**FEA in Micro-Electro-Mechanical Systems (MEMS) Applications: A Micromachined Spatial Light Modulator (μSLM)**

Hao Ren and Jun Yao

160 Finite Element Analysis – New Trends and Developments

Nadu, India, 23-24 March 2011.

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Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48532

## **1. Introduction**

Since the Nobel Prize winner, Richard Feynman gave the presentation "there is plenty of room at the bottom" [1], a variety of micromachined sensors, actuators, and systems have emerged and made encouraging progress in the past 50 years, based on technological innovations and increased market demand [2]. To date, Micro-Electro-Mechanical Systems (MEMS) have been developed into an interdisciplinary subject which involves electrical, mechanical, thermal, optical, and biological knowledge. Due to its significant potential, which has partially been demonstrated by the success of inertial MEMS devices (accelerometers, gyroscopes, *etc* [3, 4]) radio frequency (RF) MEMS devices (switches, filters, resonators, *etc* [5- 7]) and optical MEMS devices (Digital Light Processing, DLP [8, 9]), the research in MEMS has attracted worldwide interest. Figure 1 shows a typical process of a MEMS device from design goal to system integration. We can see that the structure and fabrication process of MEMS device are designed according to the design goal. Then before fabrication, we need to perform modeling to the structure. By modeling, we can estimate the performance to see if it satisfies the design goal and then optimize it to achieve the best performance. By performing modeling, substantial time and money can be saved, which increases the throughput and reduces the cost. As a result, modeling is critical for MEMS research.

Modeling applied in MEMS applications can mainly be divided into two categories, theoretical modeling and numerical modeling. The theoretical modeling is to apply exact equations to obtain exact solutions. It is a direct approach which is easy to interpret intuitively [10]. However, it has limitations that solutions can only be obtained for few standard cases, and it is incapable or difficult in the following situations: (1) shape, boundary conditions, and loadings are complex; (2) material properties are anisotropic; (3) structure has more than one material; (4) problems with material and geometric non-

© 2012 Ren and Yao, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Ren and Yao, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

linearity; (5) multiphysics situations when more than two physics are coupled together. The theoretical modeling is sometimes applied in MEMS applications when the structure is not complex, and it is also useful to verify the result of FEA.

FEA in Micro-Electro-Mechanical

Systems (MEMS) Applications: A Micromachined Spatial Light Modulator (μSLM) 163

thermoelectromechanical simulation, piezoelectric/piezoresistive simulation, microfluidics simulation, *etc*. Quite a few commercially available FEA softwares are readily used in MEMS applications, such as Intellisuite, ANSYS, COMSOL, Conventorware. Of them, Intellisuite is specially designated for MEMS simulation with quite a few modules, including Intellimask, Intellifab, MEMeterial, 3D Builder, Thermoelectromechanical modules, *etc*., which has user friendly interface that users can obtain 3D structure directly from defined masks and fabrication process, and the 3D structure can be directly applied to further simulation [12]. What is more, the simulation result is quite close to experiment. In this chapter, we use

In this chapter, we show the importance of FEA in MEMS research through an example of a micromachined spatial light modulator (μSLM). Firstly we will introduce the design and operating principle of the μSLM. Then we will introduce the modeling of the μSLM, including theoretical modeling and FEA modeling. Following is the optimization of the

μSLM according to the modeling. Finally we present the fabrication and experiment.

**Figure 2.** A schematic of the operating principle of spatial light modulator (SLM) used in Adaptive

Adaptive Optics area, researchers use SLM to correct the wavefront to be plane.

Spatial Light modulators (SLMs) play an important role in modern technology, particularly in the field of micro-optical technology. They find applications in optical communication systems, and adaptive optics (AO) systems [13]. AO systems perform closed-loop phase correction of time-varying, aberrated wavefronts using two essential components: a wavefront sensor and a SLM [14]. A schematic of the operating principle of SLM is illustrated in figure 2. When light from stars travel through the atmosphere, aberration is induced by the turbulence in atmosphere, and the wavefront of the light is no longer a plane. If this wavefront is imaged by a telescope, a very blurry image is formed, therefore in

Conventional SLMs based on piezoelectric actuators cost approximately \$1000 per actuator and therefore find limited use even at major research centers [15]. In contrast, MEMS

Optics systems, SLM is used to corrected the wavefront

Intellisuite for simulation.

**Figure 1.** A schematic of a typical process of a MEMS device

Numerical modeling is to apply exact equations to obtain approximate solutions only at discrete points called nodes. Contrary to theoretical modeling, numerical modeling can handle situations which theoretical modeling is incapable. Finite Element Method (FEM) and Finite Difference Method (FDM) are two approaches most frequently used in numerical modeling. For both methods, they start from discretization, which derives the solution domain into a number of small elements and nodes. For FDM, differential equation is written for each node, and the derivatives are replaced by difference equations. In contrast, for FEM, it uses integral formulations rather than difference equations to create a system of algebraic equations, and an approximate continuous function is assumed to represent the solution for each element. The complete solution is then generated by connecting or assembling the individual solutions, allowing for continuity at the interelemental boundaries [11]. FEM have quite a few advantages over FDM [10], such as (1) it can give values at any point, while FDM can only give value at discrete node points; (2) FEM can consider the sloping boundaries exactly, while FDM makes stair type approximation to sloping; (3) FEM needs fewer nodes to get good results while FDM needs large number of nodes; (4) FEM can handle almost all complicated problems, while FDM cannot handle complicated problems, such as multiphysics simulation which is the general case in MEMS applications.

Due to the aforementioned advantages, FEA has been widely applied in MEMS applications, including electromagnetic simulation, electrothermal simulation, thermoelectromechanical simulation, piezoelectric/piezoresistive simulation, microfluidics simulation, *etc*. Quite a few commercially available FEA softwares are readily used in MEMS applications, such as Intellisuite, ANSYS, COMSOL, Conventorware. Of them, Intellisuite is specially designated for MEMS simulation with quite a few modules, including Intellimask, Intellifab, MEMeterial, 3D Builder, Thermoelectromechanical modules, *etc*., which has user friendly interface that users can obtain 3D structure directly from defined masks and fabrication process, and the 3D structure can be directly applied to further simulation [12]. What is more, the simulation result is quite close to experiment. In this chapter, we use Intellisuite for simulation.

162 Finite Element Analysis – New Trends and Developments

complex, and it is also useful to verify the result of FEA.

D

Design Goal

Structure Design

Optimization

Device Fab

**Figure 1.** A schematic of a typical process of a MEMS device

applications.

linearity; (5) multiphysics situations when more than two physics are coupled together. The theoretical modeling is sometimes applied in MEMS applications when the structure is not

Process Design

I i

Numerical modeling is to apply exact equations to obtain approximate solutions only at discrete points called nodes. Contrary to theoretical modeling, numerical modeling can handle situations which theoretical modeling is incapable. Finite Element Method (FEM) and Finite Difference Method (FDM) are two approaches most frequently used in numerical modeling. For both methods, they start from discretization, which derives the solution domain into a number of small elements and nodes. For FDM, differential equation is written for each node, and the derivatives are replaced by difference equations. In contrast, for FEM, it uses integral formulations rather than difference equations to create a system of algebraic equations, and an approximate continuous function is assumed to represent the solution for each element. The complete solution is then generated by connecting or assembling the individual solutions, allowing for continuity at the interelemental boundaries [11]. FEM have quite a few advantages over FDM [10], such as (1) it can give values at any point, while FDM can only give value at discrete node points; (2) FEM can consider the sloping boundaries exactly, while FDM makes stair type approximation to sloping; (3) FEM needs fewer nodes to get good results while FDM needs large number of nodes; (4) FEM can handle almost all complicated problems, while FDM cannot handle complicated problems, such as multiphysics simulation which is the general case in MEMS

Sample Test System

Due to the aforementioned advantages, FEA has been widely applied in MEMS applications, including electromagnetic simulation, electrothermal simulation, In this chapter, we show the importance of FEA in MEMS research through an example of a micromachined spatial light modulator (μSLM). Firstly we will introduce the design and operating principle of the μSLM. Then we will introduce the modeling of the μSLM, including theoretical modeling and FEA modeling. Following is the optimization of the μSLM according to the modeling. Finally we present the fabrication and experiment.

**Figure 2.** A schematic of the operating principle of spatial light modulator (SLM) used in Adaptive Optics systems, SLM is used to corrected the wavefront

Spatial Light modulators (SLMs) play an important role in modern technology, particularly in the field of micro-optical technology. They find applications in optical communication systems, and adaptive optics (AO) systems [13]. AO systems perform closed-loop phase correction of time-varying, aberrated wavefronts using two essential components: a wavefront sensor and a SLM [14]. A schematic of the operating principle of SLM is illustrated in figure 2. When light from stars travel through the atmosphere, aberration is induced by the turbulence in atmosphere, and the wavefront of the light is no longer a plane. If this wavefront is imaged by a telescope, a very blurry image is formed, therefore in Adaptive Optics area, researchers use SLM to correct the wavefront to be plane.

Conventional SLMs based on piezoelectric actuators cost approximately \$1000 per actuator and therefore find limited use even at major research centers [15]. In contrast, MEMS

technology offers a potentially low cost alternative to existing SLMs: the μSLMs. A large problem for μSLMs is their small stroke (maximum displacement), which greatly undermines the performance of the whole AO system. As a result, researchers have tried quite a few approaches to enlarge the stroke, but these approaches are difficult to implement because of either fabrication difficulty or structural complexity. As a result, we come up with a μSLM based on the leverage principle in this chapter to solve the problem.

FEA in Micro-Electro-Mechanical

Systems (MEMS) Applications: A Micromachined Spatial Light Modulator (μSLM) 165

**Figure 3.** (a) A lateral view of a μSLM, it is composed of four single out-of-plane actuators and one mirror plate, each single out-of-plane acutator is composed of two anchors, two microbeams, one lower electrode, one upper electrode (serving as the short arm), and one long arm, (b) A lateral view of a single out-of-plane actuator, (c) cross-section view of the single out-of-plane actuator in figure 2(b) when a voltage is applied between the lower and upper electrode, a small displacement at the end of

short arm will be amplified to be a larger displacement at the end of the long arm

## **2. Design and modeling of the μSLM**

## **2.1. Design and operating principle of the μSLM**

A schematic of the μSLM is shown in figure 3(a). It can be seen that the μSLM is composed of four single out-of-plane actuators and one mirror plate (here only 1/4 mirror plate is shown), which are connected together by a via. From figure 2(b) we can see that one single out-of-plane actuator is composed of two anchors, two microbeams, one lower electrode, one upper electrode (serving as the short arm), and one long arm. The lower electrodes and two anchors are fixed to substrate and the long arm is connected to the two anchors by two microbeams. The size of each actuator is shown in figure 3(b). *W1*, *L1*, *a,* and *L* represent the width and the length of the upper and lower electrodes, and *l*1, *b*1, *l*2 and *b*2 denote the length and the width of the microbeams and long arms, respectively, while *h* is the thickness of the structural layer and *d* is the horizontal distance from the fulcrum to the central line of bottom electrode.

When the lower and upper electrodes are subjected to different potentials, electrostatic attractive force arises. A torque around the microbeams emerges and makes the upper electrode and the long arm rotate around the microbeams, as shown in figure 3(c). As a result, the end of the long arm goes upward, thus forming a lever mechanism, with the microbeams as a fulcrum. If the length of the long arm is much larger than that of the short arm, the downward displacement will be magnified to be a much larger upward displacement. At the same time, microbeams will bend down due to the moment from the electrostatic force. The total displacement of the mirror plate is the vector sum of the upward displacement caused by the rotation and the downward displacement of the microbeams (Here we assume the displacement of the mirror plate is the same as the displacement at the end of the long arm, because in MEMS applications the mass of the mirror plate can be neglected). By a proper design of the structure, the downward displacement of the microbeams will be much smaller than the upward displacement at the end of the long arm, resulting in a larger upward displacement of the mirror plate [16].

## **2.2. Modeling of the μSLM**

After presenting the structure and operating principle, we modeled the μSLM both by theoretical modeling and FEA. First we carried out theoretical models. Two approaches were used in the theoretical modeling: the energy method and the superposition method.

**2. Design and modeling of the μSLM**

bottom electrode.

displacement of the mirror plate [16].

