1772 ieee sensors journal, vol. 12, no. 6, june 2012...

1772 IEEE SENSORS JOURNAL, VOL. 12, NO. 6, JUNE 2012

Analysis of Nonlinear Phenomena in aThermal Micro-Actuator with a Built-In

Thermal Position SensorAli Bazaei, Member, IEEE, Yong Zhu, Member, IEEE, Reza Moheimani, Fellow, IEEE,

and Mehmet Rasit Yuce, Senior Member, IEEE

Abstract— An analysis of nonlinear effects associated with achevron thermal micro-actuator with a built in thermal positionsensor under static conditions is presented in this paper. Thenonlinearities present in both actuator and sensor are studied.The phenomena considered for the sensor include: thermalcoupling from actuator to sensor and temperature dependence ofelectrical resistivity. Those considered for the actuator include:non-uniform spatial distribution of temperature in arms, tem-perature dependency of thermal expansion coefficient, deviationof arm shape from straight line due to physical constraints, andtemperature dependence of electrical resistivity.

Index Terms— Analytical models, electrothermal positionmicrosensor, nonlinear analysis, thermoelectric microactuators.

I. INTRODUCTION

H IGH precision nanopositioning is a necessity in anumber of applications, including scanning tunneling

microscopy (STM) [1, 2], atomic force microscopy (AFM)[3], nano-metrology [4], biology [5], ultrahigh density probestorage system [6-8], and atom manipulations [9, 10]. Nanopo-sitioning in scanning probe microscopy (SPM) is traditionallyperformed by scanning stages in the form of tripods [11], tubes[12, 13], or flexures [14-16]. These nanopositioners typicallyhave high positioning accuracy with a large dynamic rangeand a wide bandwidth, enabling fast and robust closed-loopposition control [17]. Although macro-scale nanopositionerssuch as piezoelectric actuators can achieve nanometer-scalepositioning resolution and accuracy, they are relatively large,expensive, difficult to integrate in MEMS devices, and requirehigh voltage values [18, 19]. Microelectromechanical System(MEMS) nanopositioners have attracted increasing interest

Manuscript received June 10, 2011; revised October 27, 2011; acceptedNovember 15, 2011. Date of publication December 6, 2011; date of currentversion April 20, 2012. The associate editor coordinating the review of thispaper and approving it for publication was Prof. Ralph Etienne-Cummings.

A. Bazaei and S. O. R. Moheimani are with the School ofElectrical Engineering and Computer Science, University of New-castle, Callaghan 2308, Australia (e-mail: [email protected];[email protected]).

Y. Zhu is with the Griffith School of Engineering, Griffith University, GoldCoast 4222, Australia (e-mail: [email protected]).

M. R. Yuce is with the Department of Electrical and Computer Sys-tems Engineering, Monash University, Clayton 3168, Australia (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSEN.2011.2178236

recently due to their small size, low cost, fast dynamics and theemergence of applications such as probe-based data storage[20, 21]. Electrothermal MEMS actuators can provide largeforces at moderate voltage values. Hence, they have been usedin many applications, such as investigating live biological cells[22] and linear and rotary micromotors [23-26].

Recently, incorporation of a thermal sensing scheme wasreported in a MEMS electrostatic actuator [27] and a probe-based storage device [28, 29]. Micro-heaters were used tomeasure the motion of a MEMS micro-scanner with resolutionof less than 1 nm. Also, the positions of electrothermalactuators were precisely controlled in feedback loops by on-chip sensors that work based on piezoresistivity [30] or thermalvariation of electrical resistivity [31]. Compared to the capac-itive schemes, thermal actuation and sensing involve loweractuation voltages, smaller footprint, and less complicatedcircuitry and integration mechanisms [32].

Efficient design of micro actuators and sensors in MEMSand NEMS devices can be significantly improved if reliablemodels are provided in advance. As electrothermal sensingin MEMS has been under investigation for only a few years,few modeling work can be found in the literature. A singlebeam electrothermal sensor was modeled in [33]. However,There is a need to develop a better understanding of thethermal coupling from actuator to sensor and the differentialsensing scheme used in the on-chip sensing [31]. MEMS-basedelectrothermal actuation has been studied extensively [34,35]. However, in most cases, small strains and displacementare considered and the shear deformation of the beams isassumed to be negligible. These assumptions are not applicablein MEMS devices that undergo large displacements and arethus nonlinear, e.g. nanopositioning stages with large dynamicranges.

This work considers modeling of various nonlinear effectsin a chevron thermal micro-actuator with a built in thermalsensor. The nonlinearities are included in a step-by-step andcumulative manner, such that the effects of all nonlinear phe-nomena are considered simultaneously at last. The objectiveof this paper is to develop simple analytical methods thatinclude major nonlinear phenomena affecting the actuator andsensor under static conditions. Simulations are validated byexperimental results obtained from a MEMS nanopositionerequipped with electrothermal actuation and sensing.

1530–437X/$26.00 © 2011 IEEE

BAZAEI et al.: ANALYSIS OF NONLINEAR PHENOMENA IN A THERMAL MICRO-ACTUATOR 1773

Movable heat sink plate(Positioner stage)

R1

R2

Vdc

1

2

V

Act

uato

r

Vout

Fig. 1. Schematic diagram of the thermal position sensor with a differentialamplifier circuit.

Positionsensor R

1

Positionsensor R

2

Thermalactuator

1 2Vdc

Fig. 2. SEM images of the micromachined nanopositioner.

