IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

vol. 60, no. 9,

September

2013

1989

An Equivalent Network Representation of a Clamped Bimorph Piezoelectric Micromachined Ultrasonic Transducer With Circular and Annular Electrodes Using Matrix Manipulation Techniques Firas Sammoura, Katherine Smyth, and Sang-Gook Kim Abstract—An electric circuit model for a clamped circular bimorph piezoelectric micromachined ultrasonic transducer (pMUT) was developed for the first time. The pMUT consisted of two piezoelectric layers sandwiched between three thin electrodes. The top and bottom electrodes were separated into central and annular electrodes by a small gap. While the middle electrode was grounded, the central and annular electrodes were biased with two independent voltage sources. The strain mismatch between the piezoelectric layers caused the plate to vibrate and transmit a pressure wave, whereas the received echo generated electric charges resulting from plate deformation. The clamped pMUT plate was separated into a circular and an annular plate, and the respective electromechanical transformation matrices were derived. The force and velocity vectors were properly selected using Hamilton’s principle and the necessary boundary conditions were invoked. The electromechanical transformation matrix for the clamped circular pMUT was deduced using simple matrix manipulation techniques. The pMUT performance under three biasing schemes was elaborated: 1) central electrode only, 2) central and annular electrodes with voltages of the same magnitude and polarity, and 3) central and annular electrodes with voltages of the same magnitude and opposite polarity. The circuit parameters of the pMUT were extracted for each biasing scheme, including the transformer ratio, the clamped electric impedance, and the open-circuit mechanical impedance. Each pMUT scheme was characterized under different acoustic loadings using the theoretically developed model, which was verified with finite element modeling (FEM) simulation. The electrode size was optimized to maximize the electromechanical transformer ratio. As such, the developed model could provide more insight into the design, optimization, and characterization of pMUTs and allow for performance comparison with their cMUT counterparts.

Manuscript received January 28, 2013; accepted June 24, 2013. This project is supported by Masdar Institute of Science and Technology, Abu Dhabi, UAE grant no. 6923443 under the cooperative agreement between the Masdar Institute of Science and Technology and the Massachusetts Institute of Technology. Katherine Smyth appreciates support from the National Science Foundation Graduate Research Fellowship Program. F. Sammoura is with Microsystems Engineering, Masdar Institute of Science and Technology, Abu Dhabi, UAE, and the MI/MIT Cooperative Program, Massachusetts Institute of Technology, Cambridge, MA (e-mail: [email protected]). K. Smyth and S.-G. Kim are with the Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA. DOI http://dx.doi.org/10.1109/TUFFC.2013.2784

0885–3010/$25.00

I. Introduction

A

lthough the development of new software techniques and integrated circuits have enabled higher resolution medical ultrasound imaging, commercial ultrasonic transducer technology has remained largely unchanged over the past few decades [1]. Current transducers are fabricated from bulk lead zirconate titanate (PZT) using a variety of dicing and assembly steps that are labor-intensive and limit individual transducers to millimeter-sized features [2]. With micro-fabrication, micro-scale transducers can be manufactured at a fraction of the cost. Incorporated into arrays, these micro ultrasonic transducers could also greatly increase imaging resolution and acoustic power, making previously impossible advanced imaging techniques, such as 3-D real-time imaging, possible [3]. Recently, capacitive micro-machined transducers (cMUTs) have been introduced [4] to leverage the low-cost MEMS fabrication techniques for small factor transducers. Because the acoustic pressure scales with the volumetric displacement rate, larger deflection results in a high acoustic pressure. Although cMUTs have a larger bandwidth, they are limited by small plate size, gap separation, and deflection that make high-acoustic-power imaging difficult [5]. The high voltage requirement may cause charge accumulation at the plate surface, which leads to performance shifts and insulator breakdown [6]. With its high piezoelectric coupling, a piezoelectric micro-machined ultrasonic transducer (pMUT) based on a PZT thin film [7], [8] or PVDF [9] can produce the large deflection necessary for high-acoustic-pressure applications including intracranial pressure monitoring of head injuries [10]. A typical pMUT consists of a suspended plate, which is either clamped [11] or simply supported [12] at its edges. With the application of an ac voltage, the strain mismatch between the structural layer and the piezoelectric layer of which the plate is composed causes the plate to vibrate in the flexural mode. Thickness-mode PZT ultrasonic sensor and cMUT elements have been well analyzed and characterized using equivalent circuit models [13]. For instance, Yamada et al. derived the equivalent network representation for a piezoelectric plate in thickness vibration [14], whereas

