JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. …webee.technion.ac.il/Sites/People/YaelNemirovsky/Download/Publications/Journal...tensive parameter and the appropriate magnetic intensive

JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 14, NO. 6, DECEMBER 2005 1253

A Methodology and Model for the Pull-InParameters of Magnetostatic Actuators

Yael Nemirovsky, Fellow, IEEE, I. Zelniker, Ofir Degani, Student Member, IEEE, and Gilaad Sarusi

Abstract—Magnetostatic actuators exhibit bistability similarlyto the Pull-In phenomena of electrostatic actuators. In this paper amethodology and model for the extraction of the magnetic Pull-Inparameters of magnetostatic actuators are derived. The flux-con-trolled magnetostatic actuator is analyzed based on the energy rep-resentation and the magnetomotive force-controlled magnetostaticactuator is analyzed in the thermodynamic potential energy (orco-energy) representation. An algebraic equation, referred to asthe magnetic Pull-In equation, for each of the two cases (flux-con-trolled and magnetomotive force-controlled actuators) is derived.By solving these Pull-In equations either analytically or numeri-cally, the magnetic Pull-In parameters are obtained. Several casestudies, covering displacement and torsion magnetic actuators, arepresented and analyzed, illustrating the usefulness of the proposedmethodology, its relative simplicity as well as the adaptability andpractical usage in wide spectrum of magnetic actuators. [1425]

Index Terms—Actuators, magnetic, magnetostatic, model,pull-in.

I. INTRODUCTION

MAGNETOSTATIC actuators are among the oldest typesof actuators and are widely used. In contrast to electro-

static actuators, magnetostatic actuation offers the possibilityof generating repulsive forces in addition to attractive forces.Moreover, magnetostatic actuators may be robust and more ca-pable of producing large forces. However, implementation ofmagnetostatic actuators in MEMS technologies and on a sil-icon chip is much more difficult while electrostatic actuatorsare easily integrated on a chip, are easily controlled and con-sume little power [1], [2]. As a result, MEMS electrostatic ac-tuators are much more prevalent and are extensively studied inthe literature [[3], [4] and references therein]. Nevertheless, be-cause of the many possible applications and market require-ments, MEMS magnetostatic actuators are currently drawingmuch attention [5]–[20].

Electrostatic actuators often exhibit inherent instability,known as Pull-In [3], [4], which can be either harmful or useful,depending on the application. It was shown that magnetostaticactuators may exhibit similar inherent bistable phenomenon[19], [20]. This bistability is used, for example, to obtain anoptical valve that latches in the open position and requirespower only to switch between the two stable positions. Inaddition to the interesting applications, it is important to model

Manuscript received September 9, 2004; revised March 5, 2005. SubjectEditor L. Lin.

The authors are with the Kidron Microelectronics Research Center, Depart-ment of Electrical Engineering, Technion—Israel Institute of Technology, Haifa32000, Israel (e-mail: [email protected]).

Digital Object Identifier 10.1109/JMEMS.2005.859074

in a unified approach the conditions leading to bistability inmagnetostatic actuators.

In this paper, we analyze and model the bistability of mag-netostatic actuators, which we refer to as magnetic Pull-In phe-nomenon similarly to electrostatic actuators, using a general-ized approach. In [4], the authors presented a methodology andmodel for the Pull-In parameters of electrostatic actuators basedon a unified (thermodynamic) approach [21]. This approach isextended here to magnetostatic actuators. The flux-controlledmagnetostatic actuator is analyzed based on the energy rep-resentation, and the total energy (“fundamental equation”, see[21]) is derived using the appropriate magnetic extensive param-eters, which are defined here. The magnetomotive force-con-trolled magnetostatic actuator is analyzed in the co-energyrepresentation, (“thermodynamic potential”, see [21]) and thetotal co-energy equation is derived, using the mechanical ex-tensive parameter and the appropriate magnetic intensive pa-rameter. Following the thermodynamic stability criteria [21], itis shown that the magnetic Pull-In point is achieved when thefirst and second derivatives of the energy (or potential energy)equation with respect to the mechanical extensive parameterare simultaneously zero, similarly to the electrostatic Pull-Inpoint. Finally, an algebraic equation, referred to as the magneticPull-In equation, for each of the two cases (flux-controlledand magnetomotive force-controlled actuators) is derived. Bysolving these Pull-In equations either analytically or numeri-cally, the magnetic Pull-In parameters are easily obtained.