II. DESCRIPTION OF ELECTROTHERMAL

BASED NANOPOSITIONER

The conceptual schematic view of the nanopositioner ispresented in Fig. 1. The device is micro-fabricated from single-crystal silicon using a commercial bulk silicon micromachin-ing technology-SOIMUMP in MEMSCAP [36]. This processhas a 25 μm thick silicon device layer and a minimumfeature/gap of 2 μm. The Scanning Electron Microscope(SEM) image of the whole device and a section of it areprovided in Fig. 2. The position sensors are two beam-shapedresistive heaters made from the doped silicon. Application of afixed dc voltage across the heaters results in a current passingthrough them, thereby heating the beams. As a heat sink, arectangular plate is placed beside the beam heaters with a2 μm air gap. The positioner stage is actuated by a thermalactuator, as illustrated in Fig. 2.

Before applying a voltage across the actuator, the positionerstage is at the initial rest position (dashed box in Fig. 1),where the two edges of the sink plate are exactly alignedwith the middle of the two thermal resistive sensors R1 andR2. The sensors are biased by a dc voltage source Vdc,and the heat generated in the resistive heater is conductedthrough the air to the heat sink plate (positioner stage). Asthe plate is centered between the two thermal sensors, theheat fluxes out of the sensors are identical, thereby equalingthe temperature and resistance of the sensors. After applying avoltage on the actuator beams, the positioner stage is displacedtowards left, the heat flux associated with the sensor on theleft increases, while that associated with the sensor on theright decreases, resulting in a decrease in the resistance ofthe left sensor (R1), and an increase in resistance of theright sensor (R2). Thus, the displacement information of thepositioner stage can be detected by measuring the resistancedifference between the two sensors. The differential changesof the resistance result in current variations in the beam

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

Res

istiv

ity (Ω

. m

eter

)

× 10−5 Resistivity of n-type silicon (no = 1020 cm−3)

ResistivityA rough linear approximation

400 600 800 1000 1200 1400 1600Temperature (°K)

Fig. 3. Temperature profile of resistivity for n-type silicon (no = 1020cm−3).

resistors, and the currents are converted to an output voltageusing the trans-impedance amplifiers and an instrumentationamplifier. To suppress the common-mode noise, the gains ofthese two trans-impedance amplifiers must be well matched.Employing the differential topology allows the sensor outputto be immune from undesirable drift effects due to changes inambient temperature or aging effects.

III. SENSOR ANALYSIS

The objective of this section is to predict the sensor temper-ature and resistance as a function of sensor bias and actuatordisplacement. To simplify the analysis, lumped parameterapproach and constant stationary conditions are considered.This means we assume a uniform temperature distribution ineach sensing resistor.

A. Resistivity Temperature Profile

As the sensing resistors are made from highly doped Silicon,their resistivity is strongly dependent on their temperature.Since the resistivity of un-doped silicon is very high, theresistance of each sensing resistor is almost equal to that of thehighly doped layer. Let us assume that the depth of the highlydoped layer is 2μm with a uniform donor dopant concentrationof no = 1020cm−3. At room temperature (300K), the resistiv-ity model mentioned in [33] predicts an electric resistivity of1.56 × 10−5�m. Hence, the sensor beam can be consideredas a 100μm length bar whose cross section is a square with2μm sides. For this bar, the calculated resistivity yields a 390�resistance, which agrees with the experimental results (thefirst row of Table 2). With the selected dopant concentration,the resistivity model in [33] predicts a temperature profile forelectrical resistivity ρ, as shown in Fig. 3 with the thick curve.From this profile, the differential temperature coefficient ofresistivity, defined by:

α(T ) ≡ 1

ρ(T )

dρ(T )

d T, (1)

is obtained in terms of temperature T as shown in Fig. 4.The straight line in Fig. 3 is a rough linear approximation


× 10−4

−15

−10

−5

0

5(1

/°C

) Temperature coefficient of resistivity

200 400 600 800 1000 1200 1400 1600 1800Temperature (°K)

Fig. 4. Temperature profile of the differential temperature coefficient.

of resistivity for temperatures less than 1500K, which isdescribed as:

ρ(T ) ≈ ρamb [1 + αo(T − Tamb)] , (2)

where αo = 3.7822 × 10−4(1/°C) is the temperature coeffi-cient of resistivity, Tamb is the ambient temperature, and ρambrefer to resistivity at ambient temperature.

B. Linear Analysis

The results in following example, which is based on lumpedparameter analysis for prediction of temperature in terms ofthe applied voltage, will be used subsequently.

Example III-B: Electric resistance of a resistor at roomtemperature is Ro (�) and its variation with temperature canbe described by a constant temperature coefficient α (1/°C).The overall thermal conductance between the resistor and theoutside world, including its connection to a voltage source, isK (W/°C). If the internal resistance of the voltage source isnegligible compared to the resistor, what would be the steady-state temperature of resistor for a constant voltage of V (vlots)and room temperature of To?

Assuming a uniform temperature distribution T across theresistor, we can equate the electric power generated in theresistor with the thermal power flow from the resistor tothe outside world:

V 2

Ro[1 + α(T − To)] = K (T − To) (3)

When the constant α does not depend on temperature, theresistance value and its temperature can be formulated in thefollowing form:

R = Ro

1 +√

1 + 4αV 2

Ro K

2(4)

T = To +√

1 + 4αV 2

Ro K − 1

2α(5)

Displacement bythermal actuation

Platex

R1

V V

R2

x

Fig. 5. Biased thermal sensor.

