An Improved Method for Calculating the Power Consumption of Electromagnet Coil

Abstract

The electromagnet coil is an important electromagnetic conversion element, which has been used in many fields. However, due to the existence of resistance, a significant amount of energy is lost as heat, resulting in waste. So, it is necessary to study the power consumption characteristics of electromagnet coil. In order to solve the above problem, an improved calculation method of power consumption is proposed in this paper. Firstly, the power consumption model of electromagnet coil is deduced, and the influence weights of the nominal diameter, paint layer thickness and stretch ratio are analyzed by sensitivity. Secondly, an FEM finite element method simulation model and test platform of electromagnet coil are established, which are used to simulate and test the power consumption in different structures and temperatures. Finally, the results of calculation, simulation and experiment are compared. The results show that the appropriate structural parameters can effectively reduce the power consumption of electromagnet coil. Furthermore, the influence of different parameters on the power consumption is different. The relevant results can provide a theoretical basis for the design of electromagnet coil, which has a certain engineering value.

1. Introduction

The coil is a common electromagnetic conversion element, which is mainly used as the driving device of an electromagnet, and has been widely used in construction machinery, aerospace and transportation fields [1,2,3]. In practice, in order to pursue the greater driving capacity of an electromagnet, hundreds of enameled wires are tightly wound on the coil skeleton, which enhances the intensity of the magnetic field [4]. However, this kind of winding method inevitably results in large resistance of the coil, which leads to more energy wasted in the form of heat [5], increasing the power consumption. By means of the study reported in this paper, an analytical model of power consumption is obtained, and the key influencing factors of power consumption are identified. The results can provide a theoretical basis for the design of electromagnet coil.

Aiming at determining the power consumption of electromagnet coil, many scholars in this field have carried out a lot of research. In Ref. [6], an electromagnetic-mechanical-temperature coupling model of electromagnet coil was established, and the influence laws of the coil layers, turns, height, width, turn spacing and layer spacing on heat characteristics were analyzed. In Ref. [7], the relationship among the resistance, the turns and the shape of coil was analyzed, and the coil size was optimized based on the rated excitation current. In Refs. [8,9], an energy model of high temperature superconducting (HTS) coil was established, and the structural parameters, such as the radius, height and length of coil, were optimized using FEM finite element method, which targeted the maximum stored energy. In Ref. [10], the core loss, solid loss and total loss of an ultra-high speed solenoid valve were analyzed by FEM, and the power loss characteristics under different driven strategies were compared. In Ref. [11], a calculation model of resistance was indirectly derived using the slot filling rate, and the power consumption was additionally obtained. Compared with this study, more structural parameters were considered in this paper. In Ref. [12], in order to solve the problem of power consumption caused by the long-time energization of electromagnet coil, a traction control valve (TCV) was designed using FEM. The TCV was equipped with a permanent magnet, which can retain the electromagnetic force when the power supply is cut off. In Ref. [13], in order to accurately express the power consumption of proportional solenoid valves, a thermal load prediction model based on the Kalman filter algorithm was proposed, but this method required historical samples. In Ref. [14], the influence laws of three coil configurations (single, stacked and nested) on the performance of high-speed switching valves were studied. It was found that the single configuration had the lower energy loss. In Ref. [15], in order to improve the energy density of HTS coils, the parameters, such as the inner radius, number of pancakes, etc., were optimized using the MATLAB toolbox. In Ref. [16], a multiple nonintegrated coil electromagnet was proposed; the coil was changed from the conventional integrated coil to multiple independent excitation coils, which realized a reduction in power consumption. In Ref. [17], the percentages of different types of power consumption in a high-speed solenoid valve (HSV) were analyzed, with the reactive power being reduced by changing currents. Further, in Ref. [18], an energy consumption prediction model of HSV was established, which used a back-propagation neural network (BPNN). Through the optimization of the driving strategy, the joule energy was reduced by 22.49%.

