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A simple and accurate method based on the magnetic equivalent circuit (MEC) model is proposed in this paper to predict magnetic flux density (MFD) distribution of the air-gap in a Lorentz motor (LM). In conventional MEC methods, the permanent magnet (PM) is treated as one common source and all branches of MEC are coupled together to become a MEC network. In our proposed method, every PM flux source is divided into three sub-sections (the outer, the middle and the inner). Thus, the MEC of LM is divided correspondingly into three independent sub-loops. As the size of the middle sub-MEC is small enough, it can be treated as an ideal MEC and solved accurately. Combining with decoupled analysis of outer and inner MECs, MFD distribution in the air-gap can be approximated by a quadratic curve, and the complex calculation of reluctances in MECs can be avoided. The segmented magnetic equivalent circuit (SMEC) method is used to analyze a LM, and its effectiveness is demonstrated by comparison with FEA, conventional MEC and experimental results.

In recent years, the Lorentz motor (LM) has been applied widely as an actuator to generate forces with direct drive, fast response time, great precision, low noise, low vibration,

Different methods, including analytical methods, numerical methods, and magnetic equivalent circuit (MEC) methods, have been employed to model MFD distribution [

MEC, based on the Kirchhoff's law, has become an efficient magnetic analysis method [

MEC was originally proposed and developed in [

However, MEC models always treat the permanent magnet (PM) as one common source and all branches of MEC are coupled together to become a large MEC network. If flux leakage and magnetic end are also considered, the complexity of these models has to be increased and the analysis process would become extremely complicated as well.

This paper presents a segmented magnetic equivalent circuit (SMEC) method, which can be used to analyze the magnetic field of the LM with considerably reduced complexity. The quadratic MFD distribution curves based on the sub-MECs are also proposed to analyze air-gap MFD distribution and to predict the relationship between the air-gap MFD and parameters of the LM. This SMEC method and the curve prediction method are validated by comparison with FEA, conventional MEC method and experimental results.

A LM is used as the actuator of an isolator since the Lorentz force can be characterized “fast”.

The working principle of a Lorentz motor is that a Lorentz force will be exerted on the coil when an electric current flows across it. Electric current I and MFD B are perpendicular to each other, and the direction of the Lorentz force F will be decided by Fleming's left-hand rule. The layout of the LM is illustrated in

The structure of the stator is shown in

The air-gap between the two magnet poles has width _{m}_{s}_{m}

The conventional MEC model [_{ss}_{ms}_{ss}_{gg}_{ss}_{gl}_{l}_{mg}_{mm}

Using the equivalent-resistance theory, the MEC in

The reluctance _{mm}

The flux leakage tube of _{gl}_{l}

From _{gg}_{gl}

If the magnetic flux is divided by the corresponding area that the magnetic flux through, the MFD B can be obtained. In the conventional lump-parameter MEC model, the reluctance is modeled by a single constant (e.g., _{gg}

In order to model the reluctance more accurately, the air-gap reluctance has to be divided into _{ggi}_{l}_{mg}_{mm}_{li}_{mgi}_{mmi}

In a magnetic field, magnetic flux lines (MFLs) make a closed route from the north to the south and don't cross each other. Similarly, if the MFLs are separated into groups, no group crosses another. In LM, the leakage flux appears at the edge of permanent magnets. If the MEC of the LM can be divided into three sub-MECs and every sub-MEC has independent flux sources and loop, the leakage flux only appears in the lateral groups and the middle MEC is ideal. If the analysis is based on an ideal sub-MEC from all sub-MECs, it will be simple and accurate. For every independent sub-MEC, if parameter of the LM changes in the design, it only affects the corresponding sub-MEC.

In the design and optimization of motors, it is necessary yet difficult to predict accurately MFD distribution of the air-gap. In order to overcome this problem, quadratic MFD distribution curves based on the analysis of the SMEC are used to obtain the MFD distribution curve of the air-gap. The analysis also holds for nonlinear materials, because the magnetic saturation can be negligible for a LM with larger air gap. Additionally, to avoid magnetic saturation, the geometric size and material properties of the steel-yoke have been chosen elaborately (such that _{s}_{m}

According to the SMEC method, the flux source is divided into three sub-parts (_{si}_{sm}_{so}

In all sub-MECs, the middle MEC is the smallest. In its air-gap, MFLs are deemed even without leakage and spreading. The middle MEC is closest to being ideal and affected only slightly by flux leakage in fact. Additionally, the smaller is the middle MEC, the more ideal is this sub-MEC. If necessary, the outer MEC and the inner MEC can be divided further in the segmented decoupling method.