(a)

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1.254

1.256

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ista

nces

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700

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710

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720

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pera

ture

(ce

ntig

rade

)

0.5 0.55 0.6 0.65x/L

(b)

0.5 0.55 0.6 0.65x/L

(c)

0.5 0.55 0.6 0.65x/L

T1

T2

R1/R

o

Ro(R

1−1 − R

2−1)R

2/R

o

0

1

2

3

4

5

6

7

Nor

mal

ized

sen

sor

outp

ut

× 10−3

Fig. 6. Steady-state values versus displacement. (a) Resistor temperatures.(b)Resistor values. (c) Normalized sensor output.

Fig. 5 shows the schematic diagram of the heat sink plateand the sensor, which is described by two resistances. At zeroactuation, variable x is equal to 0.5L with L referring tothe length of each sensor beam. In this way, lengths x andL − x of the right and left sensor beams are not adjacent tothe heat sink, respectively. Let us define K1 and K2 as theoverall thermal conductances from electric resistors R1 andR2 of the sensor to the area around the sensor, respectively.As the plate in Fig. 5 moves to the left, K1 increases whileK2 decreases. Let us assume a simple dependency between thethermal conductances and the plate position x , in the followingform:

K1(x̃) = Kmin + (Kmax − Kmin) x̃ (6.1)

K2(x̃) = Kmin + (Kmax − Kmin) (1 − x̃) (6.2)

where x̃ ≡ x/L is the normalized position, and Kmin and Kmaxrefer to minimum and maximum thermal conductance of eachresistor. Neglecting the thermal cross-coupling between thebeams, the results in Example III-B can be used to predictthe steady-state values of resistances and their correspondingtemperatures in the following form:

R i (x̃, V ) = Ro

1 +√

1 + 4αV 2

Ro Ki (x̃)

2, i ∈ {1, 2} (7.1)

Ti (x̃, V ) = To +√

1 + 4αV 2

Ro Ki (x̃) − 1

2α, i ∈ {1, 2} (7.2)


TABLE I

PARAMETER VALUES IN SIMULATION III-B

Parameter Valueα 3.7822 × 10−4 (K−1)

Ro 390 (�)

To 27 (°C)Kmin 100 × 10−6 (W/K)

Kmax 116 × 10−6 (W/K)

V 6 (V)

TABLE II

EFFECT OF THERMAL ACTUATION ON

UN-BIASED SENSOR RESISTANCES

Actuation Voltage (V) Displacement (μm) R1 (�) R2 (�)

0 0 388.9 391.61 0.1 389.4 391.92 0.7 391.3 3943 2 393.05 395.94 3.8 398 400.75 5.8 401.7 404.26 8 405.5 408.17 10.3 410.2 412.58 12.8 414.8 416.8

where Ro is each resistance value at zero bias voltage. At zeroactuation, where x̃ = 0.5 and K1 = K2 = (Kmax + Kmin)/2,we assume that the sensing resistors match and the sensoroutput is zero. Hence, the sensor output voltage is proportionalto R−1

1 − R−12 . This difference between the electric conduc-

tances, which changes with the distance x , can be employed tomeasure the displacement. The following simulation presentsthe predicted sensor output by the foregoing simple lumpedparameter analysis.

Simulation III-B: Assuming the following parameter val-ues and using (6) and (7), the steady-state values of tempera-tures, resistances, and senor output for different position valuesare obtained as shown in Fig. 6(a)-(c), respectively.

C. Thermal Coupling from Actuator to Sensor

The foregoing simulation predicts an almost linear depen-dency between the sensor output and displacement. Insection III-B, we assumed that the displacement does not haveany effect on the temperature of the area around the sensingresistors. In other words, we assumed that the temperaturearound the sensing resistors, denoted by To, is identical to theambient temperature Tamb. However, the displacement in theactuator is caused by thermal expansion. Hence, under station-ary conditions, the sensor area temperature (To) increases withthe displacement x . Due to lack of suitable instrumentationto measure the temperature of the sensor area, we set upthe following experiment to approximate the sensor areatemperature To.

Experiment III-C (Effect of Actuation on Un-biased Sen-sor Resistances): For different actuation voltages, the valuesof sensor resistances were measured, without applying anexternal bias voltage to the sensor, as shown in Fig. 7 andTable 2. For zero actuation voltage, we can assume that

Displacement bythermal actuation

x x

R1

Ω Ω

R2

Fig. 7. Measurement of un-biased sensing resistances for different thermalactuation voltages.

all parts are at room temperature and the values of sensorresistances are almost 390 �. It is observed that the un-biasedsensor resistances increase by about 6.4 percent for a 12μmdisplacement, just due to transfer of heat from actuator tosensor. �

With the temperature coefficient of sensing resistors as givenin Table 1, the 6.4 percent resistance increase in the un-biased sensor corresponds to 169.2 °C increase in sensor areatemperature. Since the edges of the plate align with the middleof sensing resistors at zero actuation, a zero displacementcorresponds to x̃ = 0.5. With L = 100μm, the sensor areatemperature can be roughly approximated in terms of thenormalized displacement and ambient temperature as:

To ≈ Tamb + 1410(x̃ − 0.5), (°C). (8)

Using (8) and the given temperature coefficient in Table 1,the value of unbiased sensing resistances in terms of displace-ment can be written as:

Ro ∼= Ramb[1 + 0.53(x̃ − 0.5)] (9)

where Ramb refers to sensor resistance at zero actuation and nobias voltage. Now, we can approximate the effect of thermalcoupling from the actuator to the biased sensor, in stationaryconditions. To do this, we replace To and Ro in (7) by theirposition dependent values on the right hand sides of (8) and(9), respectively. With the same parameters and ambient valuesas in Simulation III-B, we obtain the results shown in Fig. 8,where effect of no thermal coupling on sensor output is alsoincluded in Fig. 8(c) for comparison.