**2.2. Modeling of the μSLM**

**2.1. Design and operating principle of the μSLM**

technology offers a potentially low cost alternative to existing SLMs: the μSLMs. A large problem for μSLMs is their small stroke (maximum displacement), which greatly undermines the performance of the whole AO system. As a result, researchers have tried quite a few approaches to enlarge the stroke, but these approaches are difficult to implement because of either fabrication difficulty or structural complexity. As a result, we come up

A schematic of the μSLM is shown in figure 3(a). It can be seen that the μSLM is composed of four single out-of-plane actuators and one mirror plate (here only 1/4 mirror plate is shown), which are connected together by a via. From figure 2(b) we can see that one single out-of-plane actuator is composed of two anchors, two microbeams, one lower electrode, one upper electrode (serving as the short arm), and one long arm. The lower electrodes and two anchors are fixed to substrate and the long arm is connected to the two anchors by two microbeams. The size of each actuator is shown in figure 3(b). *W1*, *L1*, *a,* and *L* represent the width and the length of the upper and lower electrodes, and *l*1, *b*1, *l*2 and *b*2 denote the length and the width of the microbeams and long arms, respectively, while *h* is the thickness of the structural layer and *d* is the horizontal distance from the fulcrum to the central line of

When the lower and upper electrodes are subjected to different potentials, electrostatic attractive force arises. A torque around the microbeams emerges and makes the upper electrode and the long arm rotate around the microbeams, as shown in figure 3(c). As a result, the end of the long arm goes upward, thus forming a lever mechanism, with the microbeams as a fulcrum. If the length of the long arm is much larger than that of the short arm, the downward displacement will be magnified to be a much larger upward displacement. At the same time, microbeams will bend down due to the moment from the electrostatic force. The total displacement of the mirror plate is the vector sum of the upward displacement caused by the rotation and the downward displacement of the microbeams (Here we assume the displacement of the mirror plate is the same as the displacement at the end of the long arm, because in MEMS applications the mass of the mirror plate can be neglected). By a proper design of the structure, the downward displacement of the microbeams will be much smaller than the upward displacement at the end of the long arm, resulting in a larger upward

After presenting the structure and operating principle, we modeled the μSLM both by theoretical modeling and FEA. First we carried out theoretical models. Two approaches were used in the theoretical modeling: the energy method and the superposition method.

with a μSLM based on the leverage principle in this chapter to solve the problem.

**Figure 3.** (a) A lateral view of a μSLM, it is composed of four single out-of-plane actuators and one mirror plate, each single out-of-plane acutator is composed of two anchors, two microbeams, one lower electrode, one upper electrode (serving as the short arm), and one long arm, (b) A lateral view of a single out-of-plane actuator, (c) cross-section view of the single out-of-plane actuator in figure 2(b) when a voltage is applied between the lower and upper electrode, a small displacement at the end of short arm will be amplified to be a larger displacement at the end of the long arm

In the energy method, the force and moments applied to the structure is shown in figure 4. When different potentials are subjected to the upper and lower electrodes, electrostatic force arises. As mentioned above, the upper electrodes and long arms rotate around microbeams, and at the same time microbeams bend down. The force, moment and torque of anchors tend to resist this movement and the structure will ultimately reach a balance.

When a voltage *V* is applied to the four upper and the four lower electrodes, the electrostatic attractive force can be calculated by [16]:

$$F\_e = \frac{\varepsilon \varepsilon\_0 L aV^2}{2h\_0^{\ast^2}} [1 + \frac{2d}{h\_0} \theta] \tag{1}$$

FEA in Micro-Electro-Mechanical

Systems (MEMS) Applications: A Micromachined Spatial Light Modulator (μSLM) 167

<sup>1</sup> *FF F N e* 2 8 (3)

1 2 *M M Fx M <sup>N</sup>* , *TTT* 1 2 (4)

2 12 2 12 2

(9)

1 ( 8) 8 2 *F l F l d GJ <sup>e</sup> <sup>T</sup> EI l GJl*

(7)

(8)

3 1 ( ) 2 2 4 2, *M F a x F x T Fa Fx T e Ne* <sup>3</sup> *T* 0 (5)

one long arm and one upper electrode (short arm). In our case, the width of upper electrodes is much larger than the other three parts and the length is smaller, therefore we

As shown in figure 4, the bending moment and the torque applied to two microbeams and

According to the virtual work principle, the total strain energy in one single actuator is

<sup>1</sup> 0 0 0 00 <sup>221</sup>

1 1 2 11 2 2 2 2 2 1 2 3 1 2

Where *E*, *G*, *J*, *I1*, and *I2* represent the Young's modulus, the shear modulus, the polar moment of microbeams, the inertial moment for the long arm and the microbeams,

As the total strain energy stored in the lever actuator is four times of that in single actuator, and the bending and torsional angles at point A (see in figure 4) are both zero, according to

> <sup>1</sup> , <sup>2</sup> *M F l <sup>N</sup>*

11 22 3 3 <sup>000</sup> <sup>111</sup>

By combining the above equations to (8) and letting the fictitious load F1 be zero, we have

*F l F l laG J F l aGJ l l GJ <sup>z</sup> F a*

displacement of the mirror plate and the downward displacement of the microbeams.

3 3 22 2 3

In the superposition method, the displacement of the mirror plate is the sum of the upward

1 1 1 1

2 1 1 1 11 11 1 1 <sup>2</sup> ( )( ) <sup>24</sup> ( 2) 22 2

*EI GJ EI l GJl EI EI l GJl EI l GJl*

*Mx Mx Mx Mx Mx Mx z dx dx dx EI F EI F EI F*

() () () () () ()

() () ( ) () () 2 2 2 22 *l l l ll Mx Mx M x Tx Tx <sup>U</sup> dx dx dx dx dx EI EI EI GJ GJ* (6)

assume upper electrodes are rigid.

the long arm can be given by

Castigliano's second theorem

0, *Utotal M*

<sup>0</sup> *Utotal T*

11 22 0 0 1 1

*Tx Tx Tx Tx dx dx GJ F GJ F*

() () () ()

Based on the unit-load method, equation (7) can be rewritten as

1 1

*l l*

<sup>2</sup>

11 2

*lll*

2

*e e <sup>e</sup> <sup>e</sup>*

respectively.

According to equations of force equilibrium, we get

In our design, 2*d*/*h0*=20, and *θ*<<1. In order to simplify our calculation, the second term can be omitted without bringing much error.

The electrostatic force can be simplified and rewritten as follows

$$F\_e = \frac{\varepsilon \varepsilon\_0 L a V^2}{2h^2} \tag{2}$$

In order to derive the displacement at the mirror, we used Castigliano's second theorem and set a fictitious load at the central mass, as shown in Figure 4.

**Figure 4.** Mechanical model of the energy method for the μSLM, forces, moments and torques of one single out-of-plane actuator is shown for simplification.

Considering the symmetry of the structure, the μSLM can be divided into four single out-ofplane actuators and each one can be further subdivided into four parts: two microbeams, one long arm and one upper electrode (short arm). In our case, the width of upper electrodes is much larger than the other three parts and the length is smaller, therefore we assume upper electrodes are rigid.

According to equations of force equilibrium, we get

166 Finite Element Analysis – New Trends and Developments

attractive force can be calculated by [16]:

be omitted without bringing much error.

In the energy method, the force and moments applied to the structure is shown in figure 4. When different potentials are subjected to the upper and lower electrodes, electrostatic force arises. As mentioned above, the upper electrodes and long arms rotate around microbeams, and at the same time microbeams bend down. The force, moment and torque of anchors

When a voltage *V* is applied to the four upper and the four lower electrodes, the electrostatic

2

In our design, 2*d*/*h0*=20, and *θ*<<1. In order to simplify our calculation, the second term can

0 <sup>2</sup> 2 *<sup>e</sup> LaV <sup>F</sup> h*

In order to derive the displacement at the mirror, we used Castigliano's second theorem and

**Figure 4.** Mechanical model of the energy method for the μSLM, forces, moments and torques of one

Considering the symmetry of the structure, the μSLM can be divided into four single out-ofplane actuators and each one can be further subdivided into four parts: two microbeams,

2

(2)

(1)

0 2 0 0 <sup>2</sup> [1 ] <sup>2</sup> *<sup>e</sup> LaV <sup>d</sup> <sup>F</sup> h h*

The electrostatic force can be simplified and rewritten as follows

set a fictitious load at the central mass, as shown in Figure 4.

single out-of-plane actuator is shown for simplification.

tend to resist this movement and the structure will ultimately reach a balance.

$$F\_N = F\_e \left/ 2 - F\_1 \right/ 8 \tag{3}$$

As shown in figure 4, the bending moment and the torque applied to two microbeams and the long arm can be given by

$$M\_1 = M\_2 = F\_N \text{x} - M\_\prime \ T\_1 = T\_2 = T \tag{4}$$

$$M\_3 = F\_e(a+\infty) - 2F\_N \infty - 2T = F\_e a + F\_1 \ge \left\{ 4 - 2T \right\} \ T\_3 = 0 \tag{5}$$

According to the virtual work principle, the total strain energy in one single actuator is

$$\mathrm{U}\_{1} = \int\_{0}^{l\_{1}} \frac{M\_{1}^{2} \mathrm{(x)}}{2EI\_{2}} d\mathbf{x} + \int\_{0}^{l\_{1}} \frac{M\_{2}^{2} \mathrm{(x)}}{2EI\_{2}} d\mathbf{x} + \int\_{0}^{l\_{2}} \frac{M\_{3}^{2} \mathrm{(x)}}{2EI\_{1}} d\mathbf{x} + \int\_{0}^{l\_{1}} \frac{T\_{1}^{2} \mathrm{(x)}}{2Gl} d\mathbf{x} + \int\_{0}^{l\_{1}} \frac{T\_{2}^{2} \mathrm{(x)}}{2Gl} d\mathbf{x} \tag{6}$$

Where *E*, *G*, *J*, *I1*, and *I2* represent the Young's modulus, the shear modulus, the polar moment of microbeams, the inertial moment for the long arm and the microbeams, respectively.

As the total strain energy stored in the lever actuator is four times of that in single actuator, and the bending and torsional angles at point A (see in figure 4) are both zero, according to Castigliano's second theorem

$$\frac{\partial \mathcal{U}\_{\text{total}}}{\partial M} = 0, \; \frac{\partial \mathcal{U}\_{\text{total}}}{\partial T} = 0 \implies M = \frac{1}{2} F\_N l\_{2 \cdot \prime} \; T = \frac{1}{8} \frac{\{F\_1 l\_2^{\prime} + 8 F\_c l\_2 d\} \mathcal{G}\} \tag{7}$$

Based on the unit-load method, equation (7) can be rewritten as

$$\begin{split} \mathbf{z} &= \int\_{0}^{l\_{1}} \frac{M\_{1}(\mathbf{x})}{EI} \frac{\partial M\_{1}(\mathbf{x})}{\partial F\_{1}} d\mathbf{x} + \int\_{0}^{l\_{1}} \frac{M\_{2}(\mathbf{x})}{EI} \frac{\partial M\_{2}(\mathbf{x})}{\partial F\_{1}} d\mathbf{x} + \int\_{0}^{l\_{1}} \frac{M\_{3}(\mathbf{x})}{EI} \frac{\partial M\_{3}(\mathbf{x})}{\partial F\_{1}} d\mathbf{x} \\ &+ \int\_{0}^{l\_{1}} \frac{T\_{1}(\mathbf{x})}{Gf} \frac{\partial T\_{1}(\mathbf{x})}{\partial F\_{1}} d\mathbf{x} + \int\_{0}^{l\_{1}} \frac{T\_{2}(\mathbf{x})}{Gf} \frac{\partial T\_{2}(\mathbf{x})}{\partial F\_{1}} d\mathbf{x} \end{split} \tag{8}$$

By combining the above equations to (8) and letting the fictitious load F1 be zero, we have

$$z = -\frac{F\_e l^3}{24EI\_2} + \frac{1}{Gl} \frac{F\_e l\_1^3 \text{l}aG^2 \text{J}^2}{\left(EI\_1 l + 2Gl l\_1\right)^2} + \frac{1}{EI\_1} (F\_e a - \frac{2F\_e l\_1 aG \text{J}}{EI\_1 l + 2Gl l\_1}) (\frac{l\_1^2}{2} - \frac{l\_1^3 G \text{J}}{EI\_1 l + 2Gl l\_1})\tag{9}$$

In the superposition method, the displacement of the mirror plate is the sum of the upward displacement of the mirror plate and the downward displacement of the microbeams.