Several case studies, illustrating the usefulness of the pro-posed methodology for displacement as well as torsion actu-ators, are presented and the magnetic Pull-In parameters arederived.

II. MODELING OF MAGNETOSTATIC ACTUATORS

A. The Single Source Magnetostatic Actuator System andDefinitions

Consider a magnetostatic actuator composed of a single coilwith turns driven by a current source , which is the sourceof the magnetic energy (see Fig. 1 for an example of such anactuator). A ferromagnetic core with high permeability is usedto conduct the magnetic flux and energy between a fixed partand a mechanical movable part, which are separated by variableair gap. The movable part is suspended by a mechanical springexerting a restoring force on it. The spring restoring force maybe either linear or nonlinear with the displacement. For zerocurrent or magnetic flux, the initial gap between the fixed andmovable parts is denoted by . The movable part is assumed

1057-7157/$20.00 © 2005 IEEE

1254 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 14, NO. 6, DECEMBER 2005

Fig. 1. Schematic description of a parallel magnetic actuator. The actuator suspended member (slider) moves along a single axis, x, which is perpendicular tothe plane of the member. The overlap area A is fixed and the gap, between the movable part and the electromagnet, is affected by the actuation. The initial gap xbecomes x � x. The total length of the core is l . The magnetic, F and mechanical, F forces are also shown.

to move along a specific trajectory and the coordinate along thistrajectory, i.e., the generalized displacement, is denoted by .

The magnetic flux conducted in the ferromagnetic core is de-noted by . In order to minimize the resistance to the flow ofthe magnetic flux, a generalized force is exerted on the mov-able part, which tends to reduce the air gap. The mechanicalrestoring force establishes a stable equilibrium that is character-ized by a new gap . As we increase the flux, , eitherby increasing the magnetomotive force (MMF), ,in the MMF-controlled configuration or by increasing the fluxdirectly in the flux-controlled configuration, the magnetostaticforce increases and the gap decreases to a new valuewhere . At some critical point, which we refer to as thePull-In point, the system ceases to be stable. As a result of themagnetostatic force of attraction, the gap collapses to zero andthe fixed and movable parts adhere. This is equivalent to thewell-known Pull-In phenomenon observed in electrostatic actu-ators. We are interested in modeling the equilibrium states of thesystem as well as to determine the magnetostatic Pull-In param-eters, denoted by and or , for MMF-controlledor flux-controlled devices, respectively.

Since the process under study is quasistatic, the dampingprocess and any other dissipative processes are ignored. Thus,the energy of the source is converted into mechanical andmagnetostatic energy, denoted respectively, by and andstored, respectively, in the mechanical part and the magneticpart. denotes the energy of the source, which is assumed tobe much larger than and and is therefore, practicallyconstant.

In contrast to electric terms, such as voltage, current, capaci-tance, which are familiar because they are used in everyday ac-tivities, this is not true of magnetic terms, and in Appendix I,

we therefore define all the magnetic terms that are used in theanalysis. We apply the magnetic circuit concept, which is usefulbecause of the analogy to electric circuit [22]–[26]. The analogybetween magnetic and electric actuators is also presented inAppendix I.

B. Analysis of the Flux-Controlled Magnetostatic Actuator

In this case, we assume that magnetic flux is directly con-trolled. Similar to the case of charge-controlled electrostatic ac-tuators [4], the implementation of flux-controlled magnetostaticactuators is complicated, but for the sake of completeness it isincluded.