© 2013 IEEE

1990

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

Lohfink and Eccardt presented the linear equivalent circuit model for a cMUT [15]. In these circuit models, the electric port is represented by the applied voltage and the induced current, whereas the mechanical port is described by the impinging acoustic pressure and the plate velocity. The electric domain is coupled to the mechanical domain via an electromechanical transformer [16]. These circuit representations are important to extract the transducer performance parameters such as maximum acoustic pressure output, sensitivity, plate deflection per unit voltage, resonant frequency, effective coupling factor, bandwidth, and electrical input impedance for any acoustic load. There has been very little modeling work that dealt with pMUTs. Bernstein et al. reported some theoretical material that predicts the resonant frequency of a clamped circular plate suspended in air or loaded at one-side [17]. Perçin et al. outlined the general governing equations for piezoelectrically actuated micromachined ultrasound transducers with a ring electrode [18], [19]. Akasheh et al. used finite element modeling (FEM) to design and optimize pMUTs [20], which required substantial computational time and was tailored for a specific device configuration. A simplified circuit representation for a simply-supported flexural electroacoustic transducer based on approximated lumped constants was presented by Germano, which was valid up to the first resonance, omits dissipative elements, and neglects the Poisson’s effect [21]. Ha and Kim analyzed the admittance matrix of a piezoelectric symmetric and asymmetric triple-layered annular bimorph in which the mechanical effort vector constituted of the shear forces and the flexural moments acting at the internal and external boundary circles [22], [23]. However, the force vector omitted the acoustic pressure and the velocity vector neglected the volumetric velocity, which are the two critical parameters in pMUT analysis. Recently, we developed an equivalent circuit model for a simply supported pMUT composed of a bimorph plate [24]. In contrast to the previous works, this is the first paper that investigates a circuit model for a clamped circular bimorph pMUT composed of two layers of a piezoelectric material of the same thickness and excited by two electrically independent circular and annular electrodes. The objectives of the present work are to derive the vibration equation for circular piezoelectric plates using Hamilton’s principle of virtual work, formulate the force and velocity vectors for circular and annular plates based on the boundary conditions produced while carrying out Hamilton’s process, express the electromechanical solution matrices governing the vibration of annular and circular pMUT plates, and extract the electromechanical transformation matrix of a clamped circular pMUT based on simple matrix manipulation using boundary conditions. This paper will also detail the accurate electric circuit parameters representing the clamped circular bimorph pMUT, analyze the predicted performance under varying acoustic loads, and verify model accuracy with FEM.

vol. 60, no. 9,

September

2013

II. Theoretical Analysis A. Overview of the Clamped Bimorph PMUT Fig. 1(a) shows a schematic diagram of a clamped circular pMUT. The pMUT plate is composed of two piezoelectric layers having an equal thickness h and the same polarization direction. The piezoelectric layers are sandwiched between three metal electrodes, the middle of which is grounded. The top and bottom electrodes are separated into a central circular electrode of radius a and an annular ring of outer radius b. The electrodes are electrically isolated by a small gap and can be independently operated by voltages Vc and Vn, respectively, as shown in Fig. 1(b). In the transmit mode, the applied electric potential to the top and bottom electrodes will cause the induced electric fields to have different alignments with the piezoelectric polarity of the two layers. The piezoelectric effect leads to strain mismatch as one layer expands and the other shrinks, which generates an out-of-plane plate vibration and an emitted acoustic wave. In the receive mode shown in Fig. 1(c), an impinging acoustic wave of pressure q will cause the pMUT plate to bend. The electric charges separated in each piezoelectric layer resulting from the converse piezoelectric phenomenon are collected in the charge amplifiers as Ic and In, respectively. Fig. 2(a) shows a circular pMUT plate of radius b in a polar coordinate system (r, θ, z). The plate mid-plane is placed along the z = 0 location, which coincides with the neutral axis (or zero-strain) position because of the symmetric composition of the plate. The out-of-plane displacement is designated as w0 and has the opposite orientation as the z-axis. It is assumed that either an axisymmetric electric potential or an acoustic pressure excites the plate. As such, variations in the θ-direction are ignored. In-plane stress conditions dominate because the total plate thickness 2h is small compared with the plate radius b. An extracted infinitesimal element is shown in Fig. 2(b) along with moments and shear forces in the r- and θ-directions. The clamped circular plate is separated into an annular plate and a circular plate along the top and bottom electrode isolation region, as shown in Fig. 3. The solution strategy is to find the circuit representation matrix for each plate separately and then invoke boundary conditions to find the general matrix for a clamped circular pMUT. Hamilton’s principle is used to derive the plate vibration equation because the necessary boundary conditions are automatically produced during the process. As such, the force and velocity vectors are inferred along with their proper components. B. Annular Plate Electromechanical Transformation Matrix Hamilton’s principle is a variational energy technique, in which the equation of motion and the necessary boundary conditions can be derived. As detailed in Appendix