**Figure 5.** Mechanical model of the superposition method: (a) shows the forces and moments of the long and short arm, (b) shows the forces and moments of the 2 microbeams.

First we will analyze the force and moment applied on the long arm, as shown in figure 5. According to the boundary condition

$$y\_1(0) = 0, y\_1'(0) = 0, y\_1'(l\_2) = \theta\_1, \theta\_1 = M\_1 l\_1 \;/\, \text{2GL}\_p \tag{10}$$

FEA in Micro-Electro-Mechanical

Systems (MEMS) Applications: A Micromachined Spatial Light Modulator (μSLM) 169

(15)

Where *Z2R*, *ZlL* are the section vectors of the left end of section 1 and the right end of section 2, respectively. We can calculate the displacement of the center of various section beams. By substituting parameters in to these equations, we calculated that the downward

> 2 2 1 1 1 1 1 (1 ) <sup>2</sup> <sup>2</sup> 1 /2

Finally the displacement calculated by the superposition method is derived as

*waW V l*

2 2 1 1

*<sup>z</sup> <sup>y</sup> h EI EI l l GJ*

**Figure 6.** Results of the two theoretical models and FEA results, the discrepancy between theoretical

After deriving the theoretical models, we carried FEA modeling using Intellisuite. Then the FEA result is validated the theoretical modeling by comparing them with FEA. As shown in figure 5, both the two theoretical models are in good agreement with FEA, and this validates the results of the theoretical modeling and FEA. Then we combined the theoretical modeling

displacement is 32.4 nm when applied a voltage of 20 V.

Here y2 is calculated by the transfer matrix method.

modeling and FEA is small

and FEA to optimize the structure.

*Z CZ* 2 11 *R L* (14)

We can obtain the displacement of the center of the two long arms:

$$y = \frac{1}{2} \frac{\varepsilon w (b-a)(b+a)V^2 l\_2^{-2}}{4h^2 EI} (1 - \frac{1}{1 + EI\_1 / \, 2l\_2 GI\_p}) \tag{11}$$

The mechanical model of 2 microbeams is illustrated in Figure 5(b). It is an indeterminate beam with variable sections, therefore we can derive the displacement of center of the two microbeams using transfer matrix method [17]. First we divide it into 4 sections. The transfer matrix from the left end of section 1 to the right end of section 4 can be obtained

$$C = A\_1 \times A\_2 \times B \times A\_2 \times A\_1 \tag{12}$$

Where *A1*, *A2*, and *B* are transfer matrix between left end and right of section 1, left end and right of section 2, right end of section 2 and left end of section 3, which can all be calculated by law of transfer with cross section state vector.

After applying boundary condition, *y1L*=0, *θ1L*=0, *y4R*=0, *θ4R*=0 we can calculate force and moment applied on the left end of section 1 and right end of section 4. Then substituting the force and moment to

$$C\_1 = A\_2 \times A\_1 \tag{13}$$

We can calculate the transfer matrix from the left end of section 1 to the right end of section 2. Therefore according to

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$$\mathbf{Z\_{2R}} = \mathbf{C\_1} \times \mathbf{Z\_{1L}} \tag{14}$$

Where *Z2R*, *ZlL* are the section vectors of the left end of section 1 and the right end of section 2, respectively. We can calculate the displacement of the center of various section beams. By substituting parameters in to these equations, we calculated that the downward displacement is 32.4 nm when applied a voltage of 20 V.

Finally the displacement calculated by the superposition method is derived as

$$z = \frac{1}{2} \frac{\varepsilon v a l N\_1 V^2 l\_1^{-2}}{2h^2 E l\_1} (1 - \frac{1}{1 + E l\_1 l / \, 2l\_1 G l}) - y\_2 \tag{15}$$

Here y2 is calculated by the transfer matrix method.

168 Finite Element Analysis – New Trends and Developments

(a)

(b)

**Figure 5.** Mechanical model of the superposition method: (a) shows the forces and moments of the long

First we will analyze the force and moment applied on the long arm, as shown in figure 5.

1 1 1 2 1 1 11 (0) 0, (0) 0, ( ) , /2 *<sup>p</sup> y y yl*

2 2 2

The mechanical model of 2 microbeams is illustrated in Figure 5(b). It is an indeterminate beam with variable sections, therefore we can derive the displacement of center of the two microbeams using transfer matrix method [17]. First we divide it into 4 sections. The transfer

Where *A1*, *A2*, and *B* are transfer matrix between left end and right of section 1, left end and right of section 2, right end of section 2 and left end of section 3, which can all be calculated

After applying boundary condition, *y1L*=0, *θ1L*=0, *y4R*=0, *θ4R*=0 we can calculate force and moment applied on the left end of section 1 and right end of section 4. Then substituting the

We can calculate the transfer matrix from the left end of section 1 to the right end of section

*<sup>y</sup> h EI EIl l GI*

1 ( )( ) <sup>1</sup> (1 ) <sup>2</sup> <sup>4</sup> 1 /2 *<sup>p</sup>*

1 2

*C A A BA A* 12 21 (12)

*C AA* 1 21 (13)

*M l GI* (10)

(11)

and short arm, (b) shows the forces and moments of the 2 microbeams.

We can obtain the displacement of the center of the two long arms:

by law of transfer with cross section state vector.

force and moment to

2. Therefore according to

2

matrix from the left end of section 1 to the right end of section 4 can be obtained

*wb a b aV l*

According to the boundary condition

**Figure 6.** Results of the two theoretical models and FEA results, the discrepancy between theoretical modeling and FEA is small

After deriving the theoretical models, we carried FEA modeling using Intellisuite. Then the FEA result is validated the theoretical modeling by comparing them with FEA. As shown in figure 5, both the two theoretical models are in good agreement with FEA, and this validates the results of the theoretical modeling and FEA. Then we combined the theoretical modeling and FEA to optimize the structure.

## **3. Structure optimization**

In this section, we discuss the optimization of the structure, including the optimization of the microbeams, the long arm, the end of the long arm, the connection between the four single out-of-plane actuators, and the upper electrode.

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**Figure 8.** Theoretical and simulation data of length of long arm versus displacement, as the length of

Then the influence of the length of the long arm is discussed. As it is well known, the longer the long arm is, the larger out-of-plane displacement will be. The simulation and theoretical displacement are in good agreement with what is expected, as shown in figure 8. However,

After deriving characteristics of long arm, we went on with the microbeams. In this section, the influence of the width and length of microbeams is discussed. First, we come to the width of microbeams. It's easy to see that as the width of microbeam becomes larger, so does the torsional stiffness, which will thwart the rotation of the microbeams and make the

Second, the influence of the length of microbeams is discussed. Figure 9 is the simulation data of out-of-plane displacement versus microbeam length. As the microbeam length increases, so does the out-of-plane displacement, however, the increase rate slows down. This is because when the length of microbeam is small, the torsional stiffness is relatively larger, as mentioned above, thwarting the microbeams from rotating. In contrast, as the length increases, the downward bending displacement of microbeam increases, as shown in

as mentioned above, a too long arm is very fragile and is more likely to be broken.

the long arm increases, the out-of-plane displacement increases

out-of-plane displacement smaller. This is verified by FEA.

*3.1.2. Length of the long arm*

**3.2 .The microbeams**

#### **3.1. The long arm**

#### *3.1.1. Width of the long arm*

When studying the influence of the width of the long arm, we calculated the differential of equation (9) to *b2*

**Figure 7.** Theoretical and simulation data of width of long arm versus displacement, as the with of the long arm increases, the displacement decreases

It is obvious that the differential is constantly negative. Therefore the larger the width of the long arm, the smaller the displacement is when subjected to the same voltage. The simulation and theoretical data of the mirror plate displacement versus the width of long arm is shown in figure 7. Therefore as the long arm becomes wider, the out-of-plane displacement decreases. However, if the width of the long arm is very small, it tends to be more fragile and more likely to break during fabrication and test.

**Figure 8.** Theoretical and simulation data of length of long arm versus displacement, as the length of the long arm increases, the out-of-plane displacement increases

## *3.1.2. Length of the long arm*

170 Finite Element Analysis – New Trends and Developments

long arm increases, the displacement decreases

more fragile and more likely to break during fabrication and test.

single out-of-plane actuators, and the upper electrode.

In this section, we discuss the optimization of the structure, including the optimization of the microbeams, the long arm, the end of the long arm, the connection between the four

When studying the influence of the width of the long arm, we calculated the differential of

1 1 2

*EF l l a <sup>e</sup> z zI h b Ib EI l GJl*

2 12 11 2 1

**Figure 7.** Theoretical and simulation data of width of long arm versus displacement, as the with of the

It is obvious that the differential is constantly negative. Therefore the larger the width of the long arm, the smaller the displacement is when subjected to the same voltage. The simulation and theoretical data of the mirror plate displacement versus the width of long arm is shown in figure 7. Therefore as the long arm becomes wider, the out-of-plane displacement decreases. However, if the width of the long arm is very small, it tends to be

2 2 3

2 ( 2) 12

2

(16)

**3. Structure optimization**

**3.1. The long arm**

equation (9) to *b2*

*3.1.1. Width of the long arm*

Then the influence of the length of the long arm is discussed. As it is well known, the longer the long arm is, the larger out-of-plane displacement will be. The simulation and theoretical displacement are in good agreement with what is expected, as shown in figure 8. However, as mentioned above, a too long arm is very fragile and is more likely to be broken.

## **3.2 .The microbeams**

After deriving characteristics of long arm, we went on with the microbeams. In this section, the influence of the width and length of microbeams is discussed. First, we come to the width of microbeams. It's easy to see that as the width of microbeam becomes larger, so does the torsional stiffness, which will thwart the rotation of the microbeams and make the out-of-plane displacement smaller. This is verified by FEA.

Second, the influence of the length of microbeams is discussed. Figure 9 is the simulation data of out-of-plane displacement versus microbeam length. As the microbeam length increases, so does the out-of-plane displacement, however, the increase rate slows down. This is because when the length of microbeam is small, the torsional stiffness is relatively larger, as mentioned above, thwarting the microbeams from rotating. In contrast, as the length increases, the downward bending displacement of microbeam increases, as shown in

equation (9). As a result, the upward displacement is partly offset by downward bending of microbeams.

**Figure 9.** Simulation data of length of microbeams versus displacement, displacement increases with the increase of the length of microbeams, and the rate of increment slows down gradually.

There is one thing to notice when the microbeams are too short. When applied a relatively high voltage, the microbeams may fracture due to torsional stress. According to mechanics of materials, the maximum torsional stress can be calculated as follows

$$
\sigma\_{\text{max}} = \frac{T}{\alpha b\_1 h^2} \tag{17}
$$

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α=0.231, G=65 GPa. According to ref. [18], using these two equations above, we calculate that τmax=60.2 MPa and the portion of stress-strain curve is about 0.00076. We set G=79 GPa [19] and α=000.299, other parameters are based on reference [19]. Then according to reference [19], the fracture stress of polysilicon is two to ten times smaller than single crystal silicon, therefore at this time, when microbeam is shorter than 22.9 μm, it will fracture when

According to theoretical modeling and FEA, we found that the end of the long arm greatly

First, we presented a long arm with variable sections. The structure of the long arm is shown in figure 10, which consists of two different sections, one has a width of 3 μm and the other 20 μm. The total length of the two sections is set to be 190 μm. When the length of the thin 3 μm width section changes, the theoretical calculation and simulation of the thin long arm length versus the displacement is plotted in figure 11, the theoretical data is calculated using equation (A. 4) (appendix) in reference [16]. We can see that the displacement of the mirror plate increases remarkably when the length of thin long arm varies from 0 μm to 40 μm, this is attributed to the fact that the implementation of a thin long arm at the end of long arm makes the confinement of opposite levers to decrease, therefore, it makes the long arm easier to rotate. In contrast, the displacement changes little, just from 0.5 μm to 0.54 μm, when the length of thin long arm varies from 40 μm to 120 μm. This is because when the length of thin long arm increases, the confinement of the end of long arms decreases, it makes the out-of-plane displacement to increase. However, when the thin long arm become longer, the bending of the long arm increases and this makes the out-of-plane displacement decrease. The increased displacement, which is caused by a decreased confinement, is pulled back by the decrease displacement caused by the increased bending for longer thin arms. When the thin long arm is longer than 120 μm, the out-of-plane displacement demonstrates a remarkable decrease, as this thin long arm makes too much bending. Since it doesn't change much from 40 μm to 120μm, we set the thin long arm 44 μm, for the reason that according to design rules, the shorter the thin long arm, the more robust it is in fabrication. This is the first structure after optimization

The second optimization of the structure was to add a crab-leg beam, as illustrated in figure 12. By this means, the structure is more compact while at the same time achieving a larger displacement. We can see that both the displacement and amplification factor increase as the length of the crab-leg beam becomes longer. As a result, we set the length of the crab-leg beam to be 66 μm according to design rules, and we obtained the second optimized structure (Structure 2). The third optimization was to add a gimbal-like serpentine beam, as demonstrated in figure 13 [11]. This further reduces the confinement of the end of the long

confines the displacement. As a result, we made three optimizations to the structure.

applied a high voltage.