1) The Energy Representation of the Flux-ControlledMagnetostatic Actuator: We assume that the system is in equi-librium and can be analyzed on the basis of the fundamentalequation [21], which gives the total energy as a function of theextensive parameters of the system. We focus only on the exten-sive parameters that change, namely the displacement and themagnetic flux and accordingly . Furthermore, sincethe actuator is composed of a mechanical part and a magneticpart while the energy of the source is practically constant, thetotal energy is rewritten aswhere is the energy of the mechanical part while

is the energy of the magnetic part. The differentialform of the fundamental equation is given by

(1)

NEMIROVSKY et al.: METHODOLOGY AND MODEL FOR THE PULL-IN PARAMETERS OF MAGNETOSTATIC ACTUATORS 1255

The intensive parameters are defined by the first partialderivatives of the fundamental equation and accordingly,

where is a generalized force and theminus sign is added in accordance with the definition of themechanical term of a simple system, where the work is takenpositive if it increases the energy of the system [21]. Thus, theminus sign indicates that the generalized force acts to decreasethe energy of the system. If the single degree of freedom is alinear displacement , then is a force. If the single degreeof freedom is an angular displacement , then is a torque,defined by the product of force and distance.

Similarly, we define the forces for the two subsystems,namely —the mechanical force exertedon the mechanical part and —the mag-netostatic force exerted by the magnetic part (see Fig. 1 fordirections). In addition, we define

where is the magnetomotive force, namelythe line integral of the magnetic field around the magneticcircuit (see Appendix I, Table II).

The second derivatives of the fundamental equation are alsoof physical significance and in particular

(2)

where is the reluctance of the magnetic circuit [22], [23],[Appendix I, Table II].

In linear magnetic circuits, the above simplifies to. It should be noted that in the case of permanent mag-

nets, this is not correct and a differential definition is required.Similarly, the well-known relation for the generalized springconstant [3]

is obtained. The generalized spring constant differs from themechanical spring constant due to the contribution of thedisplacement dependent magnetic force. Based on (1) and (2),it is easily seen that

(3)

2) The Stability Criteria and the Critical Point Known asthe Pull-In Point for the Flux-Controlled Magnetostatic Actu-ator: The basic extremum principle of thermodynamics [21]implies that . Accordingly, for constant, the crite-rion of an equilibrium state is .

Three different states may emerge as follows:

i) the energy is a minimum, the system is stable and;

ii) the energy is a maximum, the system is unstable and;

iii) the energy is at a critical point, the system is at an inter-mediate state and .

Similarly to the case of electrostatic actuators, where the crit-ical state is known as the Pull-In point, we define the magneticPull-In point. It is defined by the Pull-In parameters, namely, thedisplacement and the flux at this point.

3) The Flux-Controlled Magnetic Pull-In Equation: In thiscase, the total energy is given by

(4)

The Pull-In point is an equilibrium point withcritical stability . The magnetostatic Pull-Inequation in the flux-controlled configurations is derived basedon these two equations. Thus, one can easily obtain the first andsecond derivatives of (4)

(5)

(6)

By eliminating from (5), (6), the magnetostatic flux-con-trolled Pull-In equation is derived

(7)

and the Pull-In flux is given by

(8)

Again, for the sake of clarity, it should be noted that the flux-con-trolled magnetostatic actuator is analog to the charge-controlledelectrostatic actuator, since in both cases we control the corre-sponding extensive parameter, i.e., and , respectively.

C. Analysis of the Magnetomotive Force MMF-ControlledMagnetostatic Actuator

The MMF-controlled magnetostatic actuators are character-ized by a closed current source, usually a solenoid with a ferro-magnetic core, which controls the magnetomotive force

[22], [23], [25], [26]. In this case, the magnetomotive force,which is an intensive parameter as shown in Section II-B1, isdirectly controlled. This is often encountered in practical exper-iments, where the intensive parameters (such as temperature orvoltage) are the more easily measured and controlled than thecorresponding extensive parameters (entropy, charge). Accord-ingly, we need to recast the analysis in terms of ratherthan .

