# Optimization of the Wire Diameter Based on the Analytical Model of the Mean Magnetic Field for a Magnetically Driven Actuator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methods

#### 2.1. Test Principle

_{HI}is the proportional coefficient of the mean magnetic field intensity; its value belongs to (0,1), N is the number of the coil turn, and L is the coil length.

#### 2.2. Experiment Setup and Parameters

_{a}, L

_{b}, L

_{f}represented the coil length, coil thickness, and diameter of the skeleton shaft, respectively, and D

_{wire}and D

_{core}were the enameled wire diameter and copper core diameter, respectively. The coils were tightly wound by the use of standard enameled wires. Since this article focuses on the optimization of the coil itself, it is not necessary to consider the influence of the iron core or other parts in an actuator. The parameters of the coils are given in Table 1, and some necessary parameters of the skeleton and material are supplied in Table 2. Considering the value of C

_{HI}does not affect the increasing or decreasing relationship between the variables; C

_{HI}will have no effect on the optimization results. C

_{HI}is specified as 0.8 here.

## 3. Data Processing and Analysis

#### 3.1. Inherent Characteristic Parameters of Coils

#### 3.1.1. Dimension Parameters

_{wire}and the copper core diameter D

_{core}. Figure 3 shows the actual values of D

_{wire}and D

_{core}and the fitted results using linear functions. It can be seen from Figure 3 that the diameter of copper core is approximately linear vs. the external diameter of enameled wire. With and without an intercept, the fitted linear equations were determined as D

_{core}= 0.9687 D

_{wire}− 0.03214 and D

_{core}= 0.9394 D

_{wire}, respectively. The linear function with an intercept was quite accurate as the relative error was lower than 1.52% when D

_{wire}was higher than 0.3 mm and lower than 2.55 mm. In contrast, the linear function without an intercept was not so accurate since the relative error was higher than 5% under some conditions, especially when D

_{wire}was quite low.

_{wire}changed within a relatively narrow interval. The coils used in this paper were wound by the wires with diameters of 0.3~0.8 mm. Executing a simple linear fitting, Table 3 supplies the results and relative errors of the two line equations. From computation, the linear equation with intercept was D

_{core}= 0.962 D

_{wire}− 0.0277 and had a relative error lower than 0.84%. In comparison, the linear equation without an intercept D

_{core}= 0.898 D

_{wire}also had high precision as the relative error was lower than 3.2%. Thus, it is acceptable to use a positively proportional function to describe the relationship between D

_{core}and D

_{wire}when D

_{wire}changes within a narrow interval, which is quite convenient for the following optimization.

_{a}or L

_{b}was not exactly the integral multiple of D

_{wire}, the effective length L

_{a}’ and thickness L

_{b}’ were a little lower than L

_{a}and L

_{b}, respectively. Coil length or thickness was not fully utilized, and the available area was L

_{a}’×L

_{b}’, which was slightly less than the actual area.

_{f}′ was introduced to describing the winding effect, C

_{f}was the filling factor of the enameled wire. L

_{a}L

_{b}was the axis-sectional area of the coil and πD

_{wire}

^{2}/4 was the cross-sectional area of single enameled wire.

_{wire}was conditional. That was, with the effectiveness of Equation (2), determined by the weight of C

_{f}’0.155 L

_{a}/D

_{wire}in the total coil turns. Additionally, the relative error of the positively proportional function was 0.155/(1.155 L

_{b}/D

_{wire}− 0.155) × 100%, which was determined by L

_{b}/D

_{wire}.

_{b}/D

_{wire}determined the number of layers and the relative error, which are displayed in Figure 4. From the calculation results, the relative error of C

_{f}L

_{a}L

_{b}/(πD

_{wire}

^{2}/4) computing N decreased with L

_{b}/D

_{wire}increasing. To guarantee that the relative error of Equation (2) is lower than 5.0% in computing N, it should be met that L

_{b}> 2.8 D

_{wire}. That is, the coil should be wound with three layers at least. When L

_{b}< 2.8 D

_{wire}, one should use L

_{b}’ = $\lfloor \frac{{L}_{b}-{D}_{wire}}{\sqrt{3}{D}_{wire}/2}\rfloor +1$ instead of L

_{b}for computations. For the coils in this paper, the values of L

_{b}/D

_{wire}under different D

_{wire}were higher than 6.8/0.8 = 8.5 so that the approximate expression in Equation (2) has enough precision.

_{f}was determined by the mean values of N and D

_{wire}so that C

_{f}= 0.57. The effect of Equation (2) computing N is shown in Table 4. From the calculation results, the model of coil turn can predict the practical coil turn effectively as the relative error was lower than 2.8%.

_{core}, D

_{wire}and N were written as

#### 3.1.2. Static Resistance and Static Inductance

_{L}

_{0}and C

_{L}were two parameters dependent on L

_{a}, L

_{b}, L

_{f}while independent of other variables and met C

_{L}= 64 C

_{L}

_{0}(L

_{a}L

_{b})

^{2}/π; ρ was the resistivity of copper.