D. Effect of Nonlinear Temperature Coefficient

In the foregoing analysis, we assumed a linear temperatureprofile for electrical resistivity with a positive temperaturecoefficient. As shown in Fig. 3, the resistivity is a highlynonlinear function of temperature. Moreover, the differentialtemperature coefficient, shown in Fig. 4 (up to the meltingpoint of Silicon), becomes negative at high temperatures. If theeffect of the nonlinearity is to be taken into account, we cannot use closed-from solutions such as Eqs. (4), (5), and (7).Here, we use numerical solutions to consider the effect ofthe nonlinearity on the sensor under stationary conditions.


0.550.5 0.650.6700

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760

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800

820

840

860

880

900

Tem

pera

ture

(C

entig

rade

)

(a)

1.24

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ista

nces

0. 5 0.55 0. 6 0.650

1

2

3

4

5

6

7

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mal

ized

sen

sor

outp

ut

T1

T2

R1/R

0

Ramb

(R1−1−R

2−1)

R2/R

0

No coupling

With coupling

x/L0.550.5 0.650.6

x/L

(c)

x/L

×10−3

Fig. 8. Temperatures, resistances, and output of biased sensor in Simulation III-B, including effect of thermal coupling from actuator.

Let us assume that the resistivity has a known temperatureprofile, as depicted in Fig. 3. In this way, the heat transferequation in Example III-B can be written as:

Cd T

dt= f (T ) ≡ AV 2

Lρ(T )− K (T − To) (10)

where t refers to time, C is the heat capacity of the resistor,and L and A are its length and cross sectional area, which areassumed to be constants. For a constant bias voltage V , anequilibrium point of T = Te vanishes the right hand side ofEq. (10), denoted by function f (T ). To have an equilibriumpoint below the melting point Tm of silicon, the minimumvalue of function f in the interval of [To,Tm] needs to benegative. This condition can be stated as:

minT ∈[To,Tm] [ f (T )] < 0 (11)

Because of the nonlinear dependence of f on T , we mayhave more than one solution for the equilibrium. A stationarycondition, however, corresponds to a stable equilibrium. UsingEq. (10), the stability of equilibrium can be determined fromsmall temperature perturbation δ T ≡ T − Te, which satisfiesthe following linear dynamic equation:

Cd δ T

dt= −

(AV 2

Lρ2

dρ

d T+ K

)

T =Te

· δ T (12)

Since the heat capacity is positive, the equilibrium point isstable if the coefficient of δ T in Eq. (12) is negative. UsingEq. (1), we restrict the solutions to stable equilibrium pointsby imposing the following condition during the numericalcalculations:

AV 2α(T )

Lρ(T )+ K > 0 (13)

TABLE III

PARAMETER VALUES

Parameter Value

Ramb 390 (�)

Tamb 27 (°C)

Tm 1414 (°C)

Kmin 100 × 10−6 (W/K)

Kmax 116 × 10−6 (W/K)

V 6 (V)

Here, α has the temperature profile depicted in Fig. 4. Toconsider the effect of thermal coupling from actuator to sensor,the same approach which led to (8) is adopted. With thenonlinear profile in Fig. 3, a 6.4 percent increase in resistivitycorresponds to 263 °C increase in temperature. Hence, thesensor area temperature is approximated by the followingrelationship:

To(x̃) ≈ Tamb + 2192(x̃ − 0.5), (°C). (14)

Dependencies of the thermal conductances K1 and K2 onposition are assumed as stated in Eq. (6). The numericalsolutions for resistors’ temperatures are obtained from thefollowing equation:

AV 2

Lρ(Ti )− Ki (x̃)

[Ti − To(x̃)

] = 0, i ∈ {1, 2} (15)

Having determined the resistor temperatures from (15), theresistance values are obtained as:

Ri (x̃) = ρ(Ti )L

A, i ∈ {1, 2} (16)


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empe

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T1

T2

x/L

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ized

sen

sor

outp

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Ramb

(R1−1−R

2−1)

No coupling

With coupling

(c)

x/L

×10−3

0 5 10 150

0.5

1

1.5

2

2.5

3

Displacement (μm)

Sens

or o

utpu

t (V

)

(d)

ExperimentEstimate

Fig. 9. (a)–(c) Simulation results for stationary response of sensor, considering thermal coupling from actuator and a nonlinear temperature profile forsensitivity. (d) Experimental and predicted static sensor characteristics.

Using the parameter values in Table 3 and restricting thesolution to stable equilibriums below the melting point, weobtain the results shown in Fig. 9 under stationary conditions.Effect of no thermal coupling from actuator is also included inFig. 9(c). With a position dependency for sensor temperaturein the form of (14) and the selected values for parameters,conditions (11) and (13) reduce the permissible deflectionrange to x = 0.78L. This is expected because a larger deflec-tion is cause by a warmer actuator, which makes the sensorarea warmer due to thermal coupling. Our experimental dataassociated with sensor response versus displacement, whichis in the range of x̃ ∈ [0.5, 0.64], is in good agreement withthe predicted response in Fig. 9(c) within that range. Fig. 9(d)shows the predicted sensor-displacement characteristics along

with the measured data. An amplifier gain of 90 and feedbackresistance values of 330� were used to calculate the predictedsensor output in Fig. 9(d) from the normalized one in Fig. 9(c).

IV. THERMAL ACTUATOR MODELS

This part addresses modeling of the thermal actuator understationary conditions. We start from a simple linear model andthen a variety of nonlinear effects are included in a step-by-step manner.