1) The Co-Energy Representation of the MMF-ControlledMagnetostatic Actuator: Following the established procedureof Legendre Transformations [21], we define an appropriatethermodynamic potential [3]. The thermodynamic potential issimilar, up to a minus sign, to the more widely used conceptof co-energy [3]. By following the themodynamic potential for-malism, the signs of the forces are correctly derived and the


same sign convection applies to all cases (flux controlled orMMF-controlled).

In the thermodynamic potential representation, which is equalto minus the co-energy

(9)

and hence

(10)

From Table I, it is readily seen that

(11)

In this case, the total potential energy, which is equal to minusthe co-energy, is given by

(12)

2) The Stability Criteria and the Critical Point Known asthe Pull-In Point for the MMF-Controlled Magnetostatic Ac-tuator: The thermodynamic potential (or minus the co-energyfunction) is useful since the criteria for equilibrium and stabilityare retained in this representation [21]. Accordingly, forconstant, the MMF-controlled equilibrium states are defined by,

and the three different equilibrium statesare defined by

for a stable state;for an unstable state;for the critical point known as the

MMF-controlled Pull-In point, which is defined by thePull-In parameters, namely the displacement and thevoltage at this point.

3) The MMF-Controlled Pull-In Equation: Thus, one caneasily obtain

(13)

(14)

By eliminating from (13)–(14), the magnetostatic MMF-controlled Pull-In equation is derived

(15)

Moreover, the Pull-In magneto-motive force is given by

(16)

It should be noted that the MMF-controlled magnetostatic ac-tuator is analog to the voltage-controlled electrostatic actuator,since in both cases we control the corresponding intensive pa-rameter, i.e., and , respectively.

D. Extending the Flux-Controlled and MMF-ControlledPull-In Equations to Multisource Magnetostatic Actuators

As it was mentioned earlier in this paper, the Pull-In equations(7) for the flux controlled actuators and (15) for the MMF-con-trolled actuators were derived under single source system as-sumption. While extending the problem to the multisource sys-tems these equations are not valid anymore. Therefore (4) for theflux controlled actuator and (12) for the MMF-controlled actu-ators are modified by the superposition approach to include thecontribution of all energy sources to the total energy (or minusco-energy). Thus, a direct solution of modified (5) and (6) for theflux controlled actuators and (13) and (14) for the MMF-con-trolled actuators is needed. In some cases there is no analyt-ical solution for these equations available due to complexity ofthe obtained expressions that may occur. Thus, a numerical so-lution is needed. Given a specific actuator with specified geo-metric configuration, fixing magnetic sources ( is atotal number of magnetic sources), only two parameters remainvariable—the displacement and flux of the nonfixed source inthe flux controlled actuators, and displacement and of thenonfixed source in MMF-controlled actuators.

Plotting both modified (5) and (6) for the flux controlled actu-ators and (13) and (14) for the MMF-controlled actuators (usingsymbolic tool of MATLAB, for example) on the same graph theintersection point of two graphs may occur. An intersection in-dicates when Pull-In occurs, and the Pull-In parameters are ob-tained from the graph. If it does not appear—there is no Pull-Inin the actuator.

III. CASE STUDIES

In what follows several case studies of magnetostatic ac-tuators are discussed in order to apply the above formalismto practical designs of magnetostatic actuators. The Pull-Inparameters of four case studies of magnetostatic actuatorsare derived and discussed. The first case study examinesparallel magnetostatic actuator with linear displacement, avariable gap and fixed overlap area for the two cases: 1) theMMF controlled magnetostatic actuator, which is similar tothe voltage-controlled electrostatic parallel-plate actuator,2) the flux-controlled magnetostatic actuator, which is similarto the charge-controlled electrostatic parallel-plate actuator.The second case study examines a sliding slab of ferromagneticmaterial into the gap of a solenoid, where the gap is fixedand the overlap area varies. The third case study examines atorsion-like actuator controlled by MMF. The last case studyexamines the MMF controlled torsion magnetostatic actuator,which includes two energy sources.

