)^{2} } }}] \hfill \\ \end{gathered}$$

(4)

The isosurface and contour of the morphological parameter factor G are shown in Fig. 2. The results show that, the maximum value of G is 0.1426 when X = 3.051, Y = 1.864, which is consistent with previous research [10].

Fig. 2

The 3-dimensional equivalent surface of shape factor

Without considering changing the structure of the electromagnet on the magnetic circuit and the structure of the coil skeleton, keeping inner diameter r_xqinner unchanged, combined result of the shape factor G, the shape constraint space is obtained as shown in Eq. (5):

$$28.0\,{\text{mm}} < r_{xqouter} < 54.9\;{\text{mm}},\,\;\,28.1\,{\text{mm}} < L < 33.9\,{\text{mm}}$$

(5)

The approximation process and mathematical model of the relationship between wire and turns of windings are set by Fig. 4 and Eq. (6):

$$N = \frac{{2kL(r_{xqouter} - r_{xqinner} )}}{s}$$

(6)

In Eq. (6), k is stacking coefficient, s is cross-sectional area, Fig. 3 gives coil windings stacking shape, where d is the diameter of wire. Number of coil turns N is 930, and limit value of k is 0.907 [11].

Fig. 3

The stacking pattern of coil windings

Fig. 4

Electromagnetic finite element calculation result

3 Analytical Verification of Multi-Physical Field Coupling of Parametric Model

3.1 Multi-Field Coupling Theory

Field distribution only exists in circuit of electromagnet. Considering the symmetry of the structure of PSE, when establishing the finite element model of electromagnet. In order to simplify the calculation, spiral oil tank and shell on the armature were removed. For electrical interface on the body, a 2-dimensional numerical model is established in RZ coordinate. The numerical calculation is divided into two parts: steady state and transient state. Based on the sequential coupling theory, steady-state and transient electromagnetic finite element calculations were first performed on the electromagnet. The obtained transient electromagnetic calculation results are applied as loads to steady state temperature field simulation, that is, the electromagnetic calculation results are transmitted to the temperature field calculation module with average heating power, and then distribution of temperature on each component of PSE is obtained.

3.2 Coupling Field Analysis and Model Verification

The material selection corresponding to every component of PSE is shown in Table 2.

Table 2 Material selection of electromagnet parts

Figure 4 shows the parameterized scanning calculation results of armature displacement and current magnitude in steady-state electromagnetic calculations. The current interval is set 0–700 ma, displacement interval is set 0 mm to 2.5 mm, and calculation step is set to 0.5 mm.

Figure 5 shows that temperature distribution of the electromagnet calculated by the finite element method of the magneto and thermal couped field. The initial calculated temperature and ambient temperature are both set to 30 ℃.

Fig. 5

The internal and external steady-state temperature distribution of the electromagnet caused by the transient action process of the electromagnet

4 Validation Testing and Results Analysis

4.1 Electromagnetic Characteristic Test

To test displacement and current with force characteristics of the electromagnet, peripheral components that affect PSE force characteristics test were removed. The test device is shown in Fig. 6.

Fig. 6

Electromagnetic characteristic experiment devices. 1-force sensor 2-PSE 3-displacement motor 4-displacement sensor 5-cylinder fixture

4.2 Temperature Distribution Test

The above calculation model is based on analysis of the magnetothermal coupled field. However, it is hard to conduct further detailed tests on temperature inside distribution PSE. The test was conducted in high and low temperature alternating damp thermal test chamber and being shown in Fig. 7. The relationship between internal temperature simulated by magnetothermal coupled simulation and the surface temperature of PSE obtained by experiment was established, and then the distribution of the internal temperature of PSE was inferred.

Fig. 7

High and low temperature alternating damp heat test chamber

In actual operation of the electromagnet, coil is loaded with PWM modulated current for excitation, and the armature opens and closes repeatedly to achieve working heating. Short-period action process cannot quickly meet the thermodynamic steady state inside electromagnet under uncertain external thermal dissipation conditions. Experimental process in this article was under constant conditions. Test steady-state temperature of PSE is surface ambient temperatures (−20 °C, −10 °C, 0 °C, 60 °C, 80 °C, 120 °C).

4.3 Excavator Field Test

In order to truly reflect the performance of proportional valve in the installed state and obtain real, reliable, and effective data samples, the excavator field test method was used to obtain original data samples. Parameters characterizing the dynamic traction performance of the excavator are directly measured and recorded while the excavator is operating on site. The field test took a 20t medium-sized hydraulic excavator as the subject. As shown in Fig. 8.

