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This paper was originally presented at ISEF 2009 and submitted to the Special Issue: ISEF 2009.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

An electro-quasistatic analysis of an induction micromotor has been realized by using the Cell Method. We employed the direct Finite Formulation (FF) of the electromagnetic laws, hence, avoiding a further discretization. The Cell Method (CM) is used for solving the field equations at the entire domain (2D space) of the micromotor. We have reformulated the field laws in a direct FF and analyzed physical quantities to make explicit the relationship between magnitudes and laws. We applied a primal-dual barycentric discretization of the 2D space. The electric potential has been calculated on each node of the primal mesh using CM. For verification purpose, an analytical electric potential equation is introduced as reference. In frequency domain, results demonstrate the error in calculating potential quantity is neglected (<3‰). In time domain, the potential value in transient state tends to the steady state value.

Electromagnetic laws were formulated

After Maxwell’s publication, electromagnetic laws have been commonly written using differential equations. Because differential formulation is restricted to homogeneous regions—material homogeneity—heterogeneous domains are broken in homogeneous subdomains plus jump conditions. The discrete formulation of differential equations requires a discretization method, such as finite difference, finite element, boundary element, among others.

As an alternative, a direct Finite Formulation (FF) of the electromagnetic laws based on global variables accepts material discontinuities, as is the case of the micromotor interface region, which is the surface of the resistive metal sheet of the mobile part of the micromotor in contact with the air (see

The main benefit of CM is the remarkable simplification of its theoretical formulation, and therefore, the obtained equation system. The CM algebraic equation system is equivalent to the obtained in FEM using affine approximation of the electric potential inside of the triangle mesh. CM simplification is because physical laws of the electrostatic induction micromotor are expressed directly by a set of algebraic equations. However, in FEM, the algebraic equations are obtained after a discretization process using differential equations. Thus, CM requires two steps less than FEM to obtain the same algebraic system of equations.

The fundamental principle of CM is the use of finite or global measurable quantities. In the micromotor analysis, we use the voltage along a line instead of the electric field in a point. Therefore, we don’t use those quantities that are defined through a mathematical limit process as standard operations of gradient, curl and divergence. Note that a mathematical limit process involves operational difficulties in some conditions—such as discontinuities in the electrical field in the interface, due to the superficial conductivity. They are not adequate for numerical processing. Because of this, FEM involves two additional steps: first, Green’s theorem is applied; and second, the first order interpolation function of Whitney elements is used. The last step introduces a tangential continuity of the field magnitudes in the edge of the elements and, however, allows discontinuity in the normal component. The constitutive equations in CM formulation have a deep geometric interpretation based in the geometry of primal and dual meshes. This interpretation facilitates the incorporation of two types of physical properties, volumetric and superficial with electrical conductivity.

Nowadays, the design and implementation of a micromotor using MEMS technology is a great challenge [

In this work, for an electrostatic micromotor, the superficial conductivity at the interface of the mobile part plays a key role. In addition, we consider a volumetric conductivity at the mobile part. _{S} and σ_{b}, respectively. As a consequence, time dependent terms are considered in our finite formulation problem, and therefore, we carry out both frequency and transitory analysis.

The analyzed micromotor is a simple linear electrostatic induction micromachine constituted by two parallel plates—mobile part and stator—isolated by a dielectric [

The paper is organized as follows. Section 2 contains the reformulation of the field laws in a direct FF for the micromotor. Initially, we introduce global variables by analyzing physical quantities in order to make explicit the maximum of information. Both topological and constitutive equations are explained in detail. Then, we present the final global equation of the electrostatic induction micromotor. In Section 3, we provide an analytical equation of the electric potential—global variable—at the interface of the micromotor. For verification purpose, electric potential values are calculated by solving field equations with CM. Both frequency and time domain comparisons are introduced. Finally, Section 4 provides conclusions of the work.