sammoura et al.: an equivalent network representation of a clamped bimorph pmut

1991

Fig. 2. (a) A circular pMUT plate of radius b in polar coordinates and (b) The axisymmetric moments and shear forces in the r- and θ-directions for an element extracted from the pMUT plate. 

velocities at the annular plate boundaries can be expressed as

Fig. 1. A schematic diagram of a clamped circular bimorph piezoelectric micromachined ultrasonic transducer (pMUT). The pMUT plate has a radius b and is composed of two layers of a piezoelectric material of the same thickness h. The plate is excited by two independent electrodes isolated by a small gap, where the inner circular electrode has a radius a. The electrodes separate the pMUT plate into regions I and II, respectively.(a) Before actuation; (b) in transmit mode where voltages of amplitudes Vc and Vn are applied to the central and annular electrodes, respectively; and (c) in receive mode where an acoustic wave of pressure q is impinging and currents Ic and In are collected in each amplifier, respectively. 

J 0(βa) K 0(βa) Y 0(βa)   A1  U a   I 0(βa) Ω      a  = j ω  βI 1(βa) −βJ 1(βa) −βK 1(βa) −βY1(βa)   A2  U   I (βb) J 0(βb) K 0(βb) Y 0(βb)   A3   b  0  Ωb   βI 1(βb) −βJ 1(βb) −βK 1(βb) −βK 1(βb)   A4  1 jω 0 − 2   q , ω ρs  1   0  (3) where β, ω, A1, A2, A3, A4, and ρs are constants defined in Appendix B. Eq. (3) can be written in a concise form as

u n = B unA n + u qnq, (4)

where the mechanical velocity vector, un, and the mode shape vector, An, are defined as

A, the annular plate vibration equation is obtained using Hamilton’s principle, and the force vector Tn and the velocity vector Ψn are deduced from the boundary conditions as n = [−2πaQ r a 2πa M rr a 2πbQ r b −2πb M rr b q Vn ]T T (1)

Ψn = [U a Ωa U b Ωb w 0n I n ]T , (2)

where Qr|i, Mrr|i, Ui, and Ωi, are the shear stress in the radial direction, the moment in the radial direction, the translational velocity, and the rotational velocity at radius i, respectively, and w 0n is the volumetric velocity of the annular plate. Using the shape function solution of an annular plate for an arbitrary set of boundary conditions as carried out in Appendix B, the translational and angular

Fig. 3. The solution approach for the clamped circular pMUT: the plate is separated into (a) an annular plate and (b) a circular plate. A separate matrix solution is found for each region and boundary conditions are invoked to find the solution matrix for the clamped plate. 