_{wire}from the experiment and computation. From the results, it was easily reached that both L and R were monotonically decreasing functions vs. D

_{wire}. More specifically, as concluded from the expression of N in Equation (3) and D

_{core}= 0.898 D

_{wire}, both R and L were inversely proportional functions vs. D

_{wire}

^{4}(also N

^{2}). The model was in good agreement with the experiment as the relative errors of the model in computing R and L were lower than 3.1% and 2.8%, respectively.

#### 3.2. Sinusoidal Response

_{amp}sin(ωt), the current response within the coil can be calculated by I(t) = I

_{amp}sin(ωt – φ

_{I}), where ω is the angular frequency of the input and φ

_{I}is the phase lag of the coil current compared to the voltage. From the theory of the linear time-invariant system, the amplitude ratio function is A

_{I}= I

_{amp}/U

_{amp}= 1/(R

^{2}+ ω

^{2}L

^{2})

^{1/2}, and tanφ

_{I}= ωL/R. Substituting Equation (4) into these expressions, one obtains

_{H}and the lagging phase of the magnetic field φ

_{H}can be easily reached as C

_{HI}NA

_{I}/L

_{a}and φ

_{H}= φ

_{I}. By substituting Equation (5) into these two equations, one obtains

_{H}and D

_{wire}

^{2}. From Figure 8, the linear relationship between the amplitude ratio of the magnetic field and the square of the wire diameter was verified as the tested points under a certain frequency were roughly plotted in a line passing through the origin.

#### 3.3. Square-Wave Response

#### 3.3.1. Time-Domain Response

_{0}is the initial value of the coil current, U

_{st}is the steady-state amplitude of the voltage. Equation (7) was suitable to both the charging and discharging process of the coil. For charging, I

_{0}= 0. For discharging, U

_{st}= 0.

_{HI}NI(t)/L

_{a}, one obtains the transient-state response of the magnetic field

#### 3.3.2. Steady-State Value and Response Time

_{st}can be easily acquired from Equation (9), as

_{st}vs. D

_{wire}from the test and model. From the tested and calculated results, a higher D

_{wire}is helpful for a higher H

_{st}. More specifically, H

_{st}was positively proportional to D

_{wire}

^{2}, as explained in Equation (10). Thus, the change law of the H

_{st}under the square wave was the same as one of the magnetic field amplitudes under the sinusoidal voltage. It was easily illustrated that both the functions of 1/(R

^{2}+ ω

^{2}L

^{2})

^{1/2}and 1/R can be expressed by the quartic function vs. the wire diameter approximately. In addition, the relatively errors under different wire diameters were less than 1.2% thus, the model can predict the steady-state magnetic field quite effectively.

_{req}the required intensity of the coil current and substituting I

_{req}into Equation (7), the response time t

_{Iv}can be reached, as

_{req}into Equation (9), one obtains the response time to the specified magnetic field intensity t

_{Hv,}as

_{Iv}is in the function form of aln [1 + b/(cx

^{2}− b)] vs. D

_{wire}while t

_{Hv}is expressed by aln [1 + b/(cx

^{4}− b)] vs. D

_{wire}.

_{req}= p(U

_{st}/R) or H

_{req}= p[C

_{HI}N(U

_{st}/R)/L

_{a}]. By substituting the two expressions into Equations (10) and (11), one obtains the response time to a specified proportion, as

_{Hv}, the factors influencing t

_{Hp}were almost independent of D

_{wire}. More specifically, t

_{Hp}was just determined by the ratio of L/R. The value of L/R was only slightly influenced by D

_{wire}; optimizing the wire diameter would be helpless to promote this type of response speed.

_{req}were 3 kA/m, 3.5 kA/m, and 4 kA/m, and the specified proportions p were 0.7, 0.8, and 0.9. Just as predicted by the model, H

_{req}was effectively reduced by increasing D

_{wire}. Furthermore, H

_{req}declined fast first and then slowly with D

_{wire}increasing. For the value of t

_{Hp}, it changed slightly with D

_{wire}increasing. The model was verified as the calculated results were consistent with the experimental data.

## 4. Conclusions

- (1)
- The resistance and inductance are inversely proportional functions vs. the quartic of the enameled wire diameter. Under the sinusoidal voltage, a wider wire diameter is quite helpful for a higher magnetic field amplitude while it has little influence on the phase lag of the magnetic field. Under the square-wave voltage, the steady-state magnetic field was positively proportional to the square of the wire diameter, as a wider wire diameter is helpful for a higher steady-state magnetic field. Regarding the response speed, increasing the wire’s diameter is helpful for reducing the response time from 0 to the specified intensity, while it is helpless to improve the response speed from 0 to the steady-state or any other proportional value.
- (2)
- The proposed model was verified as the calculated results from the model were in good agreement with the experimental results. Specifically, the relative errors of the model in computing the resistance and the inductance were lower than 3.1% and 2.8%, respectively. For predicting the sinusoidal response, the errors were lower than 6.4% (lower than 2.0% under most conditions) in computing the amplitude and lower than 3.2% in computing the lagging phase. For predicting the square-wave response, the model calculated the amplitudes with errors lower than 1.2% and described the curve shape effectively.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Experimental system and wound coils: (

**a**) block diagram of the experimental system; (

**b**) dimensioned sectional drawing of the coil, (

**c**) photograph of the experimental system; (

**d**) photograph of the coils.

**Figure 3.**Actual and fitted values of D

_{core}simultaneously supplied the relative errors of the fitting lines with and without an intercept.