A. The Parallel Magnetostatic Actuator WithLinear Displacement

1) The MMF-Controlled Parallel Magnetostatic Actuator:Fig. 1 exhibits a schematic view of the parallel-membersmagnetostatic actuator configuration. This actuator is the mag-netostatic equivalent of the electrostatic parallel-plate actuator.


Fig. 2. Equivalent magnetic circuit of Fig. 1. V = N � i (where N is thenumber of coil turns) and R ;R are reluctance of the core and the gaps,respectively (both are defined in the text.).

Fig. 2 exhibits the equivalent magnetic circuit. Assuming thatthe nominal gap, is much smaller than, , the reluctanceof the air-gap is given by ([22, Table I], [23]–[26])

(17)

where is initial gap between members and is a cross-sec-tional area of the gap. Given that the total flux path length in thecore is , the reluctance of the core is

(18)

where and are the permeability and cross-sectionalarea of the core, respectively. Thus, the total reluctance apparentto the MMF source is given by

Using the last equation and (11) the magnetic co-energy is givenby

(19)

Assuming that the spring’s constant is , the mechanical energyof the spring is given by (linear spring assumption)

(20)

Using (12) and (19), (20), the total co-energy of the actuator isobtained:

(21)

Using the MMF-controlled Pull-In equation (15):

(22)

The solution of (22) yields the Pull-In point,

(23)

The Pull-In magneto-motive force is obtained by combining(16) and (23)

(24)

For most cases , therefore , thus,assuming (23) becomes

and (24) becomes

These results are similar to those obtained for the electro-static parallel-plate voltage controlled actuator [4]. Never-theless, the advantages of magnetostatic actuators becomeapparent when we compare practical currents for magnetostaticMMF-controlled actuator and practical voltages for electro-static voltage-controlled actuators that may produce the Pull-In.

For example, let us assume a magnetostatic parallel-memberMMF-controlled actuator, with the following specifications:

The electrostatic Pull-In voltage for a gap of 500 m is about200 V. From this example,the advantages of magnetostatic ac-tuators in the case of large gaps become evident.

2) The Flux-Controlled Parallel Magnetostatic Actuator: Inthis case, direct control on the flux is assumed. Using (3), themagnetic energy is derived

(25)

The expressions for , and are the same as for theMMF-controlled parallel magnetostatic actuator. The mechan-ical energy remains the same, thus, the total energy is given by(4)

(26)

Using the flux-controlled Pull-In equation (7)

(27)


Fig. 3. Schematic description of a sliding slab actuator. The actuatorsuspended member (slider) moves along a single axis, x, which is perpendicularto the plane of the member. The overlap area is A � (x=x ). It depends on theextent of penetration of the movable part and is affected by the actuation.

Fig. 4. Equivalent magnetic circuit of Fig. 3. V = N � i (where N is thenumber of coil turns) and R ;R ;R ; r are reluctances derived from Fig. 3(defined in the text).

From (27) it is obvious that there is no Pull-In in this config-uration (for both and ), and again the re-sult is similar to the electrostatic parallel-plate charge controlledactuator [4].

B. The Sliding Slab Actuator

1) The MMF-Controlled Sliding Slab Actuator [3]: Fig. 3exhibits a schematic view of the sliding slab magnetostaticactuator configuration. Fig. 4 exhibits the equivalent magneticcircuit.