1992



IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

B wn =

B in = (j ω2πM Pβ)[bI 1(βb) − aI 1(βa) −bJ 1(βb) + aJ 1(βa) −bK 1(βb) + aK 1(βa) −bY1(βb) + aY1(βa)] (16)

u n = [U a Ωa U b Ωb ]T (5)



A n = [A1 A2 A3 A4]T (6)

The moments and shear forces in the radial direction can be related to the angular and translational velocities using (B1), (B2), and (B5) as in (7), see above, where the flexural rigidity, D, and the piezoelectric moment coefficient, MP, are defined in Appendix B and the functions II(x), JJ(x), KK(x), YY(x), III(x), JJJ(x), KKK(x), YYY(x) are defined in Appendix C. Eq. (7) can be rewritten in a simplified form as Fn = B fnA n + C vnV n, (8) where the mechanical force vector Fn is defined as Fn = [−2πaQ r

a

2πa M rr

a

2πbQ r

b

T

−2πb M rr b ] . (9)

The volumetric velocity of the annular plate, w 0n, is the time derivative of the volumetric displacement defined in (A24). It can be related to the mode shape vector, An, and the acoustic pressure q using (3): w 0n = B wnA n + p wnq , (10)

where the matrix Bwn and the constant pwn are defined as in (11), see above, and

p wn = −

j ωπ(b 2 − a 2) . (12) ω 2ρ s

The current In collected at the charge amplifier is the time derivative of the charge Qn separated at the piezoelectric layer surfaces in (17). The displacement current Dz can be rewritten using (A4)–(A6), (A8), and (A9):

2013

j ω2π [bI 1(βb) − aI 1(βa) bJ 1(βb) − aJ 1(βa) −bK 1(βb) + aK 1(βa) bY1(βb) − aY1(βa)] (11) β





September

 −2πaQ r a   −βIII (βa) −βJJJ (βa) −βKKK (βa) −βYYY (βa)   A1   0    II (βa)  A    2 π a M JJ ( β a ) KK ( β a ) YY ( β a )  rr a     2  + 2πM  a  V (7)  2πbQ  = −2πD β  βIII (βb) P  0 n βJJJ (βb) βKKK (βb) βYYY (βb)   A3  r b       −2πb M   −II (βb)  −b  −JJ (βb) −KK (βb) −YY (βb)   A4  rr b  





vol. 60, no. 9,

 ∂ 2w 0 1 ∂w 0   + εz′E z , (13) D z = −z(d 31mY 0m )  +  ∂r 2 r ∂r 

where d31m and Y0m are the modified piezoelectric coefficient and Young’s modulus as defined in Appendix A. The modified dielectric constant of the piezoelectric layer is defined as

 2Y d 2  εz′ = εz  1 − 0m 31m  . (14)  (1 + v)εz 

Using (A17), the current In can be expressed as

I n = B inA n + j ωC 0nV n, (15)

where the matrix Bin and the capacitance C0n are defined as (16), see above, and

C 0n =

εz′π(b 2 − a 2) . (17) h/2

The solution matrix can be formed using (4), (10), and (15) as

0   An  u qn B  un   un     w B = p 0  wn wn   q  . (18)  0n     I   B in 0 j ωC 0n   V n   n 

The shape function matrix An can be determined using (8) as

A n = B −fn1Fn − B −fn1C wnV n. (19)

The force vector Tn can be related to the velocity vector Ψn by eliminating the shape function matrix An using (18) and (19):  F   B unB −fn1 u qn −B unB −fn1C vn  un    n   −1 −1    q  . (20) −B wnB fn C vn  w 0n  =  B wnB fn p vn    I  − − 1 1 B B 0 j ωC 0n − B inB fn C vn  V n   n   in fn

sammoura et al.: an equivalent network representation of a clamped bimorph pmut

C. Circular Plate Electromechanical Transformation Matrix



For a circular plate, infinite vibration amplitudes at the center are not allowed. As such, the displacement shape function for the first vibration mode cannot contain the second-order Bessel function and modified second-order Bessel function. The displacement for a circular plate is expressed as

w 0(r ) = A1I 0(βr ) + A2J 0(βr ) −

q . (21) ω 2ρ s

Using (A28), the force vector Tc and the velocity vector Ψc for the circular plate are defined as

Tc = [−2πaQ r

2πa M rr

a

a

0 0 q V c ]T (22)







Fc = [−2πaQ r

a

2πa M rr

a

0 0]T (25)

A c = [A1 A2 A3 A4]T . (26)