A. Linear Model

For the linear model, the following assumptions areconsidered:


Actuatordisplacement

Thermallyexpanded arm

d

h

do

s � 0

s � La

s-axis

Lo

θo

θ

θ

Fig. 10. Schematic of one arm of actuator in the linear model.

1) The overall behavior of each arm of the actuator isidentical to the other arms.

2) The temperature profile along the beam is uniform.3) The temperature coefficient of expansion does not vary

with temperature.4) The beam profile remains linear during thermal expan-

sion.5) The temperature coefficient of resistivity is constant.6) The thermal conductance between each arm and the

surrounding environment is constant.

The first assumption considers the same electric current andtemperature profile for all 20 arms of the actuator. The secondassumption considers a uniform temperature distribution alongthe beam so that we can use the results of lumped parameteranalysis in Example III-B. By the fourth assumption, weassume that the beam remains a straight line after the thermalexpansion. In this way, the analysis can be reduced to that ofa rigid and linear single beam as shown in Fig. 10.

Using the first assumption and neglecting the voltage dropin the junction area between upper and lower arms, thevoltage V across each arm is half of the actuation voltage Va.Assuming a constant thermal conductance Ka between the armand the surrounding environment and using Eq. (5), the armtemperature in terms of the actuator voltage is obtained as:

Ta = Tamb +√

1 + αV 2a

Roa Ka− 1

2α(17)

where Roa is the electrical resistance of the arm at ambienttemperature (no actuation). Having found the arm temperatureand assuming a constant linear thermal expansion coefficientfor the arm defined by [37]:

αl = 1

Lo

d L a

d Ta(18a)

where Lo = √d2

o + h2 is the arm length at ambienttemperature (zero actuation voltage), the arm length La isobtained as:

La = Lo [1 + αl(Ta − Tamb)] (18b)

Vertical projection of arm, denoted by h in Fig. 10, remainsconstant after actuation. Hence, the actuator displacement d is

obtained in the following form.

d =√

L2a − h2 − do (19)

where do is horizontal projection of the arm at zero actuation.

B. Nonlinear Temperature Profile

In the linear model, we considered a uniform temperatureprofile for the beam, which is a very rough approximation.In [38], analytical expressions for temperature profile wereobtained for fixed micromachined polysilicon beams carryingelectrical current; however, the effect of thermal expansion wasnot considered and the equations are only valid for sufficientlyhigh current values. Another analysis was also presentedfor a bidirectional vertical thermal bimorph actuator in [39];however, no temperature dependency was considered for theelectrical resistivity. Under stationary conditions, temperatureprofile of the arm can be approximated by the following onedimensional heat equation:

I 2

Aeρ(T̃ ) + k A

d2T̃

ds2 = ga T̃ (20)

where T̃ = T (s) − Tamb is the temperature increase of thearm at position s on the beam (the s-axis coincides on thebeam as shown in Fig. 10), Ae is the effective cross sectionalarea through which the constant electric current I flows,k is thermal conductivity of the arm, ga refers to thermalconductance per unit length between the arm and ambientregion, A is the cross sectional area of the beam, and ρ is theelectrical resistivity. To solve the differential equation (20), weassume the following boundary conditions:

GL T̃ (La) + k AdT̃

ds

∣∣∣∣∣s=La

= 0 (21)

2k Ad T̃

ds

∣∣∣∣∣s=0

= G0T̃ (0) (22)

where GL refers to the overall thermal conductance betweenthe lower end of the arm in Fig. 10 and the ambient region,and G0 is the thermal conductance to ambient region for thejunction zone between the arm and its upper pair. In (22), acoefficient of 2 appears because two arms are connected to theaforementioned junction zone, where electric heat dissipationis neglected for simplicity.

Let us assume that the electrical resistivity can be approxi-mated by a linear temperature profile described by Eq. (2) andthe straight line in Fig. 3. In this case, analytical solutions forthe boundary value problem (20-22) are available. The form ofthe solution depends on the sign of the following parameter:

η := k−1 A−1(

I 2ρambαo A−1e − ga

)(23)

For positive values of η, the solution is presented as:

T̃ (s) = C1 sin(√

ηs) + C2 cos

(√ηs

) − γ

η(24)

whereγ := I 2ρambk−1 A−1 A−1

e , (25)


and the constants C1 and C2 satisfy the following linearalgebraic equations to meet the boundary conditions:

k A√

η[C2 sin

(√ηLa

) − C1 cos(√

ηLa)]

= GL

[C1 sin

(√ηLa

) + C2 cos(√

ηLa) − η−1γ

], (26)

2k A√

ηC1 = G0

(C2 − η−1γ

)(27)

For negative values of η, we can define β := √−η and thesolution is presented as:

T̃ (s) = C1 sinh (βs) + C2 cosh (βs) + γ

β2 (28)

where

−k Aβ [C1 cosh (βLa) + C2 sinh (βLa)]

= GL

[C1 sinh (βLa) + C2 cosh (βLa) + β−2γ

], (29)

2k AβC1 = G0

(C2 + β−2γ

)(30)

In the singular case of η = 0, the solution is presented as:

T̃ (s) = C2 + C1s − γ

2s2 (31)

where

k A (γ La − C1) = GL

(LaC1 + C2 − γ L2

a

/2)

(32)

2k AC1 = G0C2 (33)

So far, the solution for temperature profile is in terms ofthe current I and arm length La. However, the arm lengthafter actuation is not known in advance because it dependson temperature profile of the arm. This dependence can bewritten analytically if we partition the arm into infinitesimalsections si , i = 1, . . ., N , which have temperature increasesT̃i , i = 1, . . ., N , respectively. Assuming a constant thermalexpansion coefficient of αl and denoting the un-actuated lengthof si by soi , we can write a relationship similar to Eq. (18b)in the following form:

soi = si

1 + αl T̃i(34)

When the infinitesimal sections tend to zero while N tendsto infinity, a summation over i = 1, . . ., N , from both sides of(34) leads to the following constraint on profile solution:

s=La∫

s=0

ds

1 + αl T̃ (s)= Lo (35)

In this way, after choosing a solution for temperature profilebased on sign of η, the arm length La must be adjustedto satisfy constraint (35). The actuator displacement is thenobtained from Eq. (19).