The reformulation of field laws in a direct FF begins with an analysis of physical quantities. Physical measurements deal with global variables against field variables. In differential formulation, field variables are utilized because the notion of derivative refers to a point function. Contrariwise, global variables refer to a system, at a space or time element—global variables concern to oriented geometrical elements like points, lines, surfaces, volumes and time elements like instant and interval.

According to FF, global variables are also classified into configuration, source, and energy variables [

CM requires the use of a pair of oriented cell complexes, one dual to each other, endowed with inner orientation (see I,J,K cell in

According to electromagnetism FF, the first principle [

The field equation of the micromotor is enforced, on the cell complexes, in exact discrete form by using incidence matrices

The following equations represent the counterparts of the differential Maxwell′s laws.

Gauss law:

Faraday law (for quasi–electrostatic conditions):

Charge conservation law:

The duality between the oriented primal and dual space cell complexes leads, in general, to the following relationships [

While field equations in direct FF describe the physical laws exactly, the constitutive equations describe the physical laws approximately. For the micromotor the integral potential and flux state variables, which are allocated on two different cell complexes, are related to each other by the constitutive material equations. These equations are matrix equations. They contain the average information of the material and grid dimension [

Since

Volumetric properties—volumetric conductivity and permittivity—and superficial properties—superficial conductivity—are considered constitutive equations of the micromotor. Therefore, two classes of cells for the discrete constitutive material equations rise. For a bidimensional form, the volume cell and face cell are transformed in face cell and edge cell, respectively.

The constitutive equations for a simple primal–dual cell (see

The expressions for the face element are:

^{e}^{e}^{e}

Δ_{1}, Δ_{2}, and Δ_{3} expressions are:

(_{1x}, _{1y}), (_{2x}, _{2y}) and (_{3x}, _{3y}), are vectors associated to the primal edges:
_{i}_{i}^{e}

The permittivity and conductivity tensors are:
_{1}, _{2} and _{3} are the voltage associated to the edges _{1}, _{2} and _{3} (see _{1}, _{2}, _{3} and _{1}, _{2}, _{3} are the electric flow and the electric intensity associated to the surfaces _{1}, _{2} and _{3}, respectively, of the simple dual cell (see

^{e}

The components of the vectors associated to the dual surfaces _{1}, _{2} and _{3} (see _{b}_{1}_{x}_{b}_{1}_{y}_{b}_{2}_{x}_{b}_{2}_{y}_{b}_{3}_{x}_{b}_{3}_{y}

The centers (_{1}, _{2} _{3}) of the edges 1, 2, and 3, respectively, are the coordinates:

Point

We obtain the local fundamental matrix by substituting in ^{e} is the incidence matrix of one element and is expressed as follows:

For computational purpose, processing cells one by one is convenient. To obtain the global fundamental matrix all the local fundamental matrices on the reference cell are assembled (see

For a bidimensional formulation, in case of triangular elements under the hypothesis of uniform field and using a dual mesh with barycentric subdivision, the resulting matrix for one element is symmetric. Moreover, this matrix is coincident with the element matrix obtained with finite elements with affine approximation of the electric potential within the triangle [

First,

Next equation represents the analytical electric potential at the interface of the micromotor [

We have calculated the potential at the interface, applying CM and the analytical equations, for five different values of the conductivity. The mismatch between the results obtained using analytical equations and the CM are neglected, as can be seen in ^{6}) 1/Ω.

We have also calculated the electric field at the interface. CM results and analytical solution results are illustrated in ^{6}) 1/Ω. The error between the results obtained using analytical equations and the CM is neglected.

CM convergence has been guaranteed with the refining of the meshes of the micromotor as can be seen in ^{6}) 1/Ω.