Assuming that the nominal gap, is much smaller than ,the reluctances are given by ([22, Table I], [23])

(28)

When are permeability of the core and the slider, respec-tively. Thus the total reluctance apparent to the MMF source isgiven by (29) at the bottom of the page. Combining (29) and (11)the magnetic thermodynamic potential (or minus the co-energy)is derived

(30)

Assuming that the spring’s constant is , the mechanical co-en-ergy of the spring is the same as (20), and combining (30) and(20) the total co-energy expression is obtained by (31) at thebottom of the page. Using the MMF-controlled Pull-In equa-tion (15), the solution for the is given by

(32)

Assuming , (33) becomes:

(33)

(29)

(31)


Fig. 5. Schematic description of MMF-controlled torsion actuator. The actuator suspended member, beam, fixed at one side, while the other side is free to movealong a single polar axis, �. The overlap area and the gap between the movable part and the electromagnet are both functions of �, which is affected by the actuation.b is the width (in z direction) of the core and the beam (both are assumed to have same width), and l is the total length of the core.

Thus, for any it is shown that the solution is not physicalfor the ( cannot be negative). Hence, there is no Pull-Inpoint in this configuration. This case is similar to the case of theelectrostatic comb-drive actuator.

2) The Flux-Controlled Sliding Slab Actuator: Using (4),(29), and (20) the total energy for the problem is obtained by(34) shown at the bottom of the page. Using the flux-controlledPull-In equation (7), the solution for is given by

(35)

Thus, for any it is shown that the solution is not physicalfor the (cannot be negative). Hence, there is no Pull-In pointin this configuration.

C. The MMF Controlled Torsion Actuator

This group of actuators is based on torsion movement, andwidely described in literature (see [8], [11]–[13], [15], [17]).The configuration of the torsion actuator shown in Fig. 5 is sim-ilar to configuration of actuator presented in [11], with slight im-provements (mainly in obtaining expressions for the magneticreluctances), as well as finding the Pull-In point of the actuator.Fig. 5 exhibits a schematic view of the MMF controlled torsionactuator configuration. Fig. 6 exhibits the equivalent magneticcircuit.

Assuming that the initial distance, is much smaller than( is initial total area of the air-gap in X-Z plane) the re-

luctance of the air-gap is given by

Fig. 6. Equivalent magnetic circuit of Fig. 5. V = N � i (where N isthe number of coil turns) and R ;R are reluctances derived from Fig. 5(defined in the text).

while is calculated in Appendix II.

(36)

Therefore, the total reluctance of the magnetic circuit is

where (not function of ).The mechanical energy is given by [4]

(37)

where

(34)


Fig. 7. Schematic description of MMF controlled torsion actuator. The actuator suspended member moves along a single polar axis, �. The overlap area and thegap between the movable part and the electromagnet are both functions of �, which is affected by the actuation. Since both parts have attractive forces (independentof current direction), the part with the higher current has stronger attractive force than the other. Thus the suspended part is tilted toward the coil with the highercurrent and/or more coil turns (here, in this example we assume the same turns number for each coil). b is the width (in z direction) of each core and l is a totallength of each core.

Thus, the total co-energy expression is obtained

(38)

Solving the Pull-In equations analytically is somewhat diffi-cult, since no close solution can be obtained. Using a numer-ical solver (MATLAB) to solve (15), Pull-In point can be found.Neglecting the reluctance of the core , the Pull-In pointhappens at , where .Again, this result is similar to the result obtained for the elec-trostatic torsion actuator [4]. For example, let us assume mag-netostatic MMF-controlled torsion actuator with the followingspecifications:

Solving in MATLAB these results obtained

This example exhibits the usefulness of magnetostatic actuators.

D. The MMF Controlled Bidirectional Torsion Actuator

This group of actuators is based on angle bi-directional move-ment, and described in literature ([6], [7], [10], [20]). Here,the bidirectional torsion actuator is discussed, omitting the per-manent magnet (due to difficulties in analytical analysis, seeSection II-B1).

Fig. 8. Equivalent magnetic circuit of Fig. 7. V = N � i ; V =N � i (where N is the number of coil turns (the same N assumed for eachcoil)) and R ;R ;R (�); R (�); R (�);R (�) are reluctancesderived from Fig. 7 (defined in the text).

