3D micro-machined inductive contactless suspension: Testing and modeling

Sensors and Actuators A 220 (2014) 134–143

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

3D micro-machined inductive contactless suspension: Testing and modeling Zhiqiu Lu a,1 , Kirill Poletkin a,∗,1 , Bartjan den Hartogh b , Ulrike Wallrabe a , Vlad Badilita a,∗∗ a Laboratory for Microactuators, Department of Microsystems Engineering – IMTEK, University of Freiburg, Georges-Köhler-Allee 102, D-79110 Freiburg, Germany b FemtoTools AG, Furtbachstrasse 4, CH 8107 Buchs/ZH, Switzerland

a r t i c l e

i n f o

Article history: Received 12 May 2014 Received in revised form 2 September 2014 Accepted 16 September 2014 Keywords: 3D micro-coil Contactless suspension Induction Dynamics Stability Levitation

a b s t r a c t We present herewith detailed theoretical modeling coupled with experimental analysis of a micromachined inductive suspension (MIS). The reported MIS is based on two coaxial 3D solenoidal microcoils realized using our wirebonding technology. The two coils are excited using an AC signal with 180◦ phaseshift and a conductive proof mass (PM) is stably levitated on top of the coils. Using a micromechanical displacement sensor, we experimentally derive the lateral, vertical and angular stiffness constants of our MIS. Based on the analytical model presented here, we discuss the stability of the levitated proof mass as a function of the geometrical parameters of the design. We further test our model by applying it to another previously reported MIS structure realized using planar technology, providing stability diagrams as well as design guidelines for further developments of the micromachined inductive suspension as, for instance, miniature rotating gyroscopes. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Micromachined inductive suspensions (MIS) exploit the phenomenon of electromagnetic induction in order to achieve the levitation of a proof mass (PM). The development of such suspensions attracted a lot of interest during the past decades on one hand being facilitated by the advances in the MEMS-based technologies, and on the other hand being fueled by the whole range of potential applications such as gyroscopes, micromotors, micro-positioning systems, all of these in a frictionless, dust-free environment. In 1995, Shearwood et al. pioneered a MIS prototype fabricated using only surface micromachining techniques. Levitation, stabilization and rotation micro-coils have been deposited on a substrate wafer by standard metalization and photolithography. Stable levitation and rotation of a disk-shaped rotor have been demonstrated and this prototype has been proposed for application as a rotating micro-gyroscope [1–3]. Shearwood et al. [4] reported 1000 rpm rotation speed and showed that this limit is attributed to a slight

∗ Corresponding author. Tel.: +49 (0) 761 203 7435. ∗∗ Corresponding author. E-mail addresses: [email protected] (K. Poletkin), [email protected] (V. Badilita). 1 These authors contributed equally to this work. http://dx.doi.org/10.1016/j.sna.2014.09.017 0924-4247/© 2014 Elsevier B.V. All rights reserved.

wobble during rotation that excited a lateral resonant mode. In 2006 Zhang et al. [5] proposed an improved MIS design, in which the coils for levitation and rotation were separated, and the maximum reported speed was 3000 rpm. Even this result is still far from the maximum theoretical rotation speed in air which is of the order of 100,000 rpm [6]. An alternative coil design in the shape of a rectangular spiral, which provides stable levitation, was employed in a microgyroscope prototype reported in [7]. In this prototype fabricated using surface micromachining techniques, the observed rotation speed was 2000 rpm. More recently, we developed a coil winding technology which is able to produce perfect 3D solenoidal microcoils using an automatic wirebonder [8]. Using this technology we reported a 3D micromachined inductive suspension (3D MIS), which was preliminarily characterized [9] demonstrating a dramatically reduced threshold current necessary to achieve levitation, along with increased levitation height compared to Shearwood et al. [4], due to the superior number of ampere-turns in the case of 3D solenoidal microcoils as opposed to the coils obtained through planar technology. We also demonstrated that MIS may become an even more promising technology, which gives rise to a new generation of micro-machined actuators and sensors [10]. Moreover, MIS opens new opportunities for further improvements, for instance, by means of using a new type of suspension with zero spring constant

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Fig. 1. (a) Design of the 3D MIS: XYZ (red color) and xyz (dark blue color) are the fixed and movable CF, respectively; h is the levitation height of the PM. (b) 3D MIS glued on a PCB for levitation experiment.

proposed in [11] and realizing the dynamically tuned condition as reported in [12]. The previous reports on structures based on electromagnetic levitation exploit the principle and present experimental characterization of the device, sometimes along with basic design rules. An in-depth theoretical characterization is lagging behind, therefore an in-depth understanding of the mechanism of stable levitation is still missing, and as a consequence it comes as no surprise the fact that, in spite of its potential usefulness due to the complete elimination of friction and wear, electromagnetic levitation has not ignited much interest so far in fields such as microrobotics, microsurgery or micromanipulation. We have elaborated an analytical model for MIS [13] which describes the dynamics of the levitated disk-shaped proof mass in space near an equilibrium point, and the condition for the stable levitation has been developed. Based on this model, we proposed more recently an analytical approach to the qualitative analysis of MIS design from the stability and dynamics standpoint [14], elaborating MIS design rules for improved dynamics in the framework of planar coils. In particular, exploiting the approach reported in [14], the analytical model of the 3D MIS is developed and experimentally verified. The model predicts the stable levitation and dynamics in 3D MIS and can be extended to study a new MIS design. This fact fills the missing gap in previously reported pure experimental approaches to the study of MIS. In this work we are merging the pure theoretical modeling [13,14] and the previously reported MIS based on 3D solenoidal wirebonded microcoils [9]. The paper is organized as follows: in Section 2 the operating principles, design considerations and fabrication of the 3D MIS are presented. In Section 3 of the paper, the linear analytical model is introduced to describe the suspension dynamics and the condition for stable levitation. In Section 4 we present the experimental setup and, starting from experimental measurements, the derivation of mechanical parameters such as stiffness coefficients relative to lateral, vertical and angular displacements. In Section 5 of the paper we apply our theoretical model to the 3D MIS demonstrating that the values provided by the model are in good agreement with the measured values. We also provide stability diagrams for the 3D MIS prototype and we map the stiffness distribution within the stability domain. Also in Section 5, we test the analytical model presented in Section 3 on a different, two-dimensional structure which has been previously reported [4]. In the closing Section 6 of the paper, we provide a discussion and outlook with potential solutions for further improvement of systems based on micromachined inductive suspensions. 2. Operating principle, design and fabrication A schematic of our 3D MIS is shown in Fig. 1(a). It mainly consists of two coaxial coils excited with AC signals which are in anti-phase