In _{si}_{sm}_{so}_{mi}_{m}_{mo}_{si}_{sm}_{so}_{g}_{gl}_{g2}_{s}_{s1}_{s2}_{fl}_{ML}

_{s}_{s1}_{s2}

Here, L is the length of the PM in the LM as shown in _{g}_{m}

In the middle MEC, _{s}_{s1}

By flux division:

Substituting

Here, _{g1}_{g1}_{sm}_{g1}_{sm}_{g1}_{sm}_{0}_{rm}_{c}

The inner MEC can be divided further with the above method as illustrated in

By definition, _{si}_{1} and _{si2}_{si3}_{mi1}_{mi2}_{mi3}_{si}_{1}, _{si2}_{si3}_{mn}_{mm}_{smn}_{smm}_{s}

Here, it is assumed that under ideal conditions, MFLs pass through the PM evenly without leakage as illustrated in _{m}_{fl}_{fl}_{m}_{fl}

A reference frame, _{y-in}

Here, _{y-in}

The route of the flux leakage can be regarded approximately as an ellipse as shown in _{m}+ t

The MFD of the coordinate origin, as shown in

Similar to the inner MEC, the outer MEC can be divided further. Since the four sub-MECs are all similar, only one of them (in the dashed box) is divided further as in

Referring to the literatures [_{y-out}

Here, _{y-out}_{1}_{1}_{1}

On the boundary of the outer MEC, the MFD can be obtained by solving:

When flux leakage is weak, it can be ignored, or simplified as in the analysis of the inner MEC. Since the number of MFL is a constant, the following equation can be obtained:

By dividing every sub-MEC, the whole magnetic field of the LM can be divided into some separate and simple sub-MECs. In this way, every magnetic field of the LM can be analyzed independently and easily.

2-D FEA has been carried out to validate the SMEC method. The main parameters of the LM are given in

A group of lines that are parallel with the x axis and equally spaced with a distance 1 mm are drawn in air-gap g as shown in

Variation of the group of lines with coordinate x is plotted in

The MFD on MML is depicted in

As illustrated in the figure, results of the proposed SMEC method are in very good agreement with 2-D FEA and the difference is less than 6%, while the difference between conventional MEC analysis and 2-D FEA is much larger.

To obtain the relation between thrust force and current in the LM, the corresponding experiment is conducted as shown in

From

In the proposed SMEC method (

A simple yet accurate SMEC method for predicting air-gap MFD distribution of LMs is proposed, in which segmented sub-MECs are decoupled. The magnetic field of the LM can be analyzed with considerably reduced complexity and the relation between the air-gap MFD and the parameters of LM can be established easily. The size of middle sub-MEC is the smallest one, which approaches an ideal situation and can be solved accurately by MEC equations. In the middle part of the LM air-gap, the MFD is approximately uniform. Based on the study of the middle MEC, relationship between this part of the MFD and parameters of the LM can be obtained by analyzing the middle MEC. After analyzing sub-MECs, quadratic curves are used to predict the air-gap MFD of the LM. The calculation of complex reluctances of MECs is avoided. Prediction accuracy of the proposed method is verified by comparison with FEA results, and is less than 6%. The comparison between proposed SMEC and conventional MEC shows the advantage of the proposed SMEC. The proposed SMEC method can be used in LM design and optimization with improved simplicity and desirable accuracy.

The work was supported by the National Natural Science Foundation of China (No. 51121002, No. 51235005, No.51175196) and the Major State Basic Research Development Program of China (973 Program) (No. 2009CB724205).

Structure of the Lorentz motor.

Simplified layout of the Lorentz motor.

Structure of the stator.

Conventional MEC of the LM stator.

Conventional MEC reduced from

Proposed MEC of the LM stator.

The middle MEC.

The detailed inner MEC.

PM. (

The flux leakage model of the inner MEC.

The detailed outer MEC.

Illustration of the group of lines in the air-gap.

FEA results. (

Variation of the MFD with

Plot of _{g1}versus x

Photo of experimental setup.

Plot of F

Parameters of the LM.

_{0} |
4π × 10^{−7} |
H/m |

_{rm} |
1.02 | |

_{s} |
400 | |

_{m} |
20 | mm |

_{m} |
7.5 | mm |

_{s} |
10 | mm |

100 | mm |