(Structure 1).

arm, thus achieving a larger displacement.

**3.3. The end of the long arm**

where *τmax*, *α*, *T*, *b1* and *h* represent the maximum torsional stress, coefficient related to b1/h, torque of the microbeams, width and height of microbeams. Also we know that

$$\theta = \frac{\text{Tl}\_1}{\text{GJ}} \tag{18}$$

where *θ* and *T* are rotational angle and torque, respectively.

Assuming the end of short beam has a displacement of 0.5 μm, the rotation angle is about 0.0125 radian, then it can be calculated that *τmax*=45.97 MPa. Here the parameters we use are α=0.231, G=65 GPa. According to ref. [18], using these two equations above, we calculate that τmax=60.2 MPa and the portion of stress-strain curve is about 0.00076. We set G=79 GPa [19] and α=000.299, other parameters are based on reference [19]. Then according to reference [19], the fracture stress of polysilicon is two to ten times smaller than single crystal silicon, therefore at this time, when microbeam is shorter than 22.9 μm, it will fracture when applied a high voltage.

## **3.3. The end of the long arm**

172 Finite Element Analysis – New Trends and Developments

microbeams.

equation (9). As a result, the upward displacement is partly offset by downward bending of

**Figure 9.** Simulation data of length of microbeams versus displacement, displacement increases with

There is one thing to notice when the microbeams are too short. When applied a relatively high voltage, the microbeams may fracture due to torsional stress. According to mechanics

> max 2 1 *T b h*

where *τmax*, *α*, *T*, *b1* and *h* represent the maximum torsional stress, coefficient related to b1/h,

*Tl*<sup>1</sup> *GJ*

Assuming the end of short beam has a displacement of 0.5 μm, the rotation angle is about 0.0125 radian, then it can be calculated that *τmax*=45.97 MPa. Here the parameters we use are

(17)

(18)

the increase of the length of microbeams, and the rate of increment slows down gradually.

torque of the microbeams, width and height of microbeams. Also we know that

of materials, the maximum torsional stress can be calculated as follows

where *θ* and *T* are rotational angle and torque, respectively.

According to theoretical modeling and FEA, we found that the end of the long arm greatly confines the displacement. As a result, we made three optimizations to the structure.

First, we presented a long arm with variable sections. The structure of the long arm is shown in figure 10, which consists of two different sections, one has a width of 3 μm and the other 20 μm. The total length of the two sections is set to be 190 μm. When the length of the thin 3 μm width section changes, the theoretical calculation and simulation of the thin long arm length versus the displacement is plotted in figure 11, the theoretical data is calculated using equation (A. 4) (appendix) in reference [16]. We can see that the displacement of the mirror plate increases remarkably when the length of thin long arm varies from 0 μm to 40 μm, this is attributed to the fact that the implementation of a thin long arm at the end of long arm makes the confinement of opposite levers to decrease, therefore, it makes the long arm easier to rotate. In contrast, the displacement changes little, just from 0.5 μm to 0.54 μm, when the length of thin long arm varies from 40 μm to 120 μm. This is because when the length of thin long arm increases, the confinement of the end of long arms decreases, it makes the out-of-plane displacement to increase. However, when the thin long arm become longer, the bending of the long arm increases and this makes the out-of-plane displacement decrease. The increased displacement, which is caused by a decreased confinement, is pulled back by the decrease displacement caused by the increased bending for longer thin arms. When the thin long arm is longer than 120 μm, the out-of-plane displacement demonstrates a remarkable decrease, as this thin long arm makes too much bending. Since it doesn't change much from 40 μm to 120μm, we set the thin long arm 44 μm, for the reason that according to design rules, the shorter the thin long arm, the more robust it is in fabrication. This is the first structure after optimization (Structure 1).

The second optimization of the structure was to add a crab-leg beam, as illustrated in figure 12. By this means, the structure is more compact while at the same time achieving a larger displacement. We can see that both the displacement and amplification factor increase as the length of the crab-leg beam becomes longer. As a result, we set the length of the crab-leg beam to be 66 μm according to design rules, and we obtained the second optimized structure (Structure 2). The third optimization was to add a gimbal-like serpentine beam, as demonstrated in figure 13 [11]. This further reduces the confinement of the end of the long arm, thus achieving a larger displacement.

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**3.4. The connection between the four single out-of-plane actuators**

**Figure 12.** Displacement and Magnification factor of the second optimized structure

Then we made an optimization to the upper electrode by introducing the third layer polysilicon to the upper electrode, thus enlarging the gap from 2 μm to 2.75 μm, as illustrated

in figure 13, by this approach, we obtained the third optimized structure (Structure 3).

**3.5. The upper electrode**

first mode, which had a resonant frequency of 4.8 kHz.

After optimizing the end of the long arm, we investigated the connection of four actuators. It is found that the first mode of the natural frequency of the structure 3 was the rotation along the dotted line l1 and l2 in figure 13, other than the out-of-plane movement. This was because there was only one connection in the structure, and the restriction to this mode was smaller than the mode of out-of-plane movement. This was not desirable for inducing mechanical instability when working at a high frequency. We made an optimization to the structure, by making four separate connections to connect the four actuators to the mirror, as depicted in figure 13. Through this method, we can enlarge the restriction of the first mode, thus making out-of-plane mode to be the first mode. After this optimization, the piston mode became the

**Figure 10.** Schematic of the first optimized structure: (a) Lateral view of the structure in optimization, (b) Top view of a single actuator in optimization

**Figure 11.** Theoretical and simulation data of length of thin long arm versus displacement, as the length of the thin long arm increases, the displacement firstly increase, then after hitting a maximum, it decreases as length of thin long arm increase

## **3.4. The connection between the four single out-of-plane actuators**

After optimizing the end of the long arm, we investigated the connection of four actuators. It is found that the first mode of the natural frequency of the structure 3 was the rotation along the dotted line l1 and l2 in figure 13, other than the out-of-plane movement. This was because there was only one connection in the structure, and the restriction to this mode was smaller than the mode of out-of-plane movement. This was not desirable for inducing mechanical instability when working at a high frequency. We made an optimization to the structure, by making four separate connections to connect the four actuators to the mirror, as depicted in figure 13. Through this method, we can enlarge the restriction of the first mode, thus making out-of-plane mode to be the first mode. After this optimization, the piston mode became the first mode, which had a resonant frequency of 4.8 kHz.

**Figure 12.** Displacement and Magnification factor of the second optimized structure

### **3.5. The upper electrode**

174 Finite Element Analysis – New Trends and Developments

(b) Top view of a single actuator in optimization

decreases as length of thin long arm increase

**Figure 10.** Schematic of the first optimized structure: (a) Lateral view of the structure in optimization,

(a) (b)

**Figure 11.** Theoretical and simulation data of length of thin long arm versus displacement, as the length of the thin long arm increases, the displacement firstly increase, then after hitting a maximum, it

Then we made an optimization to the upper electrode by introducing the third layer polysilicon to the upper electrode, thus enlarging the gap from 2 μm to 2.75 μm, as illustrated in figure 13, by this approach, we obtained the third optimized structure (Structure 3).

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The three optimized structures were fabricated by a three-layer polysilicon surface microfabrication process. First of all, 600 nm low-stress silicon nitride is deposited on an ntype (100) wafer with a diameter of 150 mm to form electrical isolation layer. Then 500nm polysilicon film is deposited as the first polysilicon layer (Poly0). Afterward, Poly0 is patterned by photolithography and etched. Then 2 μm of phosphosilicate glass (PSG) is deposited as sacrificial layer. Then the first silicon dioxide layer (Oxide1) is patterned by lithography and etched to form dimples. The following step is to deposit a 2 μm polysilicon layer (Poly1) as the second polysilicon layer, which is etched afterwards to form the leverage mechanism. At last the Oxide1 layer is sacrificed in a bath of 49% HF to release the structural layer and the structure is dried by supercritical CO2 drying technique. Figure 16 shows the SEM photograph of the structures. We can see the shapes of the structures are good with little curvatures, indicating that the stress gradient and stress variation along the

**Figure 15.** Fabrication process of the μSLM, this is a three-layer surface microfabrication process, and

Then we use an optical interferometer to measure the displacement versus voltage for the three optimized structures. The test is performed using the Zygo Newview 7300 (Zygo Inc., CT, USA) A light source, in this case an incoherent broadband LED light source is split at the objective so that some of the light passes to a reference mirror and some is focused onto the surface of the sample under measurement. Light from the mirror (embedded into the interference lens) and the sample surface is reflected back into the instrument and imaged onto a camera. If the distances from the light splitter to the mirror and from the splitter to the surface are equal so that there is no optical path difference (OPD) then the camera will observe an interference pattern. This occurs when the objective is held so that the focal plane of the objective lies in the same plane as the surface. In order to perform a measurement of

this schematic is based on the fabrication of the first optimized structure

**4. Fabrication and test**

beams is negligible.

**Figure 13.** Lateral view of the third optimized μSLM

#### **3.6. Results after optimization**

The results after optimization is shown in figure 14. From this figure we can see that after optimization, the maximum displacement are 1.58 μm , 1.87 μm, and 4.5 μm, which are 3.04, 3.6, 8.65 times, respectively, higher than the structure before optimization. After optimization, we did experiment on the fabrication and test of the structures.

**Figure 14.** Simulation results before and after optimization, after optimization, the maximum displacement are 1.58 μm , 1.87 μm, and 4.5 μm, which are 3.04, 3.6, 8.65 times higher than the structure before optimization

## **4. Fabrication and test**

176 Finite Element Analysis – New Trends and Developments

**Figure 13.** Lateral view of the third optimized μSLM

The results after optimization is shown in figure 14. From this figure we can see that after optimization, the maximum displacement are 1.58 μm , 1.87 μm, and 4.5 μm, which are 3.04, 3.6, 8.65 times, respectively, higher than the structure before optimization. After

> **Before Optimization Optimized Structure 1 Optimized Structure 2 Optimized Structure 3**

optimization, we did experiment on the fabrication and test of the structures.

**Figure 14.** Simulation results before and after optimization, after optimization, the maximum

displacement are 1.58 μm , 1.87 μm, and 4.5 μm, which are 3.04, 3.6, 8.65 times higher than the structure

0 10 20 30 40 50

**Voltage (V)**

**3.6. Results after optimization**

before optimization

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

**Displacement (**

**μm)**

The three optimized structures were fabricated by a three-layer polysilicon surface microfabrication process. First of all, 600 nm low-stress silicon nitride is deposited on an ntype (100) wafer with a diameter of 150 mm to form electrical isolation layer. Then 500nm polysilicon film is deposited as the first polysilicon layer (Poly0). Afterward, Poly0 is patterned by photolithography and etched. Then 2 μm of phosphosilicate glass (PSG) is deposited as sacrificial layer. Then the first silicon dioxide layer (Oxide1) is patterned by lithography and etched to form dimples. The following step is to deposit a 2 μm polysilicon layer (Poly1) as the second polysilicon layer, which is etched afterwards to form the leverage mechanism. At last the Oxide1 layer is sacrificed in a bath of 49% HF to release the structural layer and the structure is dried by supercritical CO2 drying technique. Figure 16 shows the SEM photograph of the structures. We can see the shapes of the structures are good with little curvatures, indicating that the stress gradient and stress variation along the beams is negligible.