C. Nonlinear Thermal Expansion Coefficient

In the aforementioned method, we assumed a constantvalue for coefficient of linear thermal expansion. However, thethermal expansion coefficient for Silicon considerably varieswith temperature, as shown in Fig. 11 [37].

300 400 500 600 700 800 900 1000 11002.5

3

3.5

4

4.5

Temperature (K)

Coefficient of thermal expansion for silicon

α1 (

× 10

−6· K

−1)

Fig. 11. Schematic approximate temperature profile of linear thermalexpansion coefficient for Silicon.

0 100 200 300 400 500 600 700 8000

5

10

15

20

25

30

35

u(°C)

g(u)

(×

10−4

)

Fig. 12. Temperature profile for integral of thermal expansion coefficient inFig. 11.

To take this variation into account we can use Eq. (18a) toobtain the modified form of Equation (34) as:

soi = si

1 + ∫ T̃i0 αl(T̃ )dT̃

(36)

In this way, constraint (35) can be written in the followingform:

s=La∫

s=0

ds

1 + g(

T̃ (s)) = Lo (37)

where function g(·) is defined as:

g(u) :=∫ u

0αl(T̃ )dT̃ (38)

For the thermal expansion coefficient profile in Fig. 11,function g has the profile shown in Fig. 12. In this way,after selecting a parameterized form for temperature profileT̃ (s, La) based on sign of η, the arm length La needs to beadjusted to satisfy constraint (37). Having determined the armlength La, the actuator displacement is obtained from Eq. (19).


Actuatordisplacement

Thermallyexpanded arm

and s-coordinate

d

h

ϕ

dos � 0

s � La

x � 0

y(x)

x � −0.5Lx

x � 0.5Lx

x-axis

Lo

θo

θo

Fig. 13. Nonlinear profile of arm.

D. Nonlinear Arm Profile (Buckling)

So far, we have assumed that the shape of the arm remains asa straight line after thermal expansion. However, experimentsshow that the arm can experience noticeable deviation fromstraight line in its shape, known as buckling. Buckling inthermal actuators was investigated in [40] by experimentand nonlinear structural finite-element simulations withoutconsidering any analytical approaches. Analytical solutionsfor buckling were also obtained in [41] and [42]; however,the beam conditions in [41] and the buckling source in[42] are different from those considered here. Our analyticalapproach for buckling is also suitable to include effects ofthe thermal profile nonlinearities associated with the thermalexpansion coefficient and temperature coefficient of resistivity.This problem is mainly due to slender structure of the arm,its clamped connections at the two ends, and fixed verticalprojection of arms during thermal expansion (distance h inFig. 10). Since the end points of the arm are clamped (nothinged), the angle between the arm and vertical direction atthe end point junctions, denoted by θ in Fig. 10, remainsconstant during thermal expansion (equal to cold value of θo

in Fig. 10). Fig. 13 shows a typical shape of the actuatedarm, which is more consistent with the foregoing clampedboundary conditions at the beam ends. As shown in Fig. 13,the straight line connecting the beam ends is adopted as x-axis,which is used as a reference to measure the arm deflectiony(x). Under Euler-Bernoulli beam assumptions and a uniformbending stiffness EI, we can write the following differentialequation and boundary conditions:

d4 y(x)

d x4 = 0 , y|x=± L x2

= 0 ,dy

d x

∣∣∣∣x=± L x

2

= ∓ tan δ (39)

where

δ(d) = π

2−θo−arctan

(h

d + do

), Lx(d)=

√h2 + (d + do)2

(40)In this way, the following shape function is obtained for

the thermally expanded arm in terms of the actuator displace-ment d .

y(x) =(

x

2− 2x3

L2x

)tan δ (41)

0 10

d (μm)

d (μ

m)

La (μm)

La (

μm

)

20400

400.2

400.4

400.6

400.8

401

401.2

400 400.5 4010

5

10

15

20

25Nonlinear beam

Linear beam

Nonlinear beam

Linear beam

(a) (b)

Fig. 14. Profiles of the functions relating arm length La to displacement dfor h = 400 and do = 4 μm.

Having determined the shape function of the arm, thearm length La can be determined in terms of the actuatordisplacement as:

La = L(d) := 2

0.5L x∫

x=0

√1 +

[dy(x)

dx

]2

dx

= 2

0.5L x∫

x=0

√1 +

(1

2− 6x2

L2x

)2

tan2 δ dx (42)

For h = 400 μm and do = 4, the profiles of functionLa = L(d) and the inverse function d = L−1(La), computedby Eq. (42), are shown in Fig. 14 with solid lines. To comparewith the linear beam profile assumption, the correspondingresults have also been included in Fig. 14 with dotted lines,using Eq. (19).

In this way, when the s-coordinate is adopted to coincidewith the curved beam, Equations (20)-(33) and (36)-(38)are still valid for the nonlinear beam. After selecting aparametrized nonlinear temperature profile T̃ (s, La), basedon sign of η, the beam length La should be adjusted tosatisfy constraint (37), which considers the nonlinearity inthermal expansion coefficient. The actuator displacement isthen determined from the inverse function d = L−1(La),which includes the nonlinear beam profile.