Fig. 8

Field test environment and object

The temperature of electromagnetic coil rising fast due to the thermal effect during the operation, which causes the coil resistance increasing. The measured working pressure should be linearly proportional to the input current in the control loop, where the feedback loop in the control system performs feedback regulation on the input. Therefore, the linear mapping relationship between input and output can be constructed, and the measured pressure data can be converted into coil current data which is not easy to be measured directly in the installed state for thermal load calculation, as shown in Eq. (7). The heating power of the proportional electromagnet is 7.156 W, which is used as the heat source for thermal robustness analysis and calculation in the following.

$$P_{A} = 0.0591 \cdot I_{F} - 10.751$$

(7)

4.4 Result Analysis

In Fig. 9 and Table 3, comparison of the test and simulationof electromagnetic force and surface temperature distribution shows that the error is at the level of ±10%, which is within the design acceptance range. From this, the validity of the above proportional electromagnet parameterized model can be determined. In the later thermal robustness redesign process, the finite element simulation calculation results based on the numerical calculation model can be used as the system response of the orthogonal test, thus replacing the complex test process.

Fig. 9

Comparison of current-electromagnetic force experiment and simulation results

Table 3 Comparison of temperature distribution test and simulation results

5 Characteristic Parameter Analysis and Thermal Robustness Redesign of Electromagnet

Taguchi method is put forward by Dr. Taguchi. Taguchi design is an experimental design that selects a product or process that will perform more stably in an operating environment [12].

5.1 Comprehensive Performance Evaluation Method

The rated suction, linearity and steady-state temperature rise of the proportional solenoid for valves are closely related to its structural form and parameters. It is an key index to evaluate the performance of the solenoid, whether it meets the industry design specifications, and whether it meets the product-level customization design requirements of the user enterprise. In the specific design process, the above three performance indicators conflict with the determination of design parameters, which is manifested in the performance of multiple sets of design parameters, that is, the rated suction, linearity, and steady-state temperature rise cannot reach the optimal level at the same time. In order to achieve the goal of robust and customized redesign, the weighted average method is used to determine the comprehensive evaluation value of multiple performance indexes of the proportional electromagnet, as shown in Eq. (8):

$$Z = \alpha F + \beta V + \gamma T,\,\alpha + \beta + \gamma = 1$$

(8)

Among them, Z is the comprehensive performance index of the proportional electromagnet, F, V and T are the parameters of the performance index of the rated suction, linearity and steady-state temperature rise of the proportional electromagnet respectively, α, β and γ are the weight factors, and the determination of the weight value contains certain subjective factors of the designer. Standardization of performance evaluation parameters, as shown in Eq. (9).

$$F = \frac{{\overline{F} - \overline{F}_{\min } }}{{\overline{F}_{\max } - \overline{F}_{\min } }},\,V = \frac{{CV - CV{}_{\max }}}{{CV_{\min } - CV{}_{\max }}},\,T = \frac{{\overline{T} - \overline{T}_{\min } }}{{\overline{T}_{\max } - \overline{T}_{\min } }}$$

(9)

5.2 Approximate Calculation Model of Electromagnetic Characteristics

According to mechanical analysis, displacement of armature is the result of force. Total electromagnetic attraction consists two components. Electromagnetic attraction generated by main air gap permeability on the armature end face and its main magneto field energy changing. The additional electromagnetic force generated by the armature side leakage magnetic field energy change [13,14,15]. The solution process of electromagnetic force is very complex and contains many parameters and nonlinear problems. Therefore, steady-state temperature rise of the PSE surface is also a nonlinear process, which is not easy to solve using simple symbolic calculations. Therefore, by identifying and analyzing the characteristic parameters of electromagnets, the response surface method is an effective method for calculating electromagnetic forces based on structural parameters.