The vector uc is related to the vector Ac and the pressure q as u c = B ucA c + u qcq , (27)



where Buc and uqc are defined as





B uc

J (βa)  I 0(βa)  βI (βa) −βJ0 (βa) 1 = j ω  1 0  0  0 0 u qc

0 0 0 0

0 0  (28) 0  0 

1 j ω  0  = − 2   . (29) ω ρs  0   0 

The vector Fc is related to the vector Ac and the voltage Vc as

Fc = B fcA c + C vcVc , (30)

where Bfc and Cvc are defined as

0 0  (31) 0  0 

0 a  = 2πM P   . (32) 0  0 

−w 0c = B wcA c + p wcq , (33)

where the matrices Bwc and pwc are defined as

where w 0c is the volumetric velocity of the circular plate. The zero terms were added to make the size of the solution matrix for both the annular and the circular plates equal. The mechanical velocity vector uc, the mechanical force vector Fc, and the displacement shape vector Ac for the circular plate are defined as u c = [U a Ωa 0 0]T (24)

C vc

0 0 0 0

The volumetric velocity of the circular plate, −w 0c, is related to the shape function vector Ac and the pressure q as





 −βIII (βa) −βJJJ (βa)  II (βa) JJ (βa)  = −2πD β  0 0   0 0



Ψc = [U a Ωa 0 0 −w 0c −I c ]T , (23)



B fc

1993

B wc = −

j 2πω [aI 1(βa) aJ 1(βa) 0 0] (34) β p wc =

j ωπa 2 q. (35) ω 2ρ s

The current Ic collected at the charge amplifier is related to the shape function vector Ac and the voltage Vc using the following identity:

−I c = B icA c + j ωC 0cVc , (36)

where the matrices Bic and the capacitance C0c are defined as

B ic = −(2πM P β)[aI 1(βa) −aJ 1(βa) 0 0] (37) C 0c = −

εz′πa 2 . (38) h/2

Finally, the force vector is related to the velocity vector as  F   B ucB −fc1 u qc −B ucB −fc1C vc  uc    c    − 1 − 1   q  . (39)  −w 0c  =  B wcB fc p vc −B wcB fc C vc    −I  − 1 − 1  V c  B B B B C 0 j ω C −  ic fc vc   c  0c  ic fc D. Clamped Circular Plate Electromechanical Transformation Matrix The force vector and the velocity vector for a clamped plate can be defined as

T = [q V ]T (40)



Ψ = [w 0 I ]T , (41)

where w 0 is the volumetric velocity of the clamped plate, I is the total collected current, q is the external pressure, and V is the applied voltage. The shear force and moment

1994

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

vol. 60, no. 9,

2013

 B unB −fn1 − B ucB −fc1 u qn − u qc  F  B ucB −fc1C vc  0     c   − − − 1 1 1   q  (46) B wcB fc C vc  w 0  =  B wnB fn − B wcB fc p vn − p vc    I  − − 1 1   V    B B j C B B C − 0 − ω + i c f c c i c f c v c 0  



−1



  B unB −fn1 − B ucB −fc1 u qn − u qc B ucB −fc1C vc  Fc    q  =  B B −1 − B B −1 p − p −1  B wcB fc C vc   wc fc vn vc  wn fn   V  −1 −1  −B icB fc −j ωC 0c + B icB fc C vc  0 



  B unB −fn1 − B ucB −fc1 u qn − u qc −B unB −fn1C vn + B ucB −fc1C vc  Fc     q  =  B B −1 − B B −1 p − p −1 −1  B B C B B C − +   w n f n w c f c vn vc w n f n v n w c f c v c    V  − − 1 1 − 1 − 1  B B −B B 0 j ω(C 0n − C 0c) − B inB fn C vn + B icB fc C vc  ic fc  in fn

at the boundary between the annular plate and the circular plate as described in the force vectors (9) and (12) must be equal. In addition, the angular rotation and the vertical displacement at the interface between the two plates have to be the same. The clamped boundary conditions at the edge of the clamped plate force the angular velocity and the vertical velocity to be zero. The aforementioned boundary conditions can be summarized as Ua Ωa