According to the above investigation results, it can be seen that the existing research on the power consumption of electromagnet coil is mainly based on the finite element method. Moreover, in order to improve the simulation speed and reduce the modeling difficulty, the coil is usually simplified as a single entity, and the influence of internal enameled wire is ignored. In fact, the structure of enameled wire has an important effect on the power consumption of electromagnet coil, so it is very necessary to investigate it further.

In this paper, an improved calculation method for the power consumption of electromagnet coil is proposed. This method also considers the influence of the stretch ratio, which enables the calculation model to describe the state of the coil more realistically. The significance of this study is not only to propose a calculation model, but also to provide a theoretical basis for the design and manufacture of electromagnet coil. Other studies, such as those regarding the design of the size and arrangement of enameled wire under specific ampere-turns, the setting of the tension force in the winding machine, and the selection of enameled wire materials may rely on the calculation method proposed in this paper.

Based on the above ideas, the innovative contributions of this paper are as follows:

• In this paper, the influence of the stretch ratio in enameled wire is considered when modeling, which improves the existing calculation model of power consumption. This improved calculation model can be used as the theoretical basis for the design and manufacture of electromagnet coil.
• In this paper, the influence weights of different structural parameters on power consumption are analyzed, and the key influencing parameters are obtained. These findings can serve as a guide for the selection of optimized parameters.

The paper is organized as follows: Section 2 improves the calculation model of power consumption, considering the influence of the stretch ratio. Furthermore, the influence weights of the different parameters are analyzed according to their sensitivity. Section 3 establishes the FEM simulation model, and a cloud map of the total current density is obtained. Section 4 reports the testing of the power consumption of two different coils, and the test results are compared with the calculated results. Section 5 summarizes the innovative findings of this paper.

2. Calculating the Power Consumption of Electromagnet Coil

2.1. Physical Modeling

The coil is commonly used as the drive stage of an electromagnet. The structural model is shown in Figure 1.

In Figure 1, the coil is composed of enameled wire wound on the coil skeleton. When designing the coil, only the ampere-turns can be obtained [19], while parameters such as the diameter and paint layer thickness can only be designed by experience. In fact, when the electromagnet coil is wound, in order to ensure the coil is tight, the tension force will be applied to the enameled wire. However, the tension force will lead to a reduction in the diameter and paint thickness of the copper wire (due to the influence of the stretch ratio). If the influence of the tension force is not considered in the model (the diameter and paint thickness of copper wire is constant), the calculated value of the power consumption will inevitably deviate from the actual value, so it is very necessary to study it. The specific influence laws of the above parameters are discussed in detail according to sensitivity in Section 2.3.

2.2. Power Consumption Modeling

The enameled wire in the coil is usually composed of copper wire, and the power consumption is mainly copper loss. The power consumption P is as follows:

$$P={\displaystyle \frac{U^2}{R_t}}$$

(1)

where U is the excitation voltage and $R_t$ is the resistance of the electromagnet coil.

According to the calculation equation of conductor resistance, the resistance $R_t$ can be expressed as follows:

$$R_t=\rho {\displaystyle \frac{L}{S}}={\rho }_0(1+{\alpha }_d{t}_d){\displaystyle \frac{L}{S}}$$

(2)

where $\rho$ is the resistivity; ${\rho }_0$ is the resistivity at 0 °C; ${\alpha }_d$ is the temperature coefficient; $t_d$ is the working temperature; L is the total winding length; and S is the cross-sectional area.

According to a previous study by the authors’ team [20], the total length L of copper wire wound on the coil skeleton can be expressed as follows:

$$\begin{array}{cc}\hfill L=& \pi n{k}_c\left(2R_i+d_c+2t_q+{\displaystyle \frac{\sqrt{3}}{2}}\left(k_c-1\right)\left(d_c+2t_q\right)\right)\hfill \\ \hfill & +\pi \left(2R_i+\left(1+\sqrt{3}{k}_c\right)\left(d_c+2t_q\right)\right)\left(N-k_c·n\right)\hfill \end{array}$$

(3)

where $k_c$ is the winding layers; $R_i$ is the winding inner diameter; $d_c$ is the diameter of copper wire after stretching; $t_q$ is the paint layer thickness; n is the winding turns of each layer; and N is the total winding turns.