The structural parameters are selected as the armature cone angle (a), the gap of the magnetic isolation ring (b), the height of the magnetic isolation ring (c), the cone angle of the pole shoe (d), the diameter of the armature (e), the thickness of the annular yoke (f), and the non-working air gap (g). The sensitivity analysis of the parameters is carried out by the range analysis method. Through the sensitivity analysis of structural parameters, six parameters were determined to construct the response surface model of electromagnetic force. The response surface method fits the response surface equation through limited experimental data. Considering the number of design factors and the characteristics of design points, the Box-Behnken design is selected. The items with poor fitting effect can be eliminated by R-square analysis, and the final response surface function can be obtained by re-fitting. The final response surface function of the electromagnetic force is as shown in Eq. (10):

$$\begin{gathered} \overline{F} = 21.09 - 18.6a - 49.33b - 6.207c + 3.874e + 2.1678f \hfill \\ - 18.39g + 18.79a^{2} + 64.86b^{2} + 0.6592c^{2} \hfill \\ - 0.3445e^{2} - 0.1550f^{2} + 7.83g^{2} + 40.37ab + 1.353ac \hfill \\ - 0.865ae - 0.513af + 7.03ag - 1.523bc - 0.845be \hfill \\ - 0.298bf + 7.7bg + 0.25ce + 0.065cf + 1.2cg + 0.43ef \hfill \\ \end{gathered}$$

(10)

5.3 Thermal Robustness Redesign

The key structural component affecting the thermal performance of PSE is coil. The shape of coil winding rules of copper wire directly affect the steady-state temperature rise of PSE. Therefore, coil inner radius r_xqin, outer radius r_xqout, length h_xq, coil turns N, wire duty cycle k caused by different winding rules, and the total working air gap δ are set as the control factors of the test. The diameter of the enameled copper wire is set as the noise factor of the test, and the input current I is set as the signal factor. The maximum steady-state temperature rising on the surface of the solenoid valve is system response. The control factor and its level setting are shown in Table 4. The L8 (2 ^ 6) Taguchi table is designed, including 6 control factors and 1 signal factor, which are response sequences of two levels and two noise levels.

Table 4 Control factors at different level settings

The results of Taguchi test are shown in Fig. 10, which are given from three aspects: signal-to-noise ratio, slope and standard deviation. In these results, we want to make the standard deviation as small as possible, and the signal-to-noise ratio and slope as large as possible.

Fig. 10

Diagram of main effect by Taguchi test

Signal-to-noise ratio is a measure of robustness and is used to determine control factor settings that minimize the impact of noise on the response. In these results, the main effect plot of the signa-noise ratio shows that the coil length has the greatest impact on the signal-to-noise ratio, followed by the number of coil turns, and the smallest is the working air gap. In all cases we want to maximize the signal-to-noise ratio. For dynamic designs, the signal-to-noise ratio is approximated by the phase “lookout” signal–noise ratio. The main rendering shows how each factor affects the response characteristics (signal-to-noise ratio, slope, standard deviation). Main effects exist when factors at different levels have different effects on a trait. Through experimental design, it can be concluded that the combination of factor levels shown in Table 5 can make the steady-state temperature rising of PSE remain stable under the influence of noise, that is, it is robust.

Table 5 The highest signal-to-noise ratio factor combination

According to the Eq. (8), the thermal robust performance evaluation function under the influence of the rated suction, linearity and steady-state temperature rise of the proportional electromagnet is shown in Eq. (11), where α₁, α₂ and α₃ are weighting coefficients, which are 0.3,0.3 and 0.4 respectively.

$$Z = \alpha_{1} F + \alpha_{2} V + \alpha_{3} T$$

(11)

Based on the results of sensitivity analysis and thermal robustness analysis of electromagnet pull-in structure parameters, the optimal factor combination is obtained. The value of Z is 0.897, which is the closest to 1 compared with other factor combinations, indicating that it is more robust. The calculation result of the Z value depends on the numerical distribution of the weighting coefficient, and the selection rules of different electromagnets are different.

6 Conclusion

In this article, Taguchi and FE theory are combined to the processing of redesign thermal robustness of the electromagnet and the coil. From the analysis of design results, the robust parameter factor combination of steady state temperature rise of PSE is obtained on premise of ensuring the robustness of the electromagnetic characteristics. The design method improves the efficiency of redesign and analysis calculation effectively, and the operation is more flexible.

Inner and outer diameter, length, turns, filling rate and working air gap of PSE are key parameters that affect thermal characteristics of PSE. Appropriate parameter combination selection can improve the aging capability of the system and quickly achieve thermal balance, thereby reducing the excessive temperature rising of PSE caused by local heat accumulation caused by temperature inertia. The coil structure parameters get reck with the wire diameter of the enameled wire. Random error in wire diameter caused by the winding process is transmitted in the system, affecting the transfer vector of the inter-turn microstructural force in the system, resulting in excessive local stress and insulation failure.

The thermal robust redesign method is used to release sensitivity of PSE coil to factors which are uncontrollable. It could be used to solve insulation failure and stagnation caused by PSE thermal effects. The redesign theory play a key role in improving durability of PSE.