to each other. The role of the inner coil is to levitate a disk-shaped conductive proof mass placed on top of the MIS structure, while the role of the outer coil is to provide stable levitation of the proof mass. Alternating currents IL and IS passing through the levitation and stabilization coils respectively, create a magnetic flux which is variable in space. This variable flux of the inner, levitation coil intercepts the surface area of the conducting proof mass and induces eddy currents, which in turn generate a magnetic field to produce a repulsive force that levitates the proof mass. The addition of an outer coil excited with a signal which is phase-shifted with 180◦ with respect to the levitation coil, introduces a restoring force pulling the proof mass back towards the center of the structure, and therefore shaping a potential well which governs the stability of the levitating proof mass. Because the purpose of this paper is to study theoretically and to compare experimentally the proof mass stability, we will consider the topology of our micromachined inductive suspension as well as the fabrication process along with its advantages and limitations. The fabrication of the MIS presented here is very similar to that reported in detail in [9], with a couple of modifications which will be discussed below. The fabricated MIS structure used as a prototype for this work is presented in Fig. 1(b). An insulating substrate (glass or Pyrex) is metallized and pads for electrical contacts are defined by standard UV photolithography, electroplating and wet chemical etching. With respect to our previous report [9], we are using here contact pads electroplated up to 10 ␮m, thus providing better reliability for the wirebonding process, as well as decreased resistance for the current path. In a next step, 700 ␮m-thick SU-8 2150 is cast on the wafer and structured by UV lithography to define the cylindrical pillars for subsequent wirebonding of the coils. While thicker sidewalls of the cylinders or even full pillars are beneficial for the adhesion of the thick SU-8 structures on the Pyrex substrate, this provides increased stiction between the pillars and the proof mass making the onset of the levitation process very difficult. In order to compromise between these two aspects, we have increased the sidewall of the SU-8 cylinder only for the outer coil to 200 ␮m as opposed to 100 ␮m [9]. In the last step, the coils are manufactured using an automatic wirebonder which allows us to freely define the total number of windings per coil, the pitch between the windings, and the number of winding layers. Although using a wirebonder for the coil fabrication is a serial process, this process is very fast – hundreds of milliseconds per coil depending on the exact dimensions – and perfectly integrated with traditional MEMS processes [8]. While the ability to increase the number of winding layers is a very useful feature because it allows for an increase in the number of ampere-turns per coil while still using the same height of the coil, it comes with a couple of inherent drawbacks. The amount of

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Table 1 Parameters of the 3D MIS prototype. Radius of the levitation coil, rl Radius of the stabilization coil, rs The coils pitch of winding, p Number of windings for stabilization coil, N Number of windings for levitation coil, M Radius of PM, rpm Thickness of PM

1000 ␮m 1900 ␮m 25 ␮m 12 20 1600 ␮m 25 ␮m

heat generated in the same volume increases with the number of layers, i.e., the number of overlapping windings, eventually leading to damaging the wire insulation, short-circuit and failure of the device. The second limitation brought along by multiple layers of windings concerns the operation range in terms of frequency. A higher operating frequency is equivalent to a higher variation speed of the magnetic flux, therefore larger eddy currents and more efficient levitation for the same excitation current in the coils. However, multiple layers of winding decrease the self-resonant frequency of the corresponding coil, therefore the border where the coil does not behave as an inductor any longer. For this reason, in the present work we only used one single layer of windings both for the levitation and for the stabilization coils. The levitation coil is formed by 20 windings, while the stabilization coil by 12 windings. The top winding of each coil is located at the same height. This modification means an inductance value reduced by a factor of 4, therefore a reduced levitation height. However, because of less heat issues, the coils are able to withstand increased root mean square (rms) current values in excess of 160 mA as compared to 120 mA in [9], therefore compensating to some extent for the decrease in the levitation height. Laser cutting using a Trumpf TruMark Station 5000 is employed in order to fabricate the disk-shaped PMs. PMs are cut from a 25␮m-thick Aluminium foil (Advent Research Materials) glued to a glass wafer by thick AZ9260 photoresist. In this particular case, the aluminium PM was cut to a diameter of 3200 ␮m. All the parameters of the 3D MIS used in this work are summarized in Table 1.

3. Linear analytical model for MIS In this section we present a linear analytical model of the MIS based on the design described in the previous section. The aim of the model is to describe the suspension dynamics and the condition for stable levitation of the disk-shaped PM. This model is obtained by using the method developed in [14].

Fig. 2. An arbitrary position of the PM: x1 y1 z1 , x2 y2 z2 and x3 y3 z3 are auxiliary CFs (green color); s, l and ϕ are the generalized coordinates of the mechanical part of the MIS.