E. Nonlinear Resistivity Temperature Profile

In [43], effect of nonlinear resistivity temperature profilewas considered using a numerical finite-element ANSYSmodel; however, their profile did not include the high temper-ature part of the profile that has a negative slope. Moreover,their modeling procedure was not clarified by any analyticalexpressions. However, the numerical method that we offer hereis more transparent due to the analytical expressions presentedearlier. Up to this point, we assumed a linear temperatureprofile for electrical resistivity, represented by Eq. (2) andthe straight line in Fig. 3. This assumption led to closed-form expressions (23)-(33) for temperature profiles. When the


200 400 600 800 1000 1200 1400 1600 18001.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

Temperature (K)

Res

istiv

ity (�

·m)

ResistivityPolynomial

× 10−5

Fig. 15. Approximation of nonlinear resistivity profile by a fifth orderpolynomial.

TABLE IV

PARAMETER VALUES USED FOR SIMULATIONS

Parameter Valuedo 4 (μm)

h 400 (μm)

A 125 (μm)2

Ae 12.5 (μm)2

αl (at room temperature) 2.6×10−6 (1/K)

αo 3.7822×10−4 (1/°C)

k 130 (WK−1m−1)

GL 170×10−6 (W/K)

G0 35×10−6 (W/K)

ga 0.03 (WK−1m−1)

Ka 199.5×10−6 (W/K)

Tamb 300 (K)Roa 500 (�)

nonlinear dependency of resistivity on temperature, illustratedin Fig. 3, is to be taken into account, the boundary valueproblem (20)-(22) does not have any closed-form solution andrequires a numerical solution. Here, we use a MATLAB solver,“bvp4c”, which is suitable for ordinary differential equationswith two-point boundary value problems. To accelerate theconvergence and avoid unacceptable multiple solutions, weused the closed-form solutions, obtained via the linear resis-tivity assumption, as the initial guess for the solver. Moreover,the nonlinear curve in Fig. 3 is approximated by a fifth orderpolynomial with adequate precision, as shown in Fig. 15.

F. Simulation Results

Here we use simulation to evaluate the proposed models forthe thermal actuator. The parameter values were consistentlyselected among different models, as shown in Table 4. Assum-ing the resistivity profile in Fig. 3, the effective cross-sectionalarea Ae was adjusted to give Roa = 500 � for each arm withlength of 400.02 μm, which is consistent with the measuredactuator electrical resistance of 100 � at room temperature.For an actuation voltage of 9V, the temperature profiles of thearm obtained by the proposed models, are shown in Fig. 16.

0 50 100 150 200 250 300 350 400150

200

250

300

350

400

450

s−axis (μm)

Tem

pera

ture

(°C

)

Linear modelTemperature profile nonlinearityThermal expansion nonlinearityBeam shape nonlinearityResistivity nonlinearity

Fig. 16. Temperature profiles of the arm with each legend referring to thenonlinearity included in the model in addition to the nonlinearity in the upperlegends.

TABLE V

ELECTRIC CURRENT, ARM LENGTH CHANGE, ARM RESISTANCE,

AND ACTUATOR DISPLACEMENT FOR THE PROPOSED MODELS AT

ACTUATION VOLTAGE OF 9 V

Nonlinearity type I (mA) La − Lo (μm) Ra (�) d (μm)

Linear 8.4 0.2 536 9.17

Temperature profile 8.0 0.34 562 13

Thermal expansion 8.0 0.45 562.2 15.43

Beam shape 8.0 0.45 562.2 14.34

Resistivity 8.255 0.47 545 14.66

The legends from top to bottom refer to the last type of thenonlinearity included in the model and are orderly associatedwith the proposed models. In other words, the nonlinearitieshave not been considered individually, but in a cumulativemanner as was explained in the previous sections. Thus, eachlegend refers to the kind of nonlinearity included in additionto those mentioned in the upper legends. For the simulationresults in Fig. 16, the actuator displacement d , increase in armlength La − Lo, the arm electric current I , and arm resistanceRa have been tabulated in Table 5. For the nonlinear models,the arm resistance is calculated after obtaining the temperatureprofile using the following relationship:

Ra = 1

Ae

s=La∫

s=0

ρ(

T̃ (s))

ds (43)

When the thermal expansion nonlinearity is included, the tem-perature profile is weakly affected in Fig. 16. When the beamshape nonlinearity is added to the temperature profile, length,and resistance of the arm are expectedly not affected (seeTable 5). Hence, the profiles associated with the three middlelegends almost overlap. However, their displacements haveapparent differences as indicated in Table 5. The stationarydisplacement characteristics of the thermal actuator, obtainedby the proposed models, are shown in Fig. 17.


0 1 2 3 4 5 6 7 8 9−2

0

2

4

6

8

10

12

14

16

Actuation voltage (V)

Dis

plac

emen

t (μ

m)

Linear modelTemperature profile nonlinearityThermal expansion nonlinearityBeam shape nonlinearityResistivity nonlinearity

Fig. 17. Stationary displacement characteristics of the actuator predicted bythe proposed models.

0 2 4 6 80

5

10

15

Actuation voltage (V)

Dis

plac

emen

t (μ

m)

Model

Experiment

Fig. 18. Experimental and predicted static displacement characteristics ofactuator, with the model that includes all four types of nonlinearities.

Fig. 18 shows the predicted displacement characteristics,using the model described in Section IV-E that considers allnonlinearities, along with the experimental data. It is seen thatthe initial deflections predicted by the model are more than themeasured values. The reason can be explained by consideringthe un-excited supporting beams, which were used to transferthe heat outside of the actuator. Evidently, the temperaturesof these beams are lower than those of the electrically excitedarms and their smaller thermal expansions compared to thoseof the arms exhibit a resisting force against the motion.