**Figure 15.** Fabrication process of the μSLM, this is a three-layer surface microfabrication process, and this schematic is based on the fabrication of the first optimized structure

Then we use an optical interferometer to measure the displacement versus voltage for the three optimized structures. The test is performed using the Zygo Newview 7300 (Zygo Inc., CT, USA) A light source, in this case an incoherent broadband LED light source is split at the objective so that some of the light passes to a reference mirror and some is focused onto the surface of the sample under measurement. Light from the mirror (embedded into the interference lens) and the sample surface is reflected back into the instrument and imaged onto a camera. If the distances from the light splitter to the mirror and from the splitter to the surface are equal so that there is no optical path difference (OPD) then the camera will observe an interference pattern. This occurs when the objective is held so that the focal plane of the objective lies in the same plane as the surface. In order to perform a measurement of

the surface observed by the field of view of the objective, the objective lens is translated vertically and linearly so that the focal plane moves through the entire height range of the surface being measured. As it does so, the interference fringes will move and follow the height profile of the surface and this information is processed by the instrument to calculate the height profile to a very high precision (0.1 nm).

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**Figure 18.** Optical profiles for Structure 1 before and after deformation: (a) Optical profile before applying voltage, we can see the profile is approximately a plane (b) Optical profile when applied a voltage of 47 V, we can see the short arm goes down and the long arm and central mass goes up.

**Figure 19.** Optical profiles for Structure 2 before and after deformation: (a) Optical profile before applying voltage, we can see the profile is approximately a plane (b) Optical profile when applied a voltage of 24.75 V, we can see the short arm goes down and the long arm and central mass goes up.

**Figure 16.** SEM photographs of the three optimized structures

**Figure 17.** Optical configuration of the white light interferometer

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178 Finite Element Analysis – New Trends and Developments

the height profile to a very high precision (0.1 nm).

**Figure 16.** SEM photographs of the three optimized structures

**Figure 17.** Optical configuration of the white light interferometer

the surface observed by the field of view of the objective, the objective lens is translated vertically and linearly so that the focal plane moves through the entire height range of the surface being measured. As it does so, the interference fringes will move and follow the height profile of the surface and this information is processed by the instrument to calculate

> **Figure 18.** Optical profiles for Structure 1 before and after deformation: (a) Optical profile before applying voltage, we can see the profile is approximately a plane (b) Optical profile when applied a voltage of 47 V, we can see the short arm goes down and the long arm and central mass goes up.

**Figure 19.** Optical profiles for Structure 2 before and after deformation: (a) Optical profile before applying voltage, we can see the profile is approximately a plane (b) Optical profile when applied a voltage of 24.75 V, we can see the short arm goes down and the long arm and central mass goes up.

The optical profile of Structure 1 before and after applying voltage is shown in figure 18. Before applying voltage, the structure profile is approximately a plane. Then after applying a voltage of 47 V, due to the electrostatic force, the upper electrode (short arm) goes down and the long arm and central mass goes up. Similar result is observed for Structure 2, as shown in figure 19. For the Structure 3, after applying voltage, the mirror plate did not go up, instead, it went down. After analysis, we believe there may be problem for the isolation of the mirror plate and the substrate. We will fix this problem in the future.

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*State Key Lab of Optical Technologies for Microfabrication, Institute of Optics and Electronics,*

*State Key Lab of Optical Technologies for Microfabrication, Institute of Optics and Electronics,*

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[11] S. Moaveni. *Finite element analysis, Theory and Application with ANSYS*. Prentice Hall,

[13] H. Ren, Z.G. Ni, J.M. Chen, A.L. Gong and J. Yao, A micro spatial light modulator based

[14] R.K. Tyson, *Principles of Adaptive Optics* (2nd ed.), Academic Press, New York 1998 [15] N. Doble, M. Helmbrecht, M. Hart and T. Juneau, Advanced wavefront correction technology for the next generation of adaptive optics equipped ophthalmic

capacitive accelerometer. *J. Microelectromech. Syst.* 14 (2):<PHONE_NUMBER>

[9] J.M. Younse, Mirrors on a chip, *IEEE Spectrum*, 30:27-31 1993

on leverage principle, *Key Engineering Materials* 483:137-142 2011

*School of Electrical, Computer, and Energy Engineering, Arizona State University,*

**Author details**

*Tempe, AZ, USA*

**6. References**

1992

2005

*Devices* 17 14-25 2001

86(8): 1640-1659 1998,

*Series*, 86:1687-1704 1988

[12] http://www.intellisense.com/

Upper Saddle River, New Jersey 1999

instrumentation. *Proc. SPIE* 5688<PHONE_NUMBER>

*Chinese Academy of Sciences, Chengdu, China*

*Chinese Academy of Sciences, Chengdu, China*

Hao Ren

Jun Yao

The displacements versus voltage for the first two optimized structures are shown in figure 20. We can see that the two optimized structures can obtain a stroke of 1.45μm, and 2.21μm, which are more than two times, and three times larger, respectively, than the stroke before optimization. Through this example, we can clearly see the importance of FEA in MEMS research: it saves time and money, while at the same time can handle complex/nonlinear structures.

**Figure 20.** Experimental results of the first two structures after optimization in comparison with the structure before optimization, the maximum displacement is more than two and three times larger than the maximum displacement before optimization.

## **5. Conclusion**

In this chapter we mainly discussed the significance of FEA in MEMS research through an example of a micromachined spatial light modulator (μSLM). We have used FEA to model the μSLM structure, verify theoretical models, and perform optimizations. After fabrication, we found that the stroke after optimization was more than 3 times larger than the stroke before optimization. As is demonstrated, FEA makes MEMS research to be time and cost efficient and thus has been widely applied in MEMS research.

## **Author details**

## Hao Ren

180 Finite Element Analysis – New Trends and Developments

the maximum displacement before optimization.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

**Displacement (**

**μm)**

efficient and thus has been widely applied in MEMS research.

**5. Conclusion**

structures.

The optical profile of Structure 1 before and after applying voltage is shown in figure 18. Before applying voltage, the structure profile is approximately a plane. Then after applying a voltage of 47 V, due to the electrostatic force, the upper electrode (short arm) goes down and the long arm and central mass goes up. Similar result is observed for Structure 2, as shown in figure 19. For the Structure 3, after applying voltage, the mirror plate did not go up, instead, it went down. After analysis, we believe there may be problem for the isolation

The displacements versus voltage for the first two optimized structures are shown in figure 20. We can see that the two optimized structures can obtain a stroke of 1.45μm, and 2.21μm, which are more than two times, and three times larger, respectively, than the stroke before optimization. Through this example, we can clearly see the importance of FEA in MEMS research: it saves time and money, while at the same time can handle complex/nonlinear

> **Before Optimization Optimized Structure 1**

**Optimized Structure 2 Pull-in**

**Pull-in**

**Pull-in**

**Figure 20.** Experimental results of the first two structures after optimization in comparison with the structure before optimization, the maximum displacement is more than two and three times larger than

0 10 20 30 40 50

**Voltage (V)**

In this chapter we mainly discussed the significance of FEA in MEMS research through an example of a micromachined spatial light modulator (μSLM). We have used FEA to model the μSLM structure, verify theoretical models, and perform optimizations. After fabrication, we found that the stroke after optimization was more than 3 times larger than the stroke before optimization. As is demonstrated, FEA makes MEMS research to be time and cost

of the mirror plate and the substrate. We will fix this problem in the future.

*State Key Lab of Optical Technologies for Microfabrication, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, China*

*School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ, USA*

## Jun Yao

*State Key Lab of Optical Technologies for Microfabrication, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, China*

## **6. References**

	- [16] H. Ren, F.G. Tao, W.M. Wang and J. Yao, An out-of-plane electrostatic actuator based on the lever principle, *J. Micromech. Microeng.* 21<PHONE_NUMBER>

**Chapter 9**

© 2012 Kurihara, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Kurihara,, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Steady-State and Transient Performance**

Recently, global warming has become an important problem. High-efficiency machines have been needed in a large variety of industrial products in order to save electrical energy. For many applications, permanent-magnet (PM) synchronous machines can be designed which is smaller in size but more efficient as compared to induction machines [1-3]. Besides, PMs have been employed as an alternative to current carrying coils for magnetic field excitation in synchronous machines for over 50 years. The lack of slip rings, brushes and field winding losses have always been viewed as distinct advantages over that of conventional wound field machines. However, when the machine size becomes small, the efficiency becomes low. This is mainly due to the reason that the iron loss and the copper loss are large, because the iron core of the stator in the small machine generally does not have annealing and the resistance of the stator windings is

This chapter presents a successful design of the high-efficiency small but novel Interior permanent-magnet (IPM) machines using Neodymium-Boron-Iron (NdBFe) magnets. It is designed to operate with both high-efficiency line-start IPM motors [3] and generators with damper bars [4]. Time-stepping finite element analysis has been used to successfully predict the dynamic and transient performances of the prototype machines. Time-stepping finite element analysis [3-6] has been used to successfully predict the dynamic and transient performances of the prototype IPM machines. The computed performance has been

**Analysis of Permanent-Magnet**

**Machines Using Time-Stepping**

**Finite Element Technique**

Additional information is available at the end of the chapter

Kazumi Kurihara

**1. Introduction**

comparatively large.

validated by tests in the prototype machine.

http://dx.doi.org/10.5772/48426

## **Steady-State and Transient Performance Analysis of Permanent-Magnet Machines Using Time-Stepping Finite Element Technique**

## Kazumi Kurihara

182 Finite Element Analysis – New Trends and Developments

Engineering Division) 1984

[16] H. Ren, F.G. Tao, W.M. Wang and J. Yao, An out-of-plane electrostatic actuator based

[19] W. Alexander and S. Harald. Torsional stress, fatigue and fracture strength in silicon

[17] S. Timoshenko and J. N. Goodier, *Theory of Elasticity,* McGraw-Hill, New York, 1951. [18] J. M. Gere and S. P. Timoshenko *Mechanics of materials* 2nd edition (Brooks/Cole

on the lever principle, *J. Micromech. Microeng.* 21<PHONE_NUMBER>

hinges of a micro scanning mirror *Proc. of SPIE* 5342:176-185 2004

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48426

## **1. Introduction**

Recently, global warming has become an important problem. High-efficiency machines have been needed in a large variety of industrial products in order to save electrical energy. For many applications, permanent-magnet (PM) synchronous machines can be designed which is smaller in size but more efficient as compared to induction machines [1-3]. Besides, PMs have been employed as an alternative to current carrying coils for magnetic field excitation in synchronous machines for over 50 years. The lack of slip rings, brushes and field winding losses have always been viewed as distinct advantages over that of conventional wound field machines. However, when the machine size becomes small, the efficiency becomes low. This is mainly due to the reason that the iron loss and the copper loss are large, because the iron core of the stator in the small machine generally does not have annealing and the resistance of the stator windings is comparatively large.

This chapter presents a successful design of the high-efficiency small but novel Interior permanent-magnet (IPM) machines using Neodymium-Boron-Iron (NdBFe) magnets. It is designed to operate with both high-efficiency line-start IPM motors [3] and generators with damper bars [4]. Time-stepping finite element analysis has been used to successfully predict the dynamic and transient performances of the prototype machines. Time-stepping finite element analysis [3-6] has been used to successfully predict the dynamic and transient performances of the prototype IPM machines. The computed performance has been validated by tests in the prototype machine.

© 2012 Kurihara, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Kurihara,, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **2. IPM machine configuration**

The photograph of an IPM rotor, the cross section of a quarter of the high-efficiency motor and the demagnetization curve of the NdBFe magnet used for finite-element analysis are shown respectively in Figures 1-3, respectively [3].

Steady-State and Transient Performance Analysis of

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 185

Furthermore, the number and configuration of rotor slots have been successfully designed by using the finite-element method so that the waveform of the electromotive force (EMF)

The analysis for taking the eddy currents into account, in general becomes essential to solve the three-dimensional problem. In this paper, it is assumed that the eddy currents flow approximately in the axial direction, because the rotor shown in Figure 1 is equipped with end rings. This reduces the analysis to a two-dimensional problem. The fundamental equations for

*A A JJJ*

*y x*

*x y*

0 *e m*

in the PM is assumed the same as the reluctivity of free

is the conductivity, and

(1)

(2)

(3)

the magnetic field are represented in the two-dimensional rectangular co-ordinates as

*e <sup>A</sup> <sup>J</sup> <sup>t</sup>*

0

*m*

where *A* is the *z* component of magnetic vector potential *A*, *J*0 is the stator-winding current density, *Je* is the eddy current density, *Jm* is the equivalent magnetizing current density, *Mx*,

The effect of the eddy current for the rotor ends is taken into account by multiplying by the coefficient *kc* as described below. It is done to reduce the analysis to two-dimensional. The

*<sup>M</sup> <sup>M</sup> <sup>J</sup>*

*xx yy*

*My* are *x* and *y* components of the magnetization *M*, respectively.