3.2. Generalized coordinates An arbitrary position of the PM with respect to the equilibrium position O when the PM is disturbed is shown in Fig. 2. To characterize the PM position near point O, let us introduce the following auxiliary CFs, namely, x1 y1 z1 , the origin of which is coincident with point O, x2 y2 z2 and x3 y3 z3 , the origins of which are both assigned to point B. Additionally, the x1 y1 z1 CF is fixed and the z1 and Z axes are coincident. The axes of x1 y1 z1 and x2 y2 z2 are collinear, hence x2 y2 z2 participates only in the linear displacements of the PM with respect to the x1 y1 z1 CF. The position of x3 y3 z3 with respect to x2 y2 z2 is defined by an arbitrary angle ˛, which characterizes the angular misalignment of x3 y3 z3 relative to the Bz2 axis. Besides, the Bx3 axis is coincident with the x axis of the movable CF. In turn, the movable CF has an angular misalignment with respect to x3 y3 z3 defined by the angle ϕ relative to the Bz3 axis. Assuming that the PM is a rigid body and displacements are small, the generalized coordinates for describing the mechanical part of the MIS  can be introduced as follows [13]; the generalized x12 + y12 characterizes the linear displacement of coordinate s = the center of mass of the PM in the radial direction; the generalized coordinate l characterizes the linear displacement of the center of mass of the PM along the Oz1 , and the generalized coordinate ϕ characterizes the angular displacement of the PM relative to an axis coincident with the principal axis of inertia of the disk-shaped PM lying on its equatorial plane as shown in Fig. 2. The electrical part of the 3D MIS can be represented by the electrical circuit illustrated in Fig. 3. The stabilization and levitation coils are fed by the alternating currents iL = IL ejωt and iS = IS ejωt ,

3.1. Coordinate frames To describe the 3D MIS dynamics, let us assign the following coordinate frames (CF), namely, the fixed CF XYZ to the coils, and movable CF xyz to the PM, assuming that the coils are coaxial and the top surfaces of coils lie in the same plane. The Z axis of the fixed CF is coincident with the coils axis, which passes through the coil center and is perpendicular to the coil plane. The origin A of the fixed CF is located at the intersection of the plane crossing the top turn of both coils and the coils axis, this plane and the axis being mutually perpendicular. Also, the X and Y axes lie on the same plane and are fixed as illustrated in Fig. 1(a). The axes of the movable CF coincide with the principal axes of inertia of the disk PM. We assume that the origin B of the movable CF coincides with the PM center of mass and, in equilibrium state, with the origin O. The position of the origin O in space with respect to the fixed CF is characterized by a vector ro = (0, 0, h), where h is the levitation height of the PM.

RL

LLm IL

LL

iL

LSL

IS

iS

LS

LPM

i

RPM

LSm

RS Fig. 3. Electric circuit of the 3D MIS: LL , LS and LPM are self inductances of the stabilization coil, levitation coil and PM, respectively; RL , RS and RPM are the electrical resistances of the stabilization coil, levitation coil and PM, respectively; LSL is the mutual inductance between the stabilization and levitation coils.

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Fig. 4. (a) Schematic of the experiment setup for mechanical property measurement. A USB microscope was employed to capture probe motions. (b) Picture of the microforce sensing probe pushing the PM in the lateral direction. The sensing probe was tilted because of topological constraints.

respectively, where IS and IL are amplitudes, and ω is the frequency, √ j = −1 is the imaginary unit. Due to the mutual inductance LLm between the levitation coil and PM, and the mutual inductance LSm between the stabilization coil and PM, the eddy current i is induced. Hence, the currents iS and iL are taken as the velocities of the generalized coordinates of the electrical part of the 3D MIS.

According to the Sylvester criterion from (1), the condition for the stable levitation of the PM is reduced to the analysis of the following set of inequalities [14]: c0S IS ± c0L IL > 0; cllS IS ± cllL IL +

(clS IS ± clL IL ) c0S IS ± c0L IL

2

> 0;

(3a)

S L css IS ± css IL > 0;

(3b)

3.3. Linear model and stable levitation condition

S cϕϕ IS

(3c)

Using the generalized coordinates defined above and assuming a high frequency of the supply current, that the levitation height of the proof mass is small, and the fact that the function of mutual inductance between the disk shaped proof mass and the ring shaped coils can be simplified and expressed in terms of simple functions as shown in [14], the linear analytical model of the 3D MIS, which describes the behavior of the disk-shaped PM near the equilibrium point, can be written as follows:

S L S L S L (cϕϕ IS ± cϕϕ IL ) · (css IS ± css IL ) > (csϕ IS ± csϕ IL ) .

  ⎧ 2 (clS IS ± clL IL ) c S I ± c0L IL ⎪ S L ¨l + l l˙ + 0 S ⎪ m c I ± c I + · l = Fl ; L S ⎪ ll ll LPM ⎪ c0S IS ± c0L IL ⎪ ⎨ m¨s + s s˙ + ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

¨ + ϕ ϕ˙ + Jϕ

c0S IS ± c0L IL LPM

c0S IS ± c0L IL LPM

S L S L [(css IS ± css IL ) · s + (csϕ IS ± csϕ IL ) · ϕ] = Fs ;

(1)

S L S L [(csϕ IS ± csϕ IL ) · s + (cϕϕ IS ± cϕϕ IL ) · ϕ] = Mϕ .

where m is the mass of PM, J is the moment of inertia of the PM about the axis lying on its equatorial plane, l , s and ϕ are the damping coefficients of the PM relative to the appropriate velocities of the generalized coordinates l,˙ s˙ and ϕ, ˙ respectively; Fl , Fs and Mϕ are the generalized forces and torque acting on the PM relative to the appropriate generalized coordinates l, s and ϕ, respectively; c0 , cl , cnw (n = l, s, ϕ; w = l, s, ϕ) are the coefficients of the quadratic forms of mutual inductances of LSm and LLm , the coefficients superscripts L and S correspond to the stabilization and levitation coil, respectively; in the particular case, the quadratic forms for both functions LSm and LLm have the following form [14]: Lm (l, s, ϕ) = c0 + cl l +

1 2 1 1 c l + css s2 + cϕϕ ϕ2 + csϕ sϕ, 2 ll 2 2

(2)

the ± sign defines the direction of currents iL and iS , the plus sign corresponds to the case when the currents in the two coils have the same direction (without the phase shift), while the minus sign describes the case where there is the phase-shift of 180◦ between the two currents; LPM is the self induction of the PM. For the particular design of the 3D MIS under consideration, the coefficients c0 , cl , cnw (n = l, s, ϕ; w = l, s, ϕ) are calculated by using the equations presented in Appendix A.