The winding layers $k_c$ are as follows:

$$k_c=\mathrm{floor}\left({\displaystyle \frac{N}{n}}\right)$$

(4)

The cross-sectional area S is as follows:

$$S={\displaystyle \frac{\pi }{4}}{d_c}^2$$

(5)

Substituting Equations (3) and (5) into Equation (2), the resistance $R_t$ can be rewritten as follows:

$$\begin{array}{cc}\hfill R_t=& {\displaystyle \frac{4{\rho }_0(1+{\alpha }_d{t}_d)n{k}_c\left(2R_i+d_c+2t_q+\frac{\sqrt{3}}{2}\left(k_c-1\right)\left(d_c+2t_q\right)\right)}{{d_c}^2}}\hfill \\ \hfill & +{\displaystyle \frac{4{\rho }_0(1+{\alpha }_d{t}_d)\left(2R_i+\left(1+\sqrt{3}{k}_c\right)\left(d_c+2t_q\right)\right)\left(N-k_c·n\right)}{{d_c}^2}}\hfill \end{array}$$

(6)

Although Equation (1) provides a method to calculate the power consumption, it does not consider the stretch ratio of the enameled wire. When the electromagnet coil is wound, the tension force will be applied to the enameled wire, which causes the diameter of the copper wire to be stretched and reduced. Therefore, Equation (1) should be improved.

The definition of the stretch ratio $\sigma$ is as follows:

$$\sigma ={\displaystyle \frac{L_h-L_0}{L_0}}\times 100\%$$

(7)

where $L_h$ is the length of copper wire after stretching, and $L_0$ is the length of copper wire before stretching.

According to the volume invariance, the following relationship can be obtained:

$${\displaystyle \frac{\pi }{4}}{d_0}^2{L}_0={\displaystyle \frac{\pi }{4}}{d_c}^2{L}_h$$

(8)

where $d_0$ is the diameter of copper wire before stretching (referred to as the nominal diameter).

According to Equations (7) and (8), the relationship between $d_c$ and $d_0$ is as follows:

$$d_c={\displaystyle \frac{d_0}{\sqrt{1+\sigma }}}$$

(9)

Substituting Equation (9) into Equation (6), the improved resistance $R_a$ is as follows:

$$\begin{array}{cc}\hfill R_a=& {\displaystyle \frac{4{\rho }_0(1+{\alpha }_d{t}_d)(1+\sigma )n{k}_c\left(2R_i+\sqrt{\frac{1}{1+\sigma }}{d}_0+2t_q+\frac{\sqrt{3}}{2}\left(k_c-1\right)\left(\sqrt{\frac{1}{1+\sigma }}{d}_0+2t_q\right)\right)}{{d_0}^2}}\hfill \\ \hfill & +{\displaystyle \frac{4{\rho }_0(1+{\alpha }_d{t}_d)(1+\sigma )\left(2R_i+\left(1+\sqrt{3}{k}_c\right)\left(\sqrt{\frac{1}{1+\sigma }}{d}_0+2t_q\right)\right)\left(N-k_c·n\right)}{{d_0}^2}}\hfill \end{array}$$

(10)

According to Equation (1), the power consumption P can be rewritten as follows:

$$P={\displaystyle \frac{U^2}{4\lambda {\rho }_0(1+{\alpha }_d{t}_d)(1+\sigma )}}$$

(11)

In Equation (11), the constant $\lambda$ is as follows:

$$\begin{array}{cc}\hfill \lambda =& {\displaystyle \frac{k_{c}n\left(2R_i+\left(\sqrt{\frac{1}{1+\sigma }}{d}_0+2t_q\right)·\left(\frac{\sqrt{3}}{2}\left(k_c-1\right)+1\right)\right)}{{d_0}^2}}\hfill \\ \hfill & +{\displaystyle \frac{\left(2R_i+\left(1+\sqrt{3}{k}_c\right)\left(\sqrt{\frac{1}{1+\sigma }}{d}_0+2t_q\right)\right)\left(N-k_c·n\right)}{{d_0}^2}}\hfill \end{array}$$