0. *Jm* is assumed zero, outside the PM.

due to the PMs was close to the sine waveform and the cogging torque was low.

**Figure 3.** Demagnetization curve of NdBFe magnet

**3. Method for analysis**

is the reluctivity. The value of

space

A frame size of a 600 W, 3-phase, 4-pole, Y-connected, 50 Hz, 200 V squirrel- cage induction machine was used for testing the IPM rotor shown in Figure 1. The four-pole magnets arrangement in the rotor is oriented for a high-field type IPM synchronous machine. The experimentally developed rotor has the following distinctive design features [3]:

**Figure 2.** Configuration of high-efficiency IPM machine

Furthermore, the number and configuration of rotor slots have been successfully designed by using the finite-element method so that the waveform of the electromotive force (EMF) due to the PMs was close to the sine waveform and the cogging torque was low.

**Figure 3.** Demagnetization curve of NdBFe magnet

## **3. Method for analysis**

184 Finite Element Analysis – New Trends and Developments

shown respectively in Figures 1-3, respectively [3].

**Figure 2.** Configuration of high-efficiency IPM machine

The photograph of an IPM rotor, the cross section of a quarter of the high-efficiency motor and the demagnetization curve of the NdBFe magnet used for finite-element analysis are

A frame size of a 600 W, 3-phase, 4-pole, Y-connected, 50 Hz, 200 V squirrel- cage induction machine was used for testing the IPM rotor shown in Figure 1. The four-pole magnets arrangement in the rotor is oriented for a high-field type IPM synchronous machine. The

1. The fluxes from both sides of the magnet are concentrated effectively in the middle of

2. The reluctance of the d axis is larger than that of the q axis, because the d- axis flux passes across the magnet with high reluctance. Large reluctance torque can be obtained. 3. The conducting material between the magnet and the rotor core is made from

experimentally developed rotor has the following distinctive design features [3]:

aluminum and has both functions of the flux barrier and cage bar.

**2. IPM machine configuration**

the magnetic poles of the rotor.

**Figure 1.** IPM rotor

The analysis for taking the eddy currents into account, in general becomes essential to solve the three-dimensional problem. In this paper, it is assumed that the eddy currents flow approximately in the axial direction, because the rotor shown in Figure 1 is equipped with end rings. This reduces the analysis to a two-dimensional problem. The fundamental equations for the magnetic field are represented in the two-dimensional rectangular co-ordinates as

$$\frac{\partial}{\partial \mathbf{x}} \left( \nu \frac{\partial A}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( \nu \frac{\partial A}{\partial y} \right) = -J\_0 - J\_e - J\_m \tag{1}$$

$$J\_e = -\sigma \frac{\partial A}{\partial t} \tag{2}$$

$$J\_m = \nu\_0 \left(\frac{\partial \mathcal{M}\_y}{\partial \mathbf{x}} - \frac{\partial \mathcal{M}\_x}{\partial y}\right) \tag{3}$$

where *A* is the *z* component of magnetic vector potential *A*, *J*0 is the stator-winding current density, *Je* is the eddy current density, *Jm* is the equivalent magnetizing current density, *Mx*, *My* are *x* and *y* components of the magnetization *M*, respectively. is the conductivity, and is the reluctivity. The value of in the PM is assumed the same as the reluctivity of free space 0. *Jm* is assumed zero, outside the PM.

The effect of the eddy current for the rotor ends is taken into account by multiplying by the coefficient *kc* as described below. It is done to reduce the analysis to two-dimensional. The equivalent resistance *R*2 for rotor bars including the rotor end rings can be given below if the bars are distributed at equal intervals in the rotor [7].

$$R\_2 = R\_b + R\_e \frac{Z\_2}{\left(2p\pi\right)^2} \tag{4}$$

Steady-State and Transient Performance Analysis of

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 187

0. *abc vvv* (11)

(12)

(13)

(14)

(15)

(16)

*<sup>r</sup>* is the rotor

*t* is the time step. *eb*, and *ec* can be obtained similarly, as in [5].

where *At*

is *A* at time *t*.

**Figure 4.** Circuit of three-phase line-start IPM synchronous motor

*vn* can be obtained by adding each side of (6)-(8) and then applying (9) and (11)

*n*

*v*

One obtains the following equation by substituting (12) in (6)-(8) [5]:

3 3

3 3

3 3

The dynamic equation is given as [3]

calculated by using the *Bil* rule [8]. The angular speed,

. <sup>3</sup> *abc*

*eee*

1 1 <sup>2</sup> <sup>0</sup>

1 1 <sup>2</sup> <sup>0</sup>

1 1 <sup>2</sup> 0.

0

*t*

*t*

*t*

*r l*

*r* is given by

*b c a a aa ee i e ri L v*

*c a b b bb ee i e ri L v*

*a b c c cc ee i e ri L v*

*r*

angular speed, *B*0 is the friction coefficient, and *Tl* is the load torque. The torque *T* is

*d TJ B T dt*

where *T* is the instantaneous electromagnetic torque, *J* is the rotational inertia,

For operation from a balanced three-phase system,

where *Rb* is the resistance of a bar, *Re* is the resistance of the end rings, *Z*2 is the number of rotor slots and *p* is the pole pair number.

Therefore, *kc* is given by

$$k\_c = \frac{R\_b}{R\_2} \tag{5}$$

This coefficient *kc* is found effective to take into account the rotor-bar current for the fundamental space harmonic. Moreover, it has been found that the agreement between computed and measured results of the starting performance characteristics in the IPM motor is good [3]. Therefore, it is considered that design use of the *kc* is acceptable, even if the higher space harmonics exists [5]. The value of coefficient *kc* is 0.55 in this paper.

## **3.1. Voltage, current and dynamic equations and calculation steps for IPM synchronous motor**

Figure 4 shows the circuit of the three-phase line-start IPM synchronous motor. It has three stator phase windings, which are star connected with neutral. The voltage and current equations of the IPM motor are given as

$$\left(e\_a + r\_1 i\_a + L\_1 \frac{\partial i\_a}{\partial t} = \upsilon\_a - \upsilon\_n\right) \tag{6}$$

$$\left(e\_b + r\_1 i\_b + L\_1 \frac{\partial i\_b}{\partial t} = \upsilon\_b - \upsilon\_n\right) \tag{7}$$

$$\mathbf{v}\_c + r\_1 \dot{\mathbf{i}}\_c + L\_1 \frac{\partial \dot{\mathbf{i}}\_c}{\partial t} = \mathbf{v}\_c - \mathbf{v}\_n \tag{8}$$

$$
\dot{i}\_a + \dot{i}\_b + \dot{i}\_c = 0 \tag{9}
$$

where *va*, *vb*, and *vc* are the phase voltages, subscripts *a*, *b*, and *c* represent stator quantities in lines *a*, *b*, and *c*, respectively. *vn* is the potential of the neutral *n*, when the potential of the neutral of the supply source is zero, *ia*, *ib*, and *ic* are the line currents, *r*1 and *L*1 are the resistance and end-winding leakage inductance of the stator winding per phase, respectively. *ea*, *eb*, and *ec* are the induced phase voltages; and *ea* is given by the line integral of the vector potential round *ca* which is along the stator windings of phase *a* [5]

$$\oint e\_a = \oint\_{c\_s} \frac{\partial A^t}{\partial t} ds = \oint\_{c\_s} \frac{A^t - A^{t - \Delta t}}{\Delta t} ds \tag{10}$$

where *At* is *A* at time *t*. *t* is the time step. *eb*, and *ec* can be obtained similarly, as in [5].

**Figure 4.** Circuit of three-phase line-start IPM synchronous motor

For operation from a balanced three-phase system,

186 Finite Element Analysis – New Trends and Developments

rotor slots and *p* is the pole pair number.

equations of the IPM motor are given as

Therefore, *kc* is given by

**synchronous motor**

bars are distributed at equal intervals in the rotor [7].

equivalent resistance *R*2 for rotor bars including the rotor end rings can be given below if the

<sup>2</sup> <sup>2</sup> <sup>2</sup> *b e*

where *Rb* is the resistance of a bar, *Re* is the resistance of the end rings, *Z*2 is the number of

2 *b*

This coefficient *kc* is found effective to take into account the rotor-bar current for the fundamental space harmonic. Moreover, it has been found that the agreement between computed and measured results of the starting performance characteristics in the IPM motor is good [3]. Therefore, it is considered that design use of the *kc* is acceptable, even if the

Figure 4 shows the circuit of the three-phase line-start IPM synchronous motor. It has three stator phase windings, which are star connected with neutral. The voltage and current

> 1 1 *<sup>a</sup> a a an <sup>i</sup> e ri L v v t*

> 1 1 *<sup>c</sup> c c cn <sup>i</sup> e ri L v v t*

> > 0 *abc*

where *va*, *vb*, and *vc* are the phase voltages, subscripts *a*, *b*, and *c* represent stator quantities in lines *a*, *b*, and *c*, respectively. *vn* is the potential of the neutral *n*, when the potential of the neutral of the supply source is zero, *ia*, *ib*, and *ic* are the line currents, *r*1 and *L*1 are the resistance and end-winding leakage inductance of the stator winding per phase, respectively. *ea*, *eb*, and *ec* are the induced phase voltages; and *ea* is given by the line integral

*t t tt*

of the vector potential round *ca* which is along the stator windings of phase *a* [5]

*a c c*

*a a*

*A AA e ds ds t t*

*b b b bn <sup>i</sup> e ri L v v t*

1 1

*c <sup>R</sup> <sup>k</sup>*

higher space harmonics exists [5]. The value of coefficient *kc* is 0.55 in this paper.

**3.1. Voltage, current and dynamic equations and calculation steps for IPM**

*RRR*

 2

*p*

*Z*

(4)

*<sup>R</sup>* (5)

*iii* (9)

(10)

(6)

(7)

(8)

$$
\upsilon\_a + \upsilon\_b + \upsilon\_c = 0.\tag{11}
$$

*vn* can be obtained by adding each side of (6)-(8) and then applying (9) and (11)

$$
\sigma\_n = \frac{e\_a + e\_b + e\_c}{3}.\tag{12}
$$

One obtains the following equation by substituting (12) in (6)-(8) [5]:

$$\frac{\partial}{\partial \mathbf{3}} e\_a - \frac{e\_b + e\_c}{\mathbf{3}} + r\_1 \dot{\mathbf{i}}\_a + L\_1 \frac{\partial \dot{\mathbf{i}}\_a}{\partial t} - \upsilon\_a = 0 \tag{13}$$

$$\frac{2}{3}e\_b - \frac{e\_c + e\_a}{3} + r\_1 \dot{i}\_b + L\_1 \frac{\partial \dot{i}\_b}{\partial t} - \upsilon\_b = 0 \tag{14}$$

$$\frac{2}{3}e\_c - \frac{e\_a + e\_b}{3} + r\_1 \dot{i}\_c + L\_1 \frac{\partial \dot{i}\_c}{\partial t} - \upsilon\_c = 0. \tag{15}$$

The dynamic equation is given as [3]

$$T = J\frac{d\phi\_r}{dt} + B\_0\phi\_r + T\_l\tag{16}$$

where *T* is the instantaneous electromagnetic torque, *J* is the rotational inertia, *<sup>r</sup>* is the rotor angular speed, *B*0 is the friction coefficient, and *Tl* is the load torque. The torque *T* is calculated by using the *Bil* rule [8]. The angular speed, *r* is given by

$$
\rho \alpha\_r = \frac{d\theta}{dt} \tag{17}
$$

Steady-State and Transient Performance Analysis of

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 189

converges, the process returns to step 6).

can be calculated. Then, the

*<sup>t</sup>*is

2. The vector potential *A* at *t* = 0 is set, where the static field caused by only PMs is given

*<sup>t</sup>* are set. 6. The matrix equation constructed by the time-stepping finite element technique is solved

> , and *ic t* , *T t*

9. The calculation process from step 3) to step7) continues till the steady-state currents are

at new *t* is set.