This actuator is similar to the torsion actuator of Fig. 5, sincein both cases the degree of freedom is an angle . However, incontrast to the torsion actuator, where only one end of the beamis free to move, here both ends may be displaced and the axis ofrotation is in the middle of the beam. Fig. 7 exhibits a schematicview of the MMF-controlled bi-directional torsion actuator con-figuration. Fig. 8 exhibits the equivalent magnetic circuit. Theanalysis presented below is based on Fig. 7, where only two elec-tromagnets are used. This analysis can be extended for an arrayof electromagnets, as presented in [6], [7]. In this case dif-ferent magnetic circuits exist, but calculations of total reluctancefor each circuit is exactly the same. Assuming that the initial gap,

is much smaller than ( is initial total area of the air-gapin X-Z plane) the reluctance of the air-gap is given by

while is calculated in Appendix II.For the right part

(39)


Fig. 9. Graphical solution for a Pull-In point in a bidirectional torsion actuator, with two sources.

For the left part

(40)

So the total reluctance for the right part is

(41)

and for the left part

(42)

TABLE IELECTRIC AND MAGNETIC CIRCUIT ANALOGIES [22], [23]

where are reluctances of the right core and theleft core, respectively (both are independent on ), as defined in(18);

Combining (37), (41), (42) the total co-energy is obtained


TABLE IIA COMPARISON OF PARAMETERS AND TERMS, WHICH ARE REQUIRED TO ANALYZE ELECTROSTATIC AND MAGNETOSTATIC ACTUATORS [22], [23]

(43)

As it was mentioned in Section II-D, this actuator has a multipleenergy sources. Therefore (15) is not applicable in this case.

Given specific actuator with specified geometric configura-tion, and fixing one of the currents in one of the coils and con-trolling the other, the expression for the total co-energy nowhas two variable parameters—the current and the angle. Plot-ting (13) and (14) on the same graph using MATLAB Symbolictool, intersection point can be observed. This is the Pull-In point.Thus, and are simultaneously obtained.

For example, let us assume magnetostatic MMF-controlledtorsion actuator with the following specifications:

Plotting in MATLAB (13) and (14) with these parameters on thesame graph, the plot of Fig. 9 is obtained. Zooming the relevantarea on Fig. 9, these results are obtained

It is obvious from this example, that is strongly dependenton the values of the induced currents in the other coil.

IV. SUMMARY

Magnetostatic actuators exhibit bistability similarly to thewell-known Pull-In phenomena of electrostatic actuators. Inthis paper, a model for the Pull-In parameters of a general mag-netostatic actuator is derived using a methodology based onthermodynamic formalism. Two types of controlled operationare presented, the magnetomotive force-controlled actuatorand the flux-controlled actuator. The difference between thetwo types of controlled operations are defined using a rigorousthermodynamic analysis: the energy representation for theflux-controlled case and the thermodynamic potential energy(which is equal up to a minus sign to the co-energy representa-tion) for the magnetomotive force-controlled case. The analysisresults in two general algebraic equations (one for each input


control type), referred to as the magnetic Pull-In equations,which directly yield the Pull-In parameters. The equationsare general and applicable for any magnetostatic actuator,regardless of its shape or the number of mechanical degrees offreedom. A number of case studies of magnetostatic actuatorsare presented, exhibiting the straightforward calculations of themagnetic Pull-In parameters using the methodology presentedhere. Examining the Pull-In equation, it is clearly seen that onlytwo functions are needed in order to calculate the Pull-In point,the mechanical force and reluctance. These functions can beexpressed either analytically or can be calculated numerically.The case studies presented here are mainly based on analyticalsolutions, but the methodology can be extended in order toallow fast and direct calculations of the Pull-In parametersin cases where either the mechanical force or reluctance, arecalculated numerically.