(12)

2.3. Sensitivity Analysis

It can be seen from Equations (11) and (12) that the power consumption P is affected by the structure and physical characteristics of the enameled wire. Among them, because the resistivity $\rho$ and the excitation voltage U are generally standard values, and the winding inner diameter $R_i$ is determined by the structure, it is not appropriate to conduct the sensitivity analysis on the above parameters. In addition, the winding turns of each layer n and the total winding turns N are discrete quantities, which cannot be analyzed in terms of sensitivity. Therefore, in this paper, the influence weights of the nominal diameter $d_0$, the paint layer thickness $t_q$ and the stretch ratio $\sigma$ on the power consumption are analyzed.

Define the sensitivity $S(P,x_i)$ as follows:

$$S(P,x_i)={\displaystyle \frac{\partial P}{\partial x_i}}·{\displaystyle \frac{x_i}{P}}$$

(13)

where,

$$x_1=d_0,x_2=t_q,x_3=\sigma$$

(14)

The reference values of the parameters in Equation (11) are shown in

Table 1. Reference values of parameters.

Parameter	Value	Parameter	Value
$d_0$/mm	0.4	U/V	20
N	2000	$R_i$/mm	9.5
$\sigma$	0.1	$t_q$/mm	0.02
n	70	${\rho }_0$/$\Omega$· m	$1.6\times {10}^{-8}$
${\alpha }_d$/$°{\mathrm{C}}^{-1}$	$4\times {10}^{-3}$	$t_d$/$°\mathrm{C}$	30

.

When the parameters $d_0$, $t_q$ and $\sigma$ are changed by ±20% from the reference values, the change in sensitivity $S(P,x_i)$ is shown in Figure 2.

According to Figure 2, the following conclusions can be obtained:

(1) The sensitivity of the paint layer thickness $t_q$ and the stretch ratio $\sigma$ are negative, while that of the nominal diameter $d_0$ is positive. This means that when the power consumption of the designed electromagnet coil is greater than the expected value, it can be solved by appropriately reducing the nominal diameter $d_0$ or increasing the paint layer thickness $t_q$ and the stretch ratio $\sigma$. Similarly, by adjusting the parameters inversely, the power consumption can be reduced.

(2) When the parameters are changed within ±20%, the nominal diameter $d_0$ has the greatest influence weight, and the paint layer thickness $t_q$ has the least. This means that when the power consumption of the electromagnet coil changes significantly, the nominal diameter $d_0$ should be adjusted primarily. However, the stretch ratio $\sigma$ and the paint layer thickness $t_q$ can only be used to adjust the power consumption in a small range.

3. Simulating the Power Consumption of Electromagnet Coil

3.1. FEM Modeling

With regard to the electromagnet coil commonly used in industry, the diameter of enameled wire is generally less than 1 mm and the total winding turns reaches hundreds of turns [21,22], which leads to simulation being impossible due to the huge number of grids. However, Equations (11) and (12) relate to the general law of electromagnet coil, regardless of the diameter and total winding turns. So, this paper simulates the power consumption under the conditions of “large diameter” and “few turns” in FEM, and tests the power consumption under the condition of “few diameter” and “large turns” in experiment. Based on the above idea, the simulation model of the electromagnet coil is established in Figure 3.

In Figure 3, the number of meshes is about 65,000, and the skewness index is mainly distributed between 0 and 0.63, indicating that the mesh quality is satisfactory.

3.2. FEM Conditions

3.2.1. Boundary Conditions

Resistance is the inherent property of an electromagnet coil, which can be calculated indirectly by the voltage and current. Therefore, in FEM, both the start face (1 V) and end face (0 V) of the coil are defined as voltage boundaries, and the resistance is calculated through the current flowing in the coil. In addition, the resistance is related to the temperature, so this paper assumes continuous heat conduction between the coil surface and environment (the heat transfer coefficient is $100\mathrm{W}/{\mathrm{mm}}^2·$°C), ensuring that the coil temperature is maintained at the environment temperature.