, and *ic*

is tested. Unless *At*

, *ia t* , *ibt*

as the initial value. 3. At *t* = *t* + *t*, the value of

5. The initial values for *At*

7. The convergence of *At*

8. After the convergence of *At*

determined from (22).

[5].

obtained.

**generator**

*t*

, *ia t* , *ibt*

**Figure 5.** Flowchart of three-phase line-start IPM synchronous motor

current equations of the IPM generator are given as

**3.2. Voltage and current equations and calculation steps for IPM synchronous**

Figure 6 shows the circuit of the three-phase IPM synchronous generator. The voltage and

4. At *t* = *t* + *t*, each voltage at new *t* is set.

where is the rotational angle of the rotor.

One obtains the following equation by substituting (17) in (16):

$$T = J\frac{d^2\theta}{dt^2} + B\_0 \frac{d\theta}{dt} + T\_l. \tag{18}$$

In this paper, the forward difference method is used to obtain the rotational angle at time *t* because the vector potential, currents and rotational angle at time *t- t* are all known

$$\frac{d\theta^{t-\Delta t}}{dt} = \frac{\theta^t - \theta^{t-\Delta t}}{\Delta t} \tag{19}$$

$$\frac{d^2\theta^{t-\Delta t}}{dt^2} = \frac{\theta^t - 2\theta^{t-\Delta t} + \theta^{t-2\Delta t}}{\left(\Delta t\right)^2}.\tag{20}$$

One obtains the following equation by substituting (19) and (20) in (18) [6]:

$$\boldsymbol{\theta}^{t} = \frac{1}{\boldsymbol{I} + B\_0 \Delta t} \mathbf{I} (\boldsymbol{T}^{t-\Lambda t} - \boldsymbol{T}\_{\boldsymbol{l}}^{t-\Lambda t}) (\boldsymbol{\Lambda} \boldsymbol{t})^2 + (\boldsymbol{\mathcal{Q}} \boldsymbol{I} + B\_0 \boldsymbol{\Lambda} \boldsymbol{t}) \boldsymbol{\theta}^{t-\Lambda t} - \boldsymbol{J} \boldsymbol{\theta}^{t-2\Lambda t} \text{ [}. \tag{21}$$

In the case when the effect of the friction is negligibly small, the above equation can be represented simply as follows:

$$\boldsymbol{\Theta}^{t} = \frac{\left(\boldsymbol{\Delta t}\right)^{2}}{J} \left(\boldsymbol{T}^{t-\Delta t} - \boldsymbol{T}\_{l}^{t-\Delta t}\right) + 2\boldsymbol{\Theta}^{t-\Delta t} - \boldsymbol{\Theta}^{t-2\Delta t}.\tag{22}$$

One can obtain the vector potential, currents and rotational angle by solving (1), (13)-(15), and (18) using the time-stepping finite element technique [3].

Next, the calculation steps for this analysis are shown in Figure 5.

1. First, the terminal voltage *Vl*, its initial phase angle 0, *Tl*, and *t* are set, respectively. Each voltage for the three stator phase windings can be represented by

$$
\upsilon\_a{}^t = \sqrt{\frac{2}{3}} V\_l \cos \left( \alpha t + \phi\_0 \right) \tag{23}
$$

$$\left.v\_{b}\right|\_{b}^{t} = \sqrt{\frac{2}{3}}V\_{l}\cos\left(\alpha t + \phi\_{0} - \frac{2}{3}\pi\right) \tag{24}$$

$$
\psi\_c{}^t = \sqrt{\frac{2}{3}} V\_l \cos \left( \cot \phi + \phi\_0 - \frac{4}{3}\pi \right). \tag{25}
$$

is the rotational angle of the rotor.

0

represented simply as follows:

One obtains the following equation by substituting (17) in (16):

where  *r d dt*

(17)

(18)

(19)

(20)

(21)

(23)

2

because the vector potential, currents and rotational angle at time *t- t* are all known

One obtains the following equation by substituting (19) and (20) in (18) [6]:

Each voltage for the three stator phase windings can be represented by

*t*

*t*

*t*

2

3

*b l vVt*

*c l vVt*

*a l v Vt*

cos

2 2 cos 3 3

2 4 cos . 3 3

*T T*

2

*t*

*J*

Next, the calculation steps for this analysis are shown in Figure 5.

and (18) using the time-stepping finite element technique [3].

1. First, the terminal voltage *Vl*, its initial phase angle

*d d TJ B T dt dt*

In this paper, the forward difference method is used to obtain the rotational angle at time *t*

*tt t tt d dt t*

2 2 2 2

*dt t*

*tt t tt t t d*

<sup>1</sup> [( )( ) (2 ) ]. *<sup>t</sup> tt tt tt t t <sup>l</sup> T T t JBt J JBt*

In the case when the effect of the friction is negligibly small, the above equation can be

<sup>2</sup> 2 . *<sup>t</sup> tt tt tt t t l*

One can obtain the vector potential, currents and rotational angle by solving (1), (13)-(15),

<sup>2</sup> <sup>0</sup> .

<sup>2</sup> .

<sup>0</sup>

0

0

2 2 0

>

0, *Tl*, and

(24)

(25)

*t* are set, respectively.

(22)

*l*

**Figure 5.** Flowchart of three-phase line-start IPM synchronous motor

## **3.2. Voltage and current equations and calculation steps for IPM synchronous generator**

Figure 6 shows the circuit of the three-phase IPM synchronous generator. The voltage and current equations of the IPM generator are given as

**Figure 6.** Circuit of three-phase IPM synchronous generator

$$
\varepsilon\_a = r\_1 \dot{\mathbf{i}}\_a + L\_1 \frac{\partial \dot{\mathbf{i}}\_a}{\partial t} + \upsilon\_a + \upsilon\_n \tag{26}
$$

Steady-State and Transient Performance Analysis of

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 191

One can obtain the vector potential, currents by assuming a constant speed and then solving

at new *t* are set.

2. The vector potential *A* at time *t* = 0 is set, where the static field caused by only PMs is

4. The matrix equation constructed by the time-stepping finite-element technique is solved

This paper contains the steady-state synchronous and transient performance characteristics of the IPM synchronous machine shown in Figure 2. The good agreement between computed and measured results validates the proposed method for the finite-element

, *ia t* , *ibt* and *ic t*

is tested. Unless *At*

, *ia t* , *ibt* and *ic t*

3) to 5) continues till the steady-state currents are obtained.

**4. Steady-state synchronous and transient performance**

*<sup>s</sup>* are set, respectively.

converges, the process returns to step 4).

are obtained. The calculation process from step

(1), (31)-(33) using the time-stepping finite-element technique [5]. Next, the calculation steps for this analysis are shown in Figure 7.

**Figure 7.** Flowchart of three-phase IPM synchronous generator

1. First, *N*, Δ*t* and the corresponding rotational step

analysis to predict the machine performance exactly.

given as the initial value. 3. At *t* = *t* + Δ*t*, the initial values for *At*

5. The convergence of *At*

6. After the convergence of *At*

[5].

$$
\varepsilon\_b = r\_1 \dot{i}\_b + L\_1 \frac{\partial \dot{i}\_b}{\partial t} + \upsilon\_b + \upsilon\_n \tag{27}
$$

$$
\sigma\_c = r\_1 i\_c + L\_1 \frac{\partial i\_c}{\partial t} + \upsilon\_c + \upsilon\_n \tag{28}
$$

For a balanced three-phase resistance load,

$$
\upsilon\_a = R\_L \dot{\imath}\_{a'} \ \upsilon\_b = R\_L \dot{\imath}\_{b'} \ \upsilon\_c = R\_L \dot{\imath}\_c \tag{29}
$$

where *RL* is a load resistance per phase.

*vn* can be obtained by substituting (29) in (26)-(28), adding each side of (26)-(28) and then applying (9)

$$\upsilon\_{\rm{tr}} = \frac{e\_a + e\_b + e\_c}{3} \tag{30}$$

One obtains the following equation by substituting (30) in (26)-(28).

$$\frac{\mathfrak{D}}{\mathfrak{D}}e\_a - \frac{e\_b + e\_c}{\mathfrak{D}} - \left(r\_1 + R\_L\right)\dot{i}\_a - L\_1 \frac{\partial \dot{i}\_a}{\partial t} = 0\tag{31}$$

$$\frac{2}{3}e\_b - \frac{e\_c + e\_a}{3} - \left(r\_1 + R\_L\right)i\_b - L\_1\frac{\partial i\_b}{\partial t} = 0\tag{32}$$

$$\frac{\mathfrak{I}}{\mathfrak{J}}e\_c - \frac{e\_a + e\_b}{\mathfrak{J}} - \left(r\_1 + R\_L\right)i\_c - L\_1\frac{\partial i\_c}{\partial t} = 0\tag{33}$$

One can obtain the vector potential, currents by assuming a constant speed and then solving (1), (31)-(33) using the time-stepping finite-element technique [5].

Next, the calculation steps for this analysis are shown in Figure 7.

190 Finite Element Analysis – New Trends and Developments

**Figure 6.**  Flowchart of three-phase IPM synchronous generator

## **4. Steady-state synchronous and transient performance**

This paper contains the steady-state synchronous and transient performance characteristics of the IPM synchronous machine shown in Figure 2. The good agreement between computed and measured results validates the proposed method for the finite-element analysis to predict the machine performance exactly.

#### **4.1. EMF due to PMs**

Figure 8 shows the terminal voltage waveform generated by PMs in driving the IPM synchronous machine at 1500 r/min by the external motor. It is shown that the agreement between the computed and measured values of the generated voltage is excellent.

Steady-State and Transient Performance Analysis of

*t*. This value must be determined by taking

*ts*, which is to move by one stator

*/*2, respectively. It can be seen

*ts* to include

*t*. should be smaller than to

(34)

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 193

**Figure 10.** Computed and measured results of current versus output power of IPM motor

is used to determine the suitable value of the

skewing the stator by one slot pitch. Therefore,

slot pitch at synchronous speed of the motor

values for the

0 of (23-25) is

and a sixty-fourth of

motor with large load inertia.

when the starting of the IPM motor.

up and synchronizing period when

the influence of the ripple harmonics on the starting with

In the figure 10, two kinds of computed curves are given, and the agreement is also good. It

into account the effects due to the space harmonics [5]. The space harmonics effect is also the source of the cogging and ripple torques in the IPM motor. It can be compensated by

(1 / ) ( /) *<sup>s</sup> s <sup>f</sup> <sup>t</sup> N p*

*t* are chosen: 208, 104, 52 and 26 *μ*s are an eighth, a sixteenth, a thirty-second,

*/*2 in the figure. It is seen that the agreement between the curves of 52 *μ*s and

0 are 52 *μ*s and

*ts*, respectively. It is evident from Figure 10 that the choice of 208 *μ*s is

where *f i*s the line frequency, *N<sup>s</sup>* is the number of stator slots. Herein, the following four

suitable at synchronous speed. However, this value is not sufficient in starting the IPM

Figure 11 shows the computed speed-time responses at no load condition with the eddycurrent brake disc coupled to the shaft, when the stator of the motor was supplied with balanced three-phase voltages at rated frequency of 50 Hz and rated voltage 140 V. The inertia of the disc is about 18 times the experimental rotor inertia, and the initial phase angle

26 *μ*s is good and that those are superposed. The choice of a time step of 52 *μ*s is suitable

Figure 12 shows the computed and measured speed-time responses with time during run-

*t* and

that the good agreement between the measured and computed results exits.

**Figure 8.** EMF generated by PMs

## **4.2. Steady-state synchronous and transient performance of Line-start IPM synchronous motor**

Figure 9 shows the load performance characteristics at 140V. It is clear from Figure 9 that the power factor is almost unity at all loads. The efficiency and power factor of the IPM motor were 86.2% and 0.986, respectively for the output of 600 W. The efficiency-power-factor product is 85.0%. It is about 35% higher than that for the induction motor. These values of the IPM motor are very high when compared to those of the induction motor for the same 600 W nameplate rating [3]. Figure 10 shows the computed and measured results of the input current versus the output power at 140V. It is shown that the agreement between the measured and computed results is excellent.