V. CONCLUSION

Simple lumped parameter methods were used to predict thestatic behavior of a micro-machined thermal sensor. Effectsof thermal coupling from actuator to sensor and nonlinearityof resistivity profile were cumulatively included. The ther-mal actuation mechanism was first modeled by a simplelumped-parameter model. The model was gradually improvedby incorporating more nonlinear effects. The effects, which

were cumulatively considered in a step by step manner, arenon-uniform temperature profile of the arms, temperaturedependency of thermal expansion coefficient, deviation of armshape from straight line due to end point constraints, andnonlinearity in resistivity temperature profile. The rough agree-ment between the resulting sensor and actuator characteristicsensures the validity of the results.

The proposed analytical methods are helpful for predictingsensitivity and power consumption of the thermoelectric sensorand actuator. They also reveal how thermal coupling fromactuator to sensor affects the sensor resolution. Moreover,using the proposed methods, one can predict the voltagelimitations of the sensor and actuator, which are very importantto prevent damages when using high voltage ranges.

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Ali Bazaei (M’10) received the M.Sc. and B.Sc.degrees from Shiraz University, Shiraz, Iran, andthe Ph.D. degree from the University of WesternOntario, London, ON, Canada, and Tarbiat ModaresUniversity, Tehran, Iran, all in electrical engineering,in 1992, 1995, 2004, and 2009, respectively.

He was an Instructor with Yazd University, Yazd,Iran, from September 1995 to January 2000. FromSeptember 2004 to December 2005, he was aResearch Assistant with the Department of Electricaland Computer Engineering, University of Western

Ontario. He has been a Post-Doctoral Researcher with the School of ElectricalEngineering and Computer Science, University of Newcastle, Callaghan,Australia, since April 2009. His current research interests include the areaof nonlinear systems including control and modeling of structurally flexiblesystems, friction modeling, compensation, and neural networks.

Yong Zhu (M’10) received the Ph.D. degree inmicroelectronics from Peking University, Beijing,China, in 2005.

He was a Research Associate with the Departmentof Engineering, University of Cambridge, Cam-bridge, U.K., in 2006 and 2007, respectively. From2008 to 2010, he was a Research Academic withthe School of Electrical Engineering and ComputerScience, University of Newcastle, Callaghan, Aus-tralia. He is currently a Lecturer with the Schoolof Engineering, Gold Coast Campus, Griffith Uni-

versity, Gold Coast, Australia. His current research interests include micro-machined resonators, power harvesters, nanopositioners, capacitive sensors,radio frequency microelectromechanical systems (MEMS), mechatronics, andinterface circuit design for MEMS.

Reza Moheimani (M’92–SM’00–F’11) joined theUniversity of Newcastle, Callaghan, Australia, in1997, where he founded and directs the Laboratoryfor Dynamics and Control of Nanosystems, a multi-million-dollar state-of-the-art research facility dedi-cated to the advancement of nanotechnology throughinnovations in systems and control engineering. Heis a Professor of electrical engineering and an Aus-tralian Research Council Future Fellow. His currentresearch interests include the area of dynamics andcontrol at the nanometer scale, applications of con-

trol and estimation in nanopositioning systems for high-speed scanning probemicroscopy, modeling and control of micro-cantilever-based devices, controlof electrostatic microactuators in microelectromechanical systems, and controlissues related to ultrahigh-density probe-based data storage systems.

Prof. Moheimani is a fellow of the International Federation of AutomaticControl and the Institute of Physics, U.K. He is a co-recipient of theIEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY OutstandingPaper Award in 2007 and the IEEE Control Systems Technology Awardin 2009, the latter together with a group of researchers from IBM ZurichResearch Laboratories, where he has held several visiting appointments. Hehas served on the editorial board of a number of journals including the IEEETRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, the IEEE/ASMETRANSACTIONS ON MECHATRONICS, and Control Engineering Practice, andhas chaired several international conferences and workshops.


Mehmet Rasit Yuce (S’01–M’05–SM’10) receivedthe M.S. degree in electrical and computer engi-neering from the University of Florida, Gainesville,and the Ph.D. degree in electrical and computerengineering from North Carolina State University(NCSU), Raleigh, in 2001 and 2004, respectively.

He was a Research Assistant with the Departmentof Electrical and Computer Engineering, NCSU,from August 2001 to October 2004. He was a Post-Doctoral Researcher with the Electrical EngineeringDepartment, University of California, Santa Cruz,

in 2005. He was a Senior Lecturer with the School of Electrical Engineeringand Computer Science, University of Newcastle, Callaghan, Australia, untilJuly 2011. In July 2011, he joined the Department of Electrical and Computer

Systems Engineering, Monash University, Clayton, Australia. He has publishedmore than 70 technical articles in his areas of expertise. He is an author ofthe book Wireless Body Area Networks (2011). His current research interestsinclude wireless implantable telemetry, wireless body area networks, bio-sensors, microelectromechanical systems, sensors, integrated circuit technol-ogy dealing with digital, analog and radio frequency (RF) circuit designs forwireless, biomedical, and RF applications.

Dr. Yuce received the NASA Group Achievement Award for developing ansilicon-on-insulator transceiver in 2007. He received the Research ExcellenceAward in the Faculty of Engineering and Built Environment, University ofNewcastle, in 2010. He is a member of the IEEE Solid-State Circuit Society,the IEEE Engineering in Medicine and Biology Society, and the IEEE Circuitsand Systems Society.

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