L ± cϕϕ IL

> 0; 2

(3d)

The linear model (1) of the 3D MIS allows us to evaluate, on the one hand, the mechanical parameters of the suspension such as stiffness coefficients, and consequently the resonant frequencies. On the other hand, this model provides understanding of the mechanism of stable levitation in the 3D MIS by using the set of conditions (3) which must be fulfilled simultaneously. Note that model (1) predicts the cross-stiffness between the generalized coordinates ϕ and s. This cross-stiffness and its physical meaning will be discussed in Section 6. Also as it is seen from (3), the stability domain is defined by the coefficients of the quadratic forms of mutual inductances of LSm and LLm and the values of currents in coils. However, it is important to notice that in the case when both currents are the same, the stability domain is defined only by the coefficients of the quadratic forms of mutual inductances and becomes independent from the current. 4. Experiment In this experiment stiffness coefficients of the 3D MIS relative to the appropriate generalized coordinates s, l and ϕ were evaluated by direct measurements of a force, which was applied in a controlled manner in order to determine a lateral or vertical displacement of the PM, respectively. This displacement was also recorded simultaneously with the applied force. These measurements have been performed using a mechanical probe (FT-FS1000, FEMTO-TOOLS AG Switzerland) and a microforce sensing probe (FT-S100, FEMTO-TOOLS AG Switzerland) with a force and linear displacement resolution of 0.005 ␮N and 5 nm, respectively. In this section we describe the experimental setup and measurements in detail. 4.1. Experiment setup Fig. 4(a) shows the schematic of experimental setup. Each coil was fed with a square wave AC current provided by a current amplifier (LCF A093R). The amplitude and the frequency of the current in each coil were controlled by a function generator (Arbstudio 1104D) via a computer. To prevent the collision of the sensing probe tip (thickness: 50 ␮m) with SU-8 pillars or the coils, the PM must

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Lateral force (µN)

0.8

measured fitted

0.6 Slope = 0.00298

0.4 15o

0.2 0.0 0

100

200

300

Horizontal displacement (µm) Fig. 5. Lateral force and displacement of the probe in horizontal plane. The angle between horizontal plane and probe was 15◦ . The movement range was 370 ␮m.

be levitated at least at a height of 50 ␮m measured from the top of the SU-8 pillars. In order to fulfill this condition we calibrated the levitation height using a laser distance sensor (LK-G32). As a result, the measured rms currents in the stabilization and levitation coils were to be IS = 0.106 A and IL = 0.11 A, respectively, at a frequency of 12 MHz, which levitated the PM at height of 54 ␮m measured from the top of the SU-8 pillars. Taking into account that the last coil winding for this particular structure ends 60 ␮m away from the top of the SU-8 structure, the total actual levitation height, h, was 114 ␮m. The calibration of the levitation conditions has been performed and then the microforce sensing probe was moved towards the PM until mechanical contact, as shown in Fig. 4(b). Once in contact, the force applied to the PM and the linear displacement along the direction of action of the applied force was recorded. 4.2. Measurement For the 3D MIS stiffness evaluation in the radial direction relative to the generalized coordinate s, the force must be applied to the edge of the PM. The sensing probe was titled by an angle of 15◦ with respect to the plane of the PM due to geometrical limitations. The result of this measurement is shown in Fig. 5. The stiffness in the radial direction, ks , was calculated to be 3.0 × 10−3 N m−1 . For this calculation, we chose a measurement range up to 200 ␮m, exhibiting a linear dependence of the applied force on the linear displacement, as shown in Fig. 5. However, in this experimental setup only the force and linear displacement can be measured directly and due to this fact the angular stiffness, kϕ , relative to the generalized coordinate, ϕ, cannot be evaluated from one single measurement. Also, in order to

evaluate the vertical stiffness, kl , relative to the generalized coordinate, l, the probe should be applied in the vertical direction exactly in the force center of the PM. However, the force center does not coincide with the geometric center of the plate surface of the PM, which makes very problematic applying the tip exactly in the force center. To avoid the difficulties mentioned above, we evaluated kϕ and kl using an indirect method. We recorded the force versus displacement while applying the tip to two different points located at rp1 = 300 ␮m and rp2 = 1450 ␮m from the PM centre. The results of these measurements are shown in Fig. 6(a) and (b). In the particular points chosen for these measurements, the following stiffness values were calculated kp1 = 3.5 × 10−2 N m−1 and kp2 = 6.0 × 10−3 N m−1 for rp1 and rp2 , respectively. Substituting kp1 , kp2 , rp1 and rp2 into the set below:

⎧ 2 − r 2 )k (rp2 p1 p1 ⎪ ⎪ ⎨ kϕ = k /k − 1 ; p1 p2 kp2 ⎪ ⎪ , ⎩ kl = 2

(4)