**Figure 9.** Measured results of load performance characteristics of IPM motor

**Figure 10.** Computed and measured results of current versus output power of IPM motor

Figure 8 shows the terminal voltage waveform generated by PMs in driving the IPM synchronous machine at 1500 r/min by the external motor. It is shown that the agreement

between the computed and measured values of the generated voltage is excellent.

**4.2. Steady-state synchronous and transient performance of Line-start IPM**

Figure 9 shows the load performance characteristics at 140V. It is clear from Figure 9 that the power factor is almost unity at all loads. The efficiency and power factor of the IPM motor were 86.2% and 0.986, respectively for the output of 600 W. The efficiency-power-factor product is 85.0%. It is about 35% higher than that for the induction motor. These values of the IPM motor are very high when compared to those of the induction motor for the same 600 W nameplate rating [3]. Figure 10 shows the computed and measured results of the input current versus the output power at 140V. It is shown that the agreement between the

**4.1. EMF due to PMs**

**Figure 8.** EMF generated by PMs

measured and computed results is excellent.

**Figure 9.** Measured results of load performance characteristics of IPM motor

**synchronous motor**

In the figure 10, two kinds of computed curves are given, and the agreement is also good. It is used to determine the suitable value of the *t*. This value must be determined by taking into account the effects due to the space harmonics [5]. The space harmonics effect is also the source of the cogging and ripple torques in the IPM motor. It can be compensated by skewing the stator by one slot pitch. Therefore, *t*. should be smaller than to *ts* to include the influence of the ripple harmonics on the starting with *ts*, which is to move by one stator slot pitch at synchronous speed of the motor

$$
\Delta t\_s = \frac{\text{(1/ }f\text{)}}{\text{(N}\_s / p\text{)}} \tag{34}
$$

where *f i*s the line frequency, *N<sup>s</sup>* is the number of stator slots. Herein, the following four values for the *t* are chosen: 208, 104, 52 and 26 *μ*s are an eighth, a sixteenth, a thirty-second, and a sixty-fourth of *ts*, respectively. It is evident from Figure 10 that the choice of 208 *μ*s is suitable at synchronous speed. However, this value is not sufficient in starting the IPM motor with large load inertia.

Figure 11 shows the computed speed-time responses at no load condition with the eddycurrent brake disc coupled to the shaft, when the stator of the motor was supplied with balanced three-phase voltages at rated frequency of 50 Hz and rated voltage 140 V. The inertia of the disc is about 18 times the experimental rotor inertia, and the initial phase angle 0 of (23-25) is */*2 in the figure. It is seen that the agreement between the curves of 52 *μ*s and 26 *μ*s is good and that those are superposed. The choice of a time step of 52 *μ*s is suitable when the starting of the IPM motor.

Figure 12 shows the computed and measured speed-time responses with time during runup and synchronizing period when *t* and 0 are 52 *μ*s and */*2, respectively. It can be seen that the good agreement between the measured and computed results exits.

Steady-State and Transient Performance Analysis of

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 195

**Figure 13.** Experimental setup for IPM synchronous generator

when load changes rapidly.

except the difference of the phase.

Figures 14 and 15 show the measured and computed results of the terminal voltage, respectively. The phase angle of the terminal voltage in computing the terminal voltage and current is fitted to the experimental one. It is seen that the good agreement exists between the measured and computed results of the terminal voltage except the difference of the phase. This is the reason why the rotor speed lags synchronous speed in the experiment

Figures 16 and 17 show the measured and computed results of the line current, respectively. The line current is zero before *t* = 0s because of no-load. It is seen that the amplitude of the measured current was slightly pulsating because of the mechanical dynamic transient. On the other hand, a constant speed has been assumed in simulation. It is, however, seen that the good agreement exists between the measured and computed results of the current

**Figure 14.** Measured results of terminal voltage versus time in IPM generator

**Figure 11.** Computed speed-time response of IPM motor

**Figure 12.** Computed and measured speed-time response of IPM motor

## **4.3. Steady-state synchronous and transient performance of IPM synchronous generator**

Figure 13 shows the experimental setup for measuring the steady-state load performance characteristics of the IPM generator shown in Figure 2. A 2.2 kW three-phase two-pole 50 Hz 200 V squirrel-cage induction motor and a torque detector were used. The IPM generator has been driven at 1500 r/min by the PWM inverter-driven induction motor.

Figures 14-17 show the terminal voltage and line current, respectively, when the IPM generator with the cage-bars was changed from no-load to resistance load of 15 Ω per phase in Figure 6 at *t* = 0s. The values of the resistance per phase for the maximum load was 15Ω. A synchronous motor has been used as the prime mover in the experiment.

Steady-State and Transient Performance Analysis of Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 195

**Figure 13.** Experimental setup for IPM synchronous generator

194 Finite Element Analysis – New Trends and Developments

**Figure 11.** Computed speed-time response of IPM motor

**Figure 12.** Computed and measured speed-time response of IPM motor

**generator**

**4.3. Steady-state synchronous and transient performance of IPM synchronous**

has been driven at 1500 r/min by the PWM inverter-driven induction motor.

A synchronous motor has been used as the prime mover in the experiment.

Figure 13 shows the experimental setup for measuring the steady-state load performance characteristics of the IPM generator shown in Figure 2. A 2.2 kW three-phase two-pole 50 Hz 200 V squirrel-cage induction motor and a torque detector were used. The IPM generator

Figures 14-17 show the terminal voltage and line current, respectively, when the IPM generator with the cage-bars was changed from no-load to resistance load of 15 Ω per phase in Figure 6 at *t* = 0s. The values of the resistance per phase for the maximum load was 15Ω. Figures 14 and 15 show the measured and computed results of the terminal voltage, respectively. The phase angle of the terminal voltage in computing the terminal voltage and current is fitted to the experimental one. It is seen that the good agreement exists between the measured and computed results of the terminal voltage except the difference of the phase. This is the reason why the rotor speed lags synchronous speed in the experiment when load changes rapidly.

Figures 16 and 17 show the measured and computed results of the line current, respectively. The line current is zero before *t* = 0s because of no-load. It is seen that the amplitude of the measured current was slightly pulsating because of the mechanical dynamic transient. On the other hand, a constant speed has been assumed in simulation. It is, however, seen that the good agreement exists between the measured and computed results of the current except the difference of the phase.

**Figure 14.** Measured results of terminal voltage versus time in IPM generator

Steady-State and Transient Performance Analysis of

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 197

**Figure 17.** Computed results of line current versus time in IPM generator

**Figure 18.** Steady-state terminal voltage versus time in IPM generator

Figure 20 shows the measured and computed results of the terminal voltage versus the output. It can be seen that the good agreement between the measured and computed values

Figures 20-22 show the steady-state load characteristics.

exists except near maximum output.

**Figure 15.** Computed results of terminal voltage versus time in IPM generator

**Figure 16.** Measured results of line current versus time in IPM generator

Figures 18 and 19 show the measured and computed results of the steady-state terminal voltage and line current respectively. It is seen that the good agreement exists between the measured and computed results of the terminal voltage and line current. It is shown that the higher harmonic components by the higher space harmonics [5] were included.

Steady-State and Transient Performance Analysis of Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 197

**Figure 17.** Computed results of line current versus time in IPM generator

196 Finite Element Analysis – New Trends and Developments

**Figure 15.** Computed results of terminal voltage versus time in IPM generator

**Figure 16.** Measured results of line current versus time in IPM generator

Figures 18 and 19 show the measured and computed results of the steady-state terminal voltage and line current respectively. It is seen that the good agreement exists between the measured and computed results of the terminal voltage and line current. It is shown that the

higher harmonic components by the higher space harmonics [5] were included.

**Figure 18.** Steady-state terminal voltage versus time in IPM generator

Figures 20-22 show the steady-state load characteristics.

Figure 20 shows the measured and computed results of the terminal voltage versus the output. It can be seen that the good agreement between the measured and computed values exists except near maximum output.

Steady-State and Transient Performance Analysis of

Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 199

**Figure 21.** Measured and computed results of line current versus output in IPM generator

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Current (A)

**Figure 22.** Measured results of efficiency versus output in IPM generator

A successful design of a high-efficiency small but novel IPM machine with cage bars was developed and tested. It is designed to operate with both high-efficiency line-start IPM

0 100 200 300 400 500 600 700

Output [W]

**5. Conclusion**

Efficiency [%]

0

100

200

300

400

Output (W)

500

600

700

Measured Computed

800

**Figure 19.** Steady-state line current versus time in IPM generator

Figure 21 shows the measured and computed results of the line current versus the output. It can be seen that the good agreement between the measured and computed values exists except near maximum output.

Figure 22 shows the measured results of the efficiency versus output. The efficiency was 85.8% at 600 W and 90% at 100W of light load. It is found that the efficiency is very high.

**Figure 20.** Measured and computed results of terminal voltage versus output in IPM generator

Steady-State and Transient Performance Analysis of Permanent-Magnet Machines Using Time-Stepping Finite Element Technique 199

**Figure 21.** Measured and computed results of line current versus output in IPM generator

**Figure 22.** Measured results of efficiency versus output in IPM generator

#### **5. Conclusion**

198 Finite Element Analysis – New Trends and Developments

**Figure 19.** Steady-state line current versus time in IPM generator

Measured Computed

except near maximum output.

0

20

40

60

80

Terminal Voltage (V)

100

120

140

160

Figure 21 shows the measured and computed results of the line current versus the output. It can be seen that the good agreement between the measured and computed values exists

Figure 22 shows the measured results of the efficiency versus output. The efficiency was 85.8% at 600 W and 90% at 100W of light load. It is found that the efficiency is very high.

**Figure 20.** Measured and computed results of terminal voltage versus output in IPM generator

0 100 200 300 400 500 600 700 800

Output (W)

A successful design of a high-efficiency small but novel IPM machine with cage bars was developed and tested. It is designed to operate with both high-efficiency line-start IPM motor and generator with damper bars. The IPM motor can start and synchronize fully with large load inertia. Beside, the effects of the damper bars on stability during load change and efficiency were investigated. Time-stepping finite element analysis has been used to successfully predict the steady-state and transient performances of the prototype IPM machines. It is clear that cage bars are used effectively to start up in a line-start IPM motor, and to operate stably in the IPM generator with damper bars. It has been found that the proposed design has yielded successful simulation and experimental results.

**Chapter 10**

© 2012 Sun and Kosel, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Sun and Kosel, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Finite-Element Modelling**

Additional information is available at the end of the chapter

Jian Sun and Jürgen Kosel

http://dx.doi.org/10.5772/47777

current sensors or pressure sensors.

**1. Introduction**

**and Analysis of Hall Effect and**

**Extraordinary Magnetoresistance Effect**

The Hall effect was discovered in 1879 by the American physicist Edwin Herbert Hall. It is a result of the Lorentz force, which a magnetic field exerts on moving charge carriers that constitute the electric current [1, 2]. Whether the current is a movement of holes, or electrons in the opposite direction, or a mixture of the two, the Lorentz force pushes the moving electric charge carriers in the same direction sideways at right angles to both the magnetic field and the direction of current flow. As a consequence, it produces a charge accumulation at the edges of the conductor orthogonal to the current flow, which, in turn, causes a differential voltage (the Hall voltage). This effect can be modeled by an anisotropic term added to the conductivity tensor of a nominally homogeneous and isotropic conductor.

The Hall effect is widely used in magnetic field measurements due to its simplicity and sensitivity [2]. Hall sensors are readily available from a number of different manufacturers and are used in various applications as, for example, rotating speed sensors (bicycle wheels, gear-teeth, automotive speedometers, and electronic ignition systems), fluid flow sensors,

Recently, a large dependence of the resistance on magnetic fields, the so-called extraordinary magnetoresistance (EMR), was found at room temperature in a certain kind of semiconductor/metal hybrid structure [3]. Sharing a similar origin with the Hall effect, the EMR effect is mainly based on the Lorentz force generated by a perpendicularly applied magnetic field, which causes a current deflection. This results in a redistribution of the current from the metal shunt into the semiconductor causing a resistance increase. It is important to note that the fundamental principle of EMR is the change of the current path in

## **Author details**

Kazumi Kurihara *Department of Electrical and Electronic Engineering, Ibaraki University, Hitachi, Japan*

## **Acknowledgement**

The author wish to thank Dr. Marushima of the Oriental Motor Company Ltd, for technical support and T. Kubota, T. Yasui, and T. Igari of the Ibaraki University for experimental support.

## **6. References**