**Figure 4.**Relative errors of the positively proportional function of the available area of coil skeleton vs. the cross-sectional area of enameled wire.

**Figure 6.**Amplitude ratio and phase lag of the magnetic field under different wire diameters: (

**a**) curves of the amplitude ratio vs. enameled wire diameter; (

**b**) curves of the phase lag vs. enameled wire diameter.

**Figure 7.**The relative errors of the model in computing the coil current under a harmonic voltage: (

**a**) the relative errors of computing the amplitude ratio; (

**b**) the relative errors of computing the phase lag.

**Figure 9.**The dynamic magnetic field under square-wave input: (

**a**) D

_{wire}= 0.31 mm; (

**b**) D

_{wire}= 0.39 mm; (

**c**) D

_{wire}= 0.49 mm; (

**d**) D

_{wire}= 0.60 mm; (

**e**) D

_{wire}= 0.69 mm; (

**f**) D

_{wire}= 0.80 mm.

**Figure 11.**The response time of the square-wave response from the test and model: (

**a**) the time from 0 to specified intensities, respectively, of 3 kA/m, 3.5 kA/m, and 4 kA/m; (

**b**) the time from 0 to the specified proportions of the steady-state response, respectively, of 0.7, 0.8, and 0.9.

Coil Label | External Diameter (D _{wire}) [mm] | Core Diameter (D _{wire}) [mm] | Number of Coil Turns (N) [Null] | Resistance (R) [Ω] | Inductance (L) [mH] |
---|---|---|---|---|---|

Coil 1 | 0.31 | 0.27 | 837 | 18.325 | 10.933 |

Coil 2 | 0.39 | 0.35 | 537 | 7.472 | 4.487 |

Coil 3 | 0.49 | 0.44 | 342 | 2.994 | 1.789 |

Coil 4 | 0.60 | 0.55 | 229 | 1.342 | 0.801 |

Coil 5 | 0.69 | 0.64 | 175 | 0.767 | 0.459 |

Coil 6 | 0.80 | 0.74 | 124 | 0.410 | 0.243 |

Parameter (Variable) [Unit] | Value |
---|---|

Coil length (L_{a}) [mm] | 16.5 |

Coil thickness (L_{b}) [mm] | 6.8 |

Diameter of skeleton shaft (L_{f}) [mm] | 18.2 |

Resistivity of copper (ρ) [Ω·m] | 1.71 × 10^{−8} |

Proportional coefficient (C_{HI}) [null] | 0.8 |

D_{wire} | Tested D_{core} | D_{core} from0.962 D _{wire} − 0.0277 | Relative Error of 0.962 D _{wire} − 0.0277 (%) | D_{core} from0.898 D _{wire} | Relative Error of 0.898 D _{wire} (%) |
---|---|---|---|---|---|

0.31 | 0.27 | 0.2705 | 0.1926 | 0.2784 | 3.1037 |

0.39 | 0.35 | 0.3475 | −0.7200 | 0.3502 | 0.0629 |

0.49 | 0.44 | 0.4437 | 0.8364 | 0.4400 | 0.0045 |

0.6 | 0.55 | 0.5495 | −0.0909 | 0.5388 | −2.0364 |

0.69 | 0.64 | 0.6361 | −0.6125 | 0.6196 | −3.1844 |

0.8 | 0.74 | 0.7419 | 0.2568 | 0.7184 | −2.9189 |

Coil Label | Coil Turns from Test | Coil Turns from Model ^{1} | Relative Error (%) |
---|---|---|---|

1 | 837 | 847.33 | 1.23 |

2 | 537 | 535.36 | −0.30 |

3 | 342 | 339.15 | −0.83 |

4 | 229 | 226.19 | −1.23 |

5 | 175 | 171.03 | −2.27 |

6 | 124 | 127.23 | 2.61 |

^{1}Cannot be an integer.

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## Share and Cite

**MDPI and ACS Style**

Wu, Z.; Bai, H.; Xue, G.; Ren, Z.
Optimization of the Wire Diameter Based on the Analytical Model of the Mean Magnetic Field for a Magnetically Driven Actuator. *Aerospace* **2023**, *10*, 270.
https://doi.org/10.3390/aerospace10030270

**AMA Style**

Wu Z, Bai H, Xue G, Ren Z.
Optimization of the Wire Diameter Based on the Analytical Model of the Mean Magnetic Field for a Magnetically Driven Actuator. *Aerospace*. 2023; 10(3):270.
https://doi.org/10.3390/aerospace10030270

**Chicago/Turabian Style**

Wu, Zhangbin, Hongbai Bai, Guangming Xue, and Zhiying Ren.
2023. "Optimization of the Wire Diameter Based on the Analytical Model of the Mean Magnetic Field for a Magnetically Driven Actuator" *Aerospace* 10, no. 3: 270.
https://doi.org/10.3390/aerospace10030270