3.2.2. Simulation Schemes

In this paper, the resistance of the electromagnet coil at different working temperatures is first simulated, and then the power consumption P is indirectly calculated using Equation (1). The simulation schemes are shown in

Table 2. Schemes of simulation.

Number	${\mathit{d}}_0$/mm	n	N	Parameter
1	5	10	10	${\alpha }_d$ = $4\times {10}^{-3}$/$°{\mathrm{C}}^{-1}$ $\sigma$ = $t_q$ = 0 $R_i$ = 50 mm ${\rho }_0$ = $1.6\times {10}^{-8}$ $\Omega$ · m
2	5	10	20
3	10	10	10

.

In Table 2, schemes 1 and 2 are used to verify the correctness of Equation (11) under the same diameter and different turns. Schemes 1 and 3 are used to verify the correctness of Equation (11) under the same turns and different diameters.

3.3. FEM Results

Taking scheme 1 in Table 2 as an example, the other schemes can be obtained by winding the series enameled wires on top of the original one. The total current density distribution of the electromagnet coil is shown in Figure 4.

According to Figure 4, it can be seen that the total current density is not uniformly distributed in the electromagnet coil. This is mainly because the bending radius of the enameled wire is inconsistent at different positions, which results in the current flowing through different cross-sectional areas. Because the resistance needs to be calculated indirectly using the voltage and current, it is necessary to obtain the average current in the coil. According to the relationship shown in Figure 4, the average current $\overline{I}$ flowing through the enameled wire can be expressed as follows:

$$\overline{I}={\displaystyle \sum _{i=1}^k}{J}_{i}dA=J_{s1}dA+J_{s2}dA+\dots +J_{x1}dA+J_{x2}dA+\dots$$

(15)

where $J_{si}$ is the total current density above the reference cross-section; $J_{xi}$ is the total current density below the reference cross-section; and $dA$ is the area microelement.

Because the current densities $J_{si}$ and $J_{xi}$ are approximately complementary to the reference cross-section, the total current densities at the two positions are equal to twice the average current densities at the reference cross-section. So, the average current $\overline{I}$ can be further expressed as follows:

$$\overline{I}=(J_{s1}+J_{x1})dA+(J_{s2}+J_{x2})dA+\dots =2\overline{J_j}dA+2\overline{J_j}dA+\dots =\overline{J_j}A$$

(16)

where $\overline{J_j}$ is the average current density at the reference cross-section, and A is the total area of the cross-section.

In the three different schemes shown in Table 2, the change curves of power consumption with working temperature are shown in Figure 5. Among them, the calculation curves are obtained using Equations (11) and (12), and the simulation curves are obtained using FEM in Figure 4.

It can be seen from Figure 5 that the calculation curves are close to the simulation curves, and both satisfy an inverse proportional relationship. This is mainly because when the excitation voltage remains unchanged, the increase in working temperature will affect the resistance of the coil, and thereby reduce the current flowing in the coil.

In order to quantitatively analyze the change curves of calculation and simulation in Figure 5, the relative deviation e is defined in this paper, which is shown in Equation (17). According to the definition, the smaller the value of the relative error e, the higher the accuracy of the calculation result.

$$e={\displaystyle \frac{{\sum }_{t_d=30}^{150}\mid P_{std}-P_{ctd}\mid }{{\sum }_{t_d=30}^{150}{P}_{std}}}$$

(17)

where $P_{std}$ is the simulation value of power consumption obtained by FEM at the temperature $t_d$, and $P_{ctd}$ is the calculation value of power consumption obtained by Equation (11) at the temperature $t_d$.

According to Equation (17), the relative deviation e of the electromagnet coil in three schemes is shown in

Table 3. Relative deviation e of power consumption.

Number	Scheme 1	Scheme 2	Scheme 3
Relative deviation e	0.54%	0.46%	0.78%

.

It can be seen from Table 3 that all the relative deviations are less than 1%, and the maximum relative deviation is 0.78%, indicating that the results of simulation and calculation can be mutually verified.