In order to perform numerical calculations in time domain it is necessary to discretize the time axis for

_{1} = 1.92·10^{−7} s, t_{2} = 4.81·10^{−7} s and t_{3} = 3.85·10^{−6} s, respectively. As the transient analysis evolves, the maximum value approaches to the value that will be reached in permanent state, as can be seen in

As an alternative to the differential formulation of the electromagnetic laws, we have rewritten the field laws of an electrostatic induction micromotor in a direct finite formulation. For solving field equations in direct finite formulation, we applied the Cell Method (CM) and obtained a relationship between volumetric and superficial material properties at the interface of the micromotor. The micromotor was analyzed in both time and frequency domain. The electric potential global variable—configuration variable—has been calculated. For comparison purpose, an analytical solution of the electric potential is utilized as reference. Comparisons against analytical solutions of the electric potential demonstrated the cogency of our proposal. In frequency domain, the error between analytical and CM is less than 0.3‰—for a primal mesh of 2,353 nodes and 4,704 elements. In addition, transient analysis in time domain has been carried out using θ-method. Electric potential at the interface of the micromotor tends to the steady state value validating our approach, also.

Linear electrical induction micromachine.

Dual barycentric subdivision.

Simple primal-dual cell for assemble process.

^{6}) 1/Ω. ^{6}) 1/Ω.

Graphical representation of maximal potential at the interface

Electric field for a superficial conductivity of 1/(600·10^{6}) 1/Ω.

Transitory state of the potential at the interface in points

Transitory state of the micromotor: potential distribution at the interface for instant t_{1}.

Transitory state of the micromotor: potential distribution at the interface for instant t_{2}.

Transitory state of the micromotor: potential distribution at the interface for instant t_{3}.

Nomenclature.

| ||
---|---|---|

Height of the air gap | m | |

Height of insulator | m | |

Number of waves per metre | - | |

Length | m | |

Imaginary unity | - | |

_{f} |
Current density | A/m^{2} |

Slip | - | |

Thickness | m | |

Linear speed of mobile part | m/s | |

Interelectrodic potential | V | |

_{0} |
Supply potential | V |

_{a} |
Electric permittivity of the air | F/m |

_{b} |
Electric permittivity of the insulator | F/m |

_{eff} |
Effective permittivity | F/m |

Electric scalar potential | V | |

Angular frequency of the signal | Hz | |

_{a} |
Electric conductivity of the air | S/m |

_{b} |
Electric conductivity of the insulator | S/m |

_{S} |
Superficial electric conductivity | 1/Ω |

_{eff} |
Effective Conductivity | S/m |

^{b} |
Potential at the interface | V |

Physical and geometrical parameters of the micromachine.

| |||
---|---|---|---|

Length of the structure | 44 | μm | |

_{m} |
Height of the metal sheet | 0.01 | μm |

Height of dielectric 2 | 3 | μm | |

Height of dielectric 1 | 10 | μm | |

Number of waves per meter | 2π/L | μm^{−1} | |

Linear speed of mobile part | 0 | μm/s | |

Temporal frequency of excitation | 2.6 × 10^{6} |
Hz | |

_{0} |
Maximum value of excitation | 200 | V |

Interface electrical potential.

| |||
---|---|---|---|

1/(50·10^{6}) |
21.6688 | 21.6947 | −0.119 |

1/(100·10^{6}) |
37.7909 | 37.7259 | 0.172 |

1/(200·10^{6}) |
53.6311 | 53.5904 | 0.075 |

1/(600·10^{6}) |
64.2738 | 64.2748 | −0.001 |

1/(1800·10^{6}) |
65.8906 | 65.9102 | −0.029 |

Electric field in the steady state at the interface in z = 0.

| |||
---|---|---|---|

1/(50·10^{6}) |
3094307 | 3102000 | −0.248 |

1/(100·10^{6}) |
5381641 | 5389700 | −0.149 |

1/(200·10^{6}) |
7658503 | 7665400 | −0.090 |

1/(600·10^{6}) |
9178278 | 9182800 | −0.049 |

1/(1800·10^{6}) |
9409100 | 9419900 | −0.114 |

Effect of the mesh in the convergence.

2353 | 4704 | 65.89 | 65.91 | 0.030 |

613 | 1224 | 65.89 | 66.02 | 0.197 |

284 | 566 | 65.89 | 66.20 | 0.470 |

170 | 338 | 65.89 | 66.40 | 0.774 |