1 − rp2 (kp2 /kϕ )

the desired stiffness coefficients can be calculated. Hence we have the vertical stiffness kl , relative to the generalized coordinated, l: kl = 4.5 × 10−2 N m−1 and the angular stiffness kϕ , relative to the generalized coordinate, ϕ: kϕ = 1.5 × 10−8 N m rad−1 . 5. Model verification In this section we verify the model expressed by the set of equations (1) with the experimental results presented in Section 4. In a next step, we test our theoretical model with another device, which was reported previously by another group [4] and was realized mainly using traditional surface microfabrication, resulting in a 2D MIS. We perform this analysis from the stability and dynamics standpoint. 5.1. 3D MIS prototype At the beginning, let us study the stability of the 3D MIS prototype by using the set of inequalities (3). Analysis of Eq. (A.1)–(A.6) shows that d (d = rpm − rl ) and h can be considered as independent variables for this study. Note that the currents in the coils have 180◦ phase shift between each other, which corresponds to the minus sign in Eq. (1). Also we recommend the following values of coefficients of similarity such as  s = 0.3,  l = 1.2 and  sl = 0.14 as is explained in Appendix A. Substituting the parameters of the prototype shown in Table 1 and the values for the currents in the coils, IS and IL , shown in Section 4.1 into Eqs. (A.1)–(A.6), we find 0.8

Vertical force (µN)

1.5

Vertical force (µN)

measured fitted

measured fitted

1.0

Slope = 0.03458 0.5

0.0 0

10

20

30

40

Vertical displacement (µm)

(a)

50

0.6

Slope = 0.00629

0.4

0.2

0.0

0

20

40

60

80

100

Vertical displacement (µm)

(b)

Fig. 6. Vertical force and displacement of the probe: (a) The initial position of the probe was 300 ␮m from the center of the PM. The movement range was 50 ␮m. (b) The initial position of the probe was 1450 ␮m from the center of the PM. The movement range was 100 ␮m.

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Table 2 Comparison of suspension stiffness coefficients from modeling and experiment results. Stiffness coefficients

Measured values

Calculated values by model (1)

ks [N m−1 ] kl [N m−1 ] kϕ [N m rad−1 ]

3.0 × 10−3 4.5 × 10−2 1.5 × 10−8

3.0 × 10−3 4.2 × 10−2 0.8 × 10−8

where the self-inductance of the PM, LPM , can be calculated as [15]:



LPM = 0 (rl + d) ln

Fig. 7. Boundaries of inequalities (3a)–(3d) and the stability domain of d and h shown in blue-gray: arrows show the change of the sign from minus to plus of the respective inequality, when its boundary is crossed in the indicated direction.

the boundaries of inequalities (3a)–(3d) and the stability domain of d and h, where both variables are defined within the range from 0 to 900 ␮m. This stability domain is mapped in Fig. 7. The analysis of Fig. 7 shows that the stable levitation in such a prototype is possible. However, solving the inequalities (3a)–(3d) for the case when there is no phase-shift between the currents in the two coils, one can demonstrate that there is no a stability domain, i.e., stable levitation is not possible. This fact agrees with the experimental observation, which confirmed that the stable levitation in the prototype under consideration is only possible when the currents have 180◦ phase-shift. Now, let us calculate the stiffness coefficients of the 3D MIS prototype, namely, kl , ks and kϕ , which can be defined according to model (1) as follows:

kl =

ks =

kϕ =

c0S IS − c0L IL LPM c0S IS − c0L IL LPM c0S IS − c0L IL LPM

cllS IS

− cllL IL

+

(clS IS − clL IL ) c0S IS − c0L IL

2

;

(5)

S L (css IS − css IL );

(6)

S L (cϕϕ IS − cϕϕ IL ),

(7)



8(rl + d) − 1.92 , ı

(8)

where ı is the effective width in which the maximum of an induced eddy current within the PM is distributed. As shown in [16], the effective width can be evaluated as ı = 0.05 . . . 0.1 · rpm . Using Eqs. (5)–(7), the stiffness distribution within the stability domain of 3D MIS prototype is mapped and presented in Fig. 8. Fig. 8 also shows the positioning of the coils and the PM relative to the stability domain. Then, according to the approach reported in [14], the region of interest for the calculation is located at the outer edge of the PM. In this particular case, the levitation height is 114 ␮m and the diameter of the levitated PM is 3.2 mm. Hence, the coordinates for calculation are d = 600 ␮m and h = 114 ␮m. Results of both measurements and calculation are shown in Table 2. Note that the outer edge of the PM is located within the stability domain. According to approach [14], this fact confirms theoretically that the levitation of the PM used in this work is stable. The analysis of Table 2 shows that model (1) agrees well with the measurements. 5.2. 2D MIS In order to test our model on a different structure reported by another group, we chose the MIS presented previously in [4]. All the relevant parameters of this structure are shown in Table 4. Due to the fact that the coils are represented as hollow disks, in this particular case effective radii of coils are used for modeling. We suggest the following radii of levitation and stabilization coils: rl = 180 and rs = 300 ␮m, respectively, which are within the dimension of the respective coil. Since in the 2D MIS the PM stably levitated with the current frequency of 8 MHz, which is similar to our experiment, the similarity coefficients are taken the same as in previous Section 5.1. Taking into account that the currents in the coils of this prototype also have 180◦ phase-shift, the boundaries as they result from inequalities (3a)–(3d) and the stability domain of d and h, where both variables are defined within the range from 0 to 120 ␮m, are presented in Fig. 9. Also, the location of the PM and coils relative to the stability domain is shown in Fig. 9. The analysis shows that

Fig. 8. The distribution of the stiffness components, namely, kl , ks and kϕ within the stability domain.