4. Testing the Power Consumption of Electromagnet Coil

4.1. Experimental System

In order to verify the correctness of the calculation of the power consumption, a test platform of the electromagnet coil was built in this paper (referred to as the test platform for short), as shown in Figure 6.

In Figure 6, the test platform of the electromagnet coil is composed of three parts: a temperature controller, a resistance tester and a multi-channel tester. Among them, the temperature controller is used for heating the coil to the preset temperature, and the actual temperature of coil is displayed by the multi-channel tester via the temperature sensor. The resistance of the coil at the preset temperature is displayed by the resistance tester and converted to the power consumption using Equation (1). The test platform was used in a previous study by the authors’ team [20].

In this paper, two electromagnet coils are selected as representatives to test the accuracy of the calculation method proposed in this paper. The two coils have the same structure as that in Figure 1, but the size and winding turns are completely different, reflecting the adaptability of the calculation method. The specific structures are shown in Figure 7.

The main structural and physical parameters of the two different electromagnet coils are shown in

Table 4. Parameters of electromagnet coils.

Number	${\mathit{d}}_0$/mm	n	N	${\mathit{\rho }}_0$/$\mathbf{\Omega }$· m	${\mathit{R}}_{\mathit{i}}$/mm	${\mathit{\alpha }}_{\mathit{d}}$/$°{\mathbf{C}}^{-1}$	${\mathit{t}}_{\mathit{q}}$/mm
Coil 1	0.236	97	880	$1.6\times {10}^{-8}$	9.05	$4\times {10}^{-3}$	0.01975
Coil 2	0.315	81	1057	$1.6\times {10}^{-8}$	9.6	$4\times {10}^{-3}$	0.02175

.

4.2. Experimental Results

Because the test platform can only test the resistance, the power consumption P needs to be calculated indirectly by Equation (1). The change curves of power consumption with working temperature are shown in Figure 8.

In Figure 8, the calculation curves are obtained using Equation (11), and the experimental curves are obtained using the test platform in Figure 6.

The results of Figure 8 are also quantitatively analyzed using Equation (17). Among them, the relative deviation of coil 1 is 2.97% with a 10% stretch ratio and 6.31% without a stretch ratio. The relative deviation of coil 2 is 3.11% with a 10% stretch ratio and 5.84% without a stretch ratio. According to the above results, it can be seen that considering the stretch ratio of the copper wire can improve the calculation accuracy of power consumption.

5. Conclusions

Aiming at electromagnet coils commonly used in industry, an improved calculation method of power consumption is proposed in this paper. By considering the stretch ratio, the diameter reduction of enameled wire caused by winding is effectively compensated. The main conclusions of this study are as follows:

(1) The geometric parameters of enameled wire have a significant effect on the power consumption, and the influence laws of different parameters are different. According to the results of sensitivity analysis, compared with the paint layer thickness $t_q$ and the stretch ratio $\sigma$, the nominal diameter $d_0$ has the largest influence weight and is positively correlated with the power consumption. However, the influence laws of the paint layer thickness $t_q$ and the stretch ratio $\sigma$ are opposite.

(2) The bending radius of the enameled wire is inconsistent at different area microelements, resulting in the current not being uniformly distributed. Furthermore, the power consumption obtained using Equation (16), which is from the perspective of a microelement, is similar to that obtained using Equation (11), which is from the perspective of the entirety, showing that the above two methods can be equivalent.

(3) Two coils with different configurations are tested, and the results show that the calculation method proposed in this paper can predict the power consumption of the electromagnet coil. When a 10% stretch ratio is considered, the relative deviations of coil 1 and coil 2 are reduced by more than 2.7%, indicating that the stretch ratio has a positive effect on the calculation accuracy.

Future work: The electromagnet coil studied in this paper is a common structure, and there are still many other structures which lack in-depth study. Therefore, the main work in the next stage is to study electromagnet coils connected in parallel and in series, wound in unison and in opposition, and to further improve the calculation model of power consumption.