Omnimagnet: An Omnidirectional Electromagnet for Controlled Dipole-Field Generation

这篇文章详细描述了Omnimagnet的制作和优化。通过内部含有一个球体铁芯和外部三层以一定规则计算的立方体正交嵌套的螺线圈，可以产生非常接近dipole model的磁场。最终大小只有，176mm边长的立方体，非常紧凑。

Omnimagnet: 用于受控偶极场生成的一个全向电磁铁
Omnimagnet: An Omnidirectional Electromagnet for Controlled Dipole-Field Generation [1]
Paper Link
Authors: J. J. Abbott, etc.
2014, IEEE Transactions on Magnetics

1. 摘要 Abstract

Omnimagnet是一个全向电磁铁，它包含一个在三个正交嵌套的螺线管内的球型铁磁核心。它会产生一个偶极磁矩大小和方向都可变且没有运动部件的磁偶极子场。
An Omnimagnet is an omnidirectional electromagnet comprising a spherical ferromagnetic core inside of three orthogonal nested solenoids. It generates a magnetic dipole field with both a variable dipole-moment magnitude and orientation with no moving parts.

这些通用关系用于设计最佳的Omnimagnet磁体，其约束条件是在任何方向上每个施加的电流都具有相同的偶极矩，每个螺线管对磁场都没有四极子的贡献，而球形磁芯的尺寸可最大程度地提高所产生的偶极场的强度。 使用FEA工具对这种最佳设计进行了分析，并证明其本质上类似于偶极子。
These general relationships are used to design an optimal Omnimagnet object to the constraints that it has the same dipole-moment per applied current in any direction, each solenoid has no quadrupole contribution to the magentic field, and the spherical core size maximizes the strength of the resulting dipole field. The optimal design is analyzed using FEA tool is verified to be dipole-like in nature.

2. 设计和优化 Design and Optimization

首先，我们将螺线管的形状选择为方形截面套管，以使包装紧密。下一步，我们将核心选择为一个球体，因为一个球形核心有三个期望的特性：（1）一个球形没有一个优先磁化方向。（2）当被放置在一个均匀场内，球体产生纯净偶极场。（3）在一个球体内平均外加磁场与在球心外加磁场相等。
Firstly, we choose the shape of the solenoids to be square-cross sectional sleeves to result in a dense packing. Next, we choose the core to be a sphere, because a spherical core has three desirable properties: (1) a sphere does not have a preferential magnetization direction. (2) when placed in a uniform field, the sphere produces a pure dipole field. (3) The average applied magnetic field within a sphere is equal to the applied magnetic field at the center of the sphere.

我们限制我们的设计为所有螺线管使用单根线规，这意味着相等的电流密度与相等的电流同义。
We constrain our design to use a single wire gauge for all solenoids, which means that an equal electrical current density is synonemous with an equal current.

在这篇文章中，I 将被用来表示电流，单位为{A}，J将被用来表示电流密度，单位为{ $A\cdot m^{-2}$}。因为电流密度对于电线选择来说是不变的，使用J执行对形状的优化；通常的讨论，但是，将使用I，因为从控制角度来看，它是更自然的参数。
Throughout this paper, I will be used to refer to the currents in units {A},and J will be used to refer to the current density in units { $A\cdot m^{-2}$}. Because current density is invarient to wire selection, the optimization for shape is performed using J; general discussion, however, will use I, as it is more natural parameter from a control perspective.

或者，一个解析偶极近似能被使用来对场建模。
alternatively, an analytical dipole approximation can be is used to model the field.

全磁铁越接近产生一个纯净偶极场，基于这个近似的算法将表现的更好。
The closer the Omnimagnet is to generate a pure dipole field, the better the algorithm based on the approximation will perform.

2.A. 螺线管多极场扩展 solenoid multipole field expansion

任何配置的电流密度的偶极磁矩是：
the dipole moment for a current density of any configuration:

$$\text{m}=\frac{1}{2}{\int }_{v_s}\text{r}\times \text{J}(\text{r})d\text{V}$$

方形截面螺线管的偶极磁矩是:
the dipole moment for a square-cross section solenoid:

$$\text{m}=\frac{J{L}^4}{6}({\beta }_2^3-{\beta }_1^3)\text{I}$$

具有边长为L不包含铁磁材料的边界正方体的任何电磁铁能够在一个方向产生的最大偶极磁矩能被计算，${\beta }_1=0$且${\beta }_2=1$，最大磁矩为$J\frac{L^4}{6}$。被期望用于具有边界长度L不包含铁磁材料的任何正方体全向电磁铁的最大理论偶极磁矩是三分之一的单向情况：$J\frac{L^4}{18}$；这个数据在这篇文章中被使用来标准化非量纲优化的强度，即使构建这样一个理想化的全方向电磁铁将会非常有挑战性。

2.B. 核心的偶极场贡献 core dipole field contribution

$$\text{m}=\mathbb{M}\text{I}$$

因为$\mathbb{M}$仅仅是螺线管几何形状的函数，优化问题能够被分为四步：(1）在每个方向选择最大的电流密度${\text{I}}_{max}$。（2）确定几何参数$\beta$来均等化$\mathbb{M}\text{I}$的元素。(3）优化整体尺寸达到一系列物理和运行限制。
Since $\mathbb{M}$ is only the function of solenoids’ geometries, the optimization problem can be split into four steps: (1) choose the maximum current density ${\text{I}}_{max}$ in each direction. (2) determine the geometric $\beta$ factors to equalize the components of $\mathbb{M}\text{I}$. (3) optimize the overal size to a set of physical and operational constraints.

2.C. 偶极磁矩均等化 dipole moment equalization

Omnimagnet的最佳几何形状对应于这样的几何比率，该比率使每个方向上产生的偶极矩最大化，具有相同的偶极矩比以最大化每个方向上的电流密度，并且没有四极矩。 这是一个带限制的优化问题，可以通过将所有长度除以L_max并将矩除以m_ref来进行无量纲化
The optimal geometry for the Omnimagnet corresponds to the geometric ratios, which maximize the dipole moment in each direction, have the same ratio of dipole moment to maximum current density in each direction, and have no quadrupole moment. This is a constrained optimization problem, and can be nondimentionalized by dividing all of the lengths by $L_{max}$, and the moments by $m_{ref}$.

因此，目的是使偶极磁矩大小$||m||$最大化，这取决于以下条件。
Thus, the objective is to maximize the dipole moment magnitude $||m||$, which is subjected to the following.
（1）这个配置不含有四极矩。This configuration has no quadrupole moment.
（2）$||{\text{m}}_x||/m_{ref}=||{\text{m}}_y||/m_{ref}=||{\text{m}}_z||/m_{ref}$。
（3）$W_x=2R_c$(4)$W_y=W_x+2T_x$(5)$W_z=W_y+2T_y$

[1]: Petruska, Andrew J. , and J. J. Abbott . “Omnimagnet: An Omnidirectional Electromagnet for Controlled Dipole-Field Generation.” IEEE Transactions on Magnetics 50.7(2014):1-10.