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6. Comparative results, discussions and conclusions

Fig. 9. Boundaries of inequalities (3a)–(3d) and the stability domain of d and h shown in blue-gray: arrows show the change of the sign from minus to plus of the respective inequality, when its boundary is crossed in the indicated direction.

the stable levitation in the prototype under consideration is possible, this fact is confirmed theoretically by the location of the outer edge of the PM within the stability domain. It is worth noting that the stability domain has the upper limit which restricts the levitation height up to 25 ␮m. This fact agrees well with experimental observation reported in Ref. [4, p. 472]. The stiffness distribution calculated by Eqs. (5)–(7) within the stability domain is shown in Fig. 10. We considered the thickness of the PM disk – 10 ␮m, the levitation height – 2 ␮m, and the radius of the PM – 250 ␮m. Hence the coordinates for the stiffness calculation are d = 70 ␮m and h = 7 ␮m. The results of the calculation compared to the measurements reported in [4, p. 471] are shown in Table 3. It should be noted that in the framework of the qualitative approach proposed in [14] for modeling, which describes the behavior of the suspension in general, the accuracy of calculation can be up to 20% (the accuracy is the ratio of calculated to measured value multiplied by 100). This fact is due to the simplifications, which allow us to obtain the analytical model. Hence, the analysis in Table 3 shows that model (1) is in good agreement with the measurements. It is worth noting that the accuracy of calculation also depends on how close the real structure is to the assumptions behind the model. The analytical model considers the MIS as a lumped system, therefore once the real MIS deviates from this assumption, the accuracy decreases. This fact clearly appears in our results presented in Tables 2 and 3 since the accuracy of the calculation in 3D MIS is higher than in 2D MIS, due to the fact that the coils in the 3D MIS are closer to the lumped system than in the 2D MIS. Table 3 Modeling results of MIS reported in [4]. Stiffness coefficients −1

ks [N m ] kl [N m−1 ] kϕ [m N rad−1 ]

Values reported in [4] −4

1.0 × 10 4.0 × 10−3 11.0 × 10−11

Calculated values by model (1) 1.0 × 10−4 1.0 × 10−3 2.0 × 10−11

Table 4 Parameters of MIS prototype in Ref. [4]. Inner radius of the levitation coil Outer radius of the levitation coil Inner radius of the stabilization coil Outer radius of the stabilization coil Radius of the PM, rpm The levitation height, h The currents in both coils Number of winding turns for stabilization coil, N Number of winding turns for levitation coil, M Thickness of PM

160 ␮m 240 ␮m 255 ␮m 345 ␮m 250 ␮m 2 ␮m 0.12 A 1 1 10 ␮m

First, let us note that the linear characteristical dimension of the 3D MIS is one order of magnitude larger than the linear characteristical dimension of the MIS reported in [4]. Due to this fact, the values predicted by our model as well as the measured ones for kl and ks in the 3D MIS are one order of magnitude larger than the corresponding values in the 2D MIS, while the value of kϕ is three orders of magnitude larger. This result is obvious and agrees with the scale analysis by using Eqs. (5)–(7). On the other hand the stiffness value, ks , in the lateral direction of the PM displacement of the 3D MIS corresponds to a resonance frequency around 15 Hz. At the same time the lateral resonance frequency measured in the 2D MIS is 25 Hz. Hence we can conclude that although we achieved an increase in the stiffness values in the 3D MIS, the overall dynamics is not very different. As a result, for instance, in the case of an application of the current design of the 3D MIS in a micro-motor, the rotation speed of the rotor will not be increased compared to the 2D MIS. Otherwise, one can see from the stability maps in Figs. 7 and 9 respectively, that the stability domain for the 3D MIS is larger than for the 2D MIS. This benefit gives us the opportunity to vary the size of the PM. Therefore one can try to find a relationship between the PM mass and the stiffness value, when the minimum of the PM mass corresponds to the maximum of the stiffness value. As seen from Fig. 8(b), the radius of the PM can be reduced to 1.35 mm, (d = 350 ␮m) and at the same levitation height the lateral stiffness is increased to 5 × 10−3 N m−1 . Hence, the lateral resonant frequency becomes around 23 Hz, which is still not enough for significantly higher rotation speed. Consequently a new design of 3D MIS with the improved dynamics is needed. As a solution, the design proposed in [14] can be used, in which the stabilization coil is elevated at the same height of the PM levitation. However, our model provides general design rules for future better designs of the 3D MIS where the lateral stiffness can be significantly improved in order to achieve higher rotation speed. As seen from set (1), the model predicts that there is a crossstiffness, which can be defined as ksϕ =

c0S IS − c0L IL LPM

S L (csϕ IS − csϕ IL ).

(9)

In the condition for stable levitation (3), this coefficient appears in inequality (3d), which defines the domain of stability bordered by the black line in Figs. 7 and 9. Hence, the physical meaning of this coefficient follows obviously from our study: it restricts the levitation height or, in other words, the stability domain from the top. Its existence was shown in Ref. [4], where the maximal stable levitation height was defined experimentally. Theoretical prediction of distributions of the value of ksϕ for our 3D MIS prototype and for the 2D MIS reported in [4], are shown in Fig. 11(a) and (b), respectively. This coefficient was not measured in our work because the linear and angular displacements of the PM should be simultaneously controlled, a feature which is not implemented in our current experimental setup. In this article, the known design of the 3D MIS consisting of stabilization and levitation coils was studied theoretically and experimentally. Using the approach to the analysis of the design of MIS presented in [14] the analytical model, which describes the 3D MIS dynamics and the condition for stable levitation of the diskshaped proof mass, was developed. Mechanical parameters such as stiffness coefficients relative to lateral, vertical and angular displacements of the PM were measured in the fabricated prototype of the 3D MIS. Then this model was applied and compared to the experimental measurements finding a good agreement between the experiments and the results predicted by the model. Additionally, the developed model was applied to the experimental study

Z. Lu et al. / Sensors and Actuators A 220 (2014) 134–143

141

Fig. 10. The distribution of the stiffness components for the MIS structure reported in [4], namely, kl , ks and kϕ within the stability domain.