APPENDIX

THE ANALOGY BETWEEN ELECTROSTATIC AND

MAGNETOSTATIC ACTUATORS

It is well known that there are certain analogies between qua-sistatic electric and magnetic fields as well as between electricand magnetic circuits [22], [23]. Table I summarizes the analo-gies between electric and magnetic circuits as well as all the pa-rameters used in the analysis of magnetostatic actuators. Table IIsummarizes the relevant electrostatic and magnetostatic field re-lations, parameters and terms required for the Pull-In analysisof the actuators. In Table II, the first column describes the na-ture of the relation, the second column gives the relation for theelectrostatic field, and the third column gives the correspondingrelation for the magnetostatic field.

The extensive parameter appropriate for magnetostatic ac-tuators is the magnetic flux [Webers] while B [Webers/m ]is the magnetic flux density. The permeability of the materialis denoted by [henrys/meter]. To eliminate the effect of themedium, the magnetic field intensity is defined (for linear fer-romagnetic materials only) where [Ampers/meter]measures the tendency of moving charges to produce flux den-sity; while the actual value of B produced depends on the per-meability of the medium.

It is well known that the line integral of field intensityalong any closed path is just equal to the current linked

. We may note an analogy of to(the electric field): the line integral around a closed path of thetotal electrostatic field intensity yields the electro-motiveforce (EMF) or voltage , while the line integral ofaround a closed path yields the magneto-motive force (MMF).The quantity is called the magnetomotive force(MMF) because in magnetic circuits it plays a role analogousto the electromotive force in electric circuits.

APPENDIX

THE CALCULATIONS OF USED FOR (3.3), (3.4)

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Yael Nemirovsky (SM’84–F’99) received the B.Sc.and D.Sc. degrees in chemistry from the Tech-nion–Israel Institute of Technology, Haifa, in 1966and 1971, respectively. WhenShe graduated fromTechnion in chemistry and her D.Sc. thesis was inelectrochemistry.

She joined the Department of Electrical Engi-neering at the Technion in 1980. Prior to that shewas a Research Scientist specializing in microelec-tronics, Rafael, Israel, a national R&D organization.She is a pioneer of microelectromechanical systems

(MEMS) research in Israel, and in the last decade, her research has focusedon microoptoelectromechanical (MOEMS) systems. She has published ap-proximately 170 papers in refereed journals, copresented over 200 talks inconferences and filed for several patents.

Dr. Nemirovsky is the recipient of an Israeli national award: “The Award forthe Security of Israel” and Technion awards for “Best Teacher” and “Novel Ap-plied Research.” She also received the Kidron Foundation award for “InnovativeApplied Research” and the USA R&D 100 2001 award recognizing the top 100new inventions and products of the year in USA. She has been a Fellow of theInstitution of Electrical Engineers (IEE–U.K.) since 1999.

I. Zelniker, photograph and biography not available at the time of publication.

Ofir Degani (S’97–M’04) was born in Ashkelon, Israel, on April 17, 1974.He received the B.Sc. degree in electrical engineering and the B.A. degree inphysics (both summa cum laude) in 1996, the M.Sc. degree in electrical engi-neering in 1999, and the Ph.D. degree in 2005, all from the Technion—IsraelInstitute of Technology, Haifa, Israel.

He is currently in the microsystems R&D group at RAFAEL, where he isinvolved in the development of micromachined inertial sensors and RF-MEMStechnologies and devices. He has published more than 40 journal and conferencepapers and filed several patents.

Dr. Degani is the recipient of the prestigious 2002 Graduate Student Fellow-ship from the IEEE Electron Devices Society and was awarded the Charles-Clore Scholarship by the Charles-Clore Foundation.

Gilaad Sarusi, photograph and biography not available at the time ofpublication.

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JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. …webee.technion.ac.il/Sites/People/YaelNemirovsky/Download/Publications/Journal...tensive parameter and the appropriate magnetic intensive