Fig. 11. The distribution of the cross-stiffness ksϕ : (a) our 3D MIS prototype and (b) 2D MIS prototype reported in [4].

of the 2D MIS prototype presented in [4], and the robustness of our model was proven by the good agreement with reported experimental results. The comparative analysis of these two prototypes, namely, the studied 3D MIS in this article and MIS reported in [4] shows that, in particular the lateral resonant frequency is similar in both prototypes. Hence, for instance, an application the 3D MIS in a high speed rotating micro-motor requires a new design of 3D MIS, in which the lateral frequency can be increased. Using developed model (1), a new design of the 3D MIS with improved dynamics can be proposed, which will be our future research topic. Fig. A1. The geometrical parameters: rpm , rl and rs are radii of the PM, levitation and stabilization coil, respectively; p is the coils pitch of winding; d = rpm − rl , c = rs − rl .

Acknowledgements Zhiqiu Lu gratefully acknowledges the support from the Siemens-DAAD Scholarship. Dr. Kirill Poletkin acknowledges with thanks the support of the Alexander von Humboldt Foundation. Dr. Vlad Badilita kindly acknowledges support from the German Research Foundation (DFG) through project number BA 4275/21.

Appendix A. Coefficient calculation To calculate the coefficients c0 , cl , cnw (n = l, s, ϕ; w = l, s, ϕ), let us introduce the geometrical parameters of the suspension as shown in Fig. A1. Hence, adapting the results obtained in [14] to the 3D MIS, the coefficients can be defined as follow:

c0S

=

N−1

as

v=0

c0L

=

M−1

a

l

N−1

−

v=0

clL =

M−1

v=0

 ln

s

v=0

clS =



l



 ln

8rs 2

(h + v · p) + (d − c)



8rl 2

(h + v · p) + d2

2

− 1.92 ;



− 1.92 ;

(A.1)

(h + v · p) as ; s (h + v · p)2 + (d − c)2 −

al (h + v · p) ; l (h + v · p)2 + d2

(A.2)

142

cllS =

Z. Lu et al. / Sensors and Actuators A 220 (2014) 134–143 N−1

2

s as

v=0 S css =

(h + v · p) − (d − c) 2

2

[(h + v · p) + (d − c) ]

; cllL = 2

M−1

2

l al

v=0

(h + v · p) − d2 2

[(h + v · p) + d2 ]

N−1 2 2

as (d − c) rs − (h + v · p) (rs − 2(c − d))

s

v=0 S cϕϕ =

2

·

2

N−1 2

as · (rl + d)

s

N−1

as sl s

v=0

2

(rs − (c − d)) · [(h + v · p) + (d − c) ]

v=0 S csϕ =

2

2

2

·

(h + v · p) − (d − c) 2

2

2

2

[(h + v · p) + (d − c) ]

L ; csϕ = 2

L ; cϕϕ = 2

M−1

v=0

M−1

al

l

v=0

2

[(h + v · p) + (d − c) ]

(rl + d)(h + v · p)(c − d)

L ; css = 2

2

2

(A.3)

;

2

·

d2 rl − (h + v · p) (rl + 2d) 2

(rl + d) · [(h + v · p) + d2 ]

M−1 2

al · (rl + d)

l

v=0

− al sl l

2

(h + v · p) − d2 2

[(h + v · p) + d2 ]

[(h + v · p) + d2 ]

where 0 = 4 × 10−7 H/m is the magnetic permeability of vacuum, as = 0 rs , al = 0 rl , rs and rl are radii of the stabilization and levitation coil, respectively, p is the coils pitch of winding, c = rs − rl , rpm is the radius of the PM, d = rpm − rl , N and M are numbers of winding turns for stabilization and levitation coil, respectively,  sl ,  s and  l are coefficients of the similarity. Coefficients of the similarity are introduced to rearrange the real contribution of each coefficients of the quadratic forms and adapt model (1) to the real condition of the suspension operation. This becomes necessary due to in reality the induced eddy current is distributed along the PM surface and this distribution is frequency dependent. However analytical model (1) is obtained by considering the suspension as the lumped system [14]. Appendix B. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.sna.2014.09.017. References [1] C. Shearwood, C. Williams, P. Mellor, R. Yates, M. Gibbs, A. Mattingley, Levitation of a micromachined rotor for application in a rotating gyroscope, Electron. Lett. 31 (21) (1995) 1845–1846. [2] C. Williams, C. Shearwood, P. Mellor, A. Mattingley, M. Gibbs, R. Yates, Initial fabrication of a micro-induction gyroscope, Microelectron. Eng. 30 (1–4) (1996) 531–534. [3] R. Yates, C. Williams, C. Shearwood, P. Mellor, A micromachined rotating gyroscope, in: IEE Colloquium on Silicon Fabricated Inertial Instruments (Digest No: 1996/227), pp. 4/1–4/6, 1996. [4] C. Williams, C. Shearwood, P. Mellor, Modeling and testing of a frictionless levitated micromotor, Sens. Actuators A: Phys. 61 (1997) 469–473. [5] W. Zhang, W. Chen, X. Zhao, X. Wu, W. Liu, X. Huang, S. Shao, The study of an electromagnetic levitating micromotor for application in a rotating gyroscope, Sens. Actuators A: Phys. 132 (2) (2006) 651–657. [6] C. Shearwood, K. Ho, C. Williams, H. Gong, Development of a levitated micromotor for application as a gyroscope, Sens. Actuators A: Phys. 83 (1–3) (2000) 85–92. [7] N.-C. Tsai, W.-M. Huan, C.-W. Chiang, Magnetic actuator design for single-axis micro-gyroscopes, Microsyst. Technol. 15 (2009) 493–503. [8] K. Kratt, V. Badilita, T. Burger, J. Korvink, U. Wallrabe, A fully MEMS-compatible process for 3D high aspect ratio micro coils obtained with an automatic wire bonder, J. Micromech. Microeng. 20 (2010) 015021. [9] V. Badilita, S. Rzesnik, K. Kratt, U. Wallrabe, Characterization of the 2nd generation magnetic microbearing with integrated stabilization for frictionless devices, in: 2011 16th IEEE International Solid-State Sensors, Actuators and Microsystems Conference (TRANSDUCERS), 2011, pp. 1456–1459. [10] K. Poletkin, C. Shearwood, A. Chernomorsky, U. Wallrabe, Micromachined contactless suspensions, in: Technologies for Smart Sensors and Sensor Fusion, Devices, Circuits, and Systems, CRC Press, 2014, pp. 211–235. [11] K.V. Poletkin, A.I. Chernomorsky, C. Shearwood, A proposal for micromachined accelerometer base on a contactless suspension with zero spring constant, IEEE Sens. J. 12 (07) (2012) 2407–2413, http://dx.doi.org/10. 1109/JSEN.2012.2188831. [12] K.V. Poletkin, A.I. Chernomorsky, C. Shearwood, A proposal for micromachined dynamically tuned gyroscope based on contactless suspension, IEEE Sens. J. 12 (06) (2012) 2164–2171, http://dx.doi.org/10.1109/JSEN.2011.2178020.

;

(A.4)

2

·

(rl + d)(h + v · p)d 2

2

2

,

2

;

(A.5)

(A.6)

[13] K. Poletkin, A.I. Chernomorsky, C. Shearwood, U. Wallrabe, An analytical model of micromachined electromagnetic inductive contactless suspension, in: The ASME 2013 International Mechanical Engineering Congress & Exposition, ASME, San Diego, CA, USA, 2013. [14] K. Poletkin, A. Chernomorsky, C. Shearwood, U. Wallrabe, A qualitative analysis of designs of micromachined electromagnetic inductive contactless suspension, Int. J. Mech. Sci. 82 (2014) 110–121, http://dx.doi.org/10.1016/j.ijmecsci. 2014.03.013 http://authors.elsevier.com/sd/article/S0020740314000897 [15] F. Grover, Inductance Calculations: Working Formulas and Tables, Dover Publications, Chicago, 2004. [16] Z. Lu, F. Jia, J. Korvink, U. Wallrabe, V. Badilita, Design optimization of an electromagnetic microlevitation system based on copper wirebonded coils, in: 2012 Power MEMS, Atlanta, GA, USA, 2012, pp. 363–366.

Biographies

Zhiqiu Lu received the B.Sc and M.Sc degrees from the Institute of Microelectronics, Peking University, Beijing, China, in 2007 and 2010, respectively. He is currently working toward the Ph.D. degree at the Department of Microsystems Engineering, University of Freiburg, Freiburg, Germany. His research interests include microfabrication, RF MEMS, and magnetic actuators.

Kirill Poletkin received the diploma (Hons.) of electromechanical engineer majoring in Aviation devices and measuring computing complexes in 2001 from Arzamas branch of Nizhny Novgorod State Technical University and the Ph.D. degree from Moscow Aviation Institute (State University of Aerospace Technologies), Russia in 2007. Currently, he is Humboldt Senior Research Fellow in the Laboratory for Microactuators, Department of Microsystems Engineering - IMTEK, University of Freiburg. He was previously with Nanyang Technological University, Giesecke & Devrient GmbH (G&D), JSC Temp-Avia, Russian Federal Nuclear Center (VNIITF). His research interests include micro- and nano-scales levels devices and processes of the energy transfer within these scales.

Bartjan den Hartogh born in 1981, holds a master degree in Material Science of ETH Zurich. After the experience of foundation and trade sale of his own start-up company he joined FemtoTools is 2013, where he started as Application engineer and now heads international sales.

Z. Lu et al. / Sensors and Actuators A 220 (2014) 134–143

Ulrike Wallrabe studied physics at Karlsruhe University, Germany. In 1992 she received her PhD degree for mechanical engineering of microturbines and micromotors. From 1989 to 2003 she was with the Institute for Microstructure Technology at Forschungszentrum Karlsruhe (today KIT) working on microactuators and Optical MEMS. She holds a Professorship for Microactuators at the Department of Microsystems Engineering, IMTEK, at the University of Freiburg, Germany, since 2003. In the year 2010 she was granted an internal fellowship at the Freiburg Institute of Advanced Studies, FRIAS. Ulrike Wallrabe has published more than 110 papers in the field of microsystems technology. Her work focus lies in magnetic microstructures including processes for magnetic materials and micro coils, in adaptive optics, using piezo actuators to tune elastic lenses and mirrors, and in micro energy harvesting. Since 2012 she is a member of the Cluster of Excellence BrainLinks-BrainTools.

143

Vlad Badilita studied at the University of Bucharest, Romania, where he obtained his BSc (1997) and MSc (1999) degrees, and at the École Polytechnique Fédérale de Lausanne (EPFL), Switzerland where he obtained his PhD degree (2004) in micro-opto-electronics with a thesis focused on the physics of coupled-cavity surface emitting lasers (CC-VCSELs). In 2007, after two years as postdoctoral research associate at the University of Maryland at College Park, USA, Dr. Badilita joined the University of Freiburg in Germany as a group leader for magnetic microsystems. His research interests span the broader area of miniaturized electromagnetic devices with a focus on electromagnetic actuators and magnetic resonance detectors for sample- and volume-limited samples. Dr. Badilita kindly acknowledges support from the German Research Foundation (DFG) through project number BA 4275/2-1.