Qr



M rr

c

−U a

n

=0

c

− Ωa

n

=0

Ub

n

=0

Ωb

n

=0

a c

= Qr

a c

= M rr

a n

(42)

2π a

w 0 =

∫∫ 0

0

 0     w 0  (47)  I   

−1

 0     w 0  (49)  I   

The solution matrix of the clamped plate is determined by subtracting the solution matrix of the circular plate in (39) from the annular plate as described in (45) using the boundary conditions outlined in (42) and (43) to obtain (46), see above. Upon inverting the solution matrix presented in (46), the force vector components of the clamped plate, q and V, are now only dependent on the volumetric velocity w 0c and the displacement current, as in (47), see above. The mechanical impedance zm, electromechanical coupling coefficient Tem, and the electrical impedance Ze can be extracted as a sub-matrix from the inverted matrix in (47) as

T   w  q  z V  = T m Zme   0  . (48)    em e  I 

(43)

a n.

The volumetric velocity for the clamped plate can be calculated as the sum of the volumetric velocity of the circular plate and the annular plate as

September

2) Case II: Vn = V and Vc = V: In this case, the total collected current is the difference between the currents collected in the circular and annular plate because they are caused by the electric potential of the same magnitude and same polarity; see (49), above.

2π b

U (r )r dr +

∫ ∫ U (r)r dr = w 0c + w 0n. (44) 0

a

1) Case I: Vn = 0 and Vc = V: The solution matrix for the annular plate degenerates into



 B B −1 u qn 0   F  u   un fn   n  n  −1  w 0n  =  B wnB fn p vn 0   q  . (45)  0   0 0 0   0   

Fig. 4. Basic electric network representation of a clamped circular bimorph pMUT. 

sammoura et al.: an equivalent network representation of a clamped bimorph pmut

1995

  B ucB −fc1 − B unB −fn1 u qc − u qn −B ucB −fc1C vc − B unB −fn1C vn  Fc     q  =  B B −1 − B B −1 p − p −1 −1  B B C B B C − −   w c f c w n f n v c v n w c f c v c w n f n v n    V   −B B −1 − B B −1 0 j ω(C 0n − C 0c) − B inB −fn1C vn + B icB −fc1C vc  in fn ic fc 



3) Case III: Vn = −V and Vc = V: In this case, the two terms Bin and Cvn have a reversed polarity to account for the voltage polarity change. The total collected current is the difference between the currents collected in the circular and annular plate because they are caused by the electric potential of the same magnitude and opposite polarity. As such, the solution matrix for the clamped plate is the difference between the annular plate and the circular plate; see (50), above. III. Equivalent Network Representation of a Clamped PMUT In the previous section, the clamped circular bimorph piezoelectric plate system was represented by a canonical matrix as shown in (48), in which the currents and volumetric velocities of the electromechanical system are established by impressed electric potential and acoustic pressure. The systems had bilateral transduction coefficients and Tem and Tme were equal. The clamped bimorph circular pMUT can be represented with an equivalent electric circuit as shown in Fig. 4. This network has the same style as the two-mesh equivalent circuit with symmetrical electromechanical coupling derived by Hunt et al. [16]. The ideal transformer with impedance ratio n shown in the circuit model has units of pascals per volt or amps per (cubic meters per second), and is defined as n =



Tme . (51) ze

In addition, the negative mechanical impedance, −zs, which accounts for the mechanical spring softening of the plate caused by the applied electric field, is expressed as zs =



2

Tem . (52) ze

−1

 0     w 0  (50)  I   

IV. Results and Discussion Using the equivalent network representation developed in this paper, the electrical input impedance and the average displacement are extracted for a clamped circular bimorph pMUT. The material properties and the geometrical dimensions are summarized in Table I, which were chosen such that the resonant frequency of the pMUT is within the ultrasonic range of 1 to 16 MHz. Both air and water loading conditions are considered. A. Case I: Active Central Electrode The transformer ratio n, or the maximum output acoustic pressure in phase with the plate volumetric velocity, was plotted versus the active central electrode radius, with the annular electrode grounded. As shown in Fig. 5, the model predicted that the maximum output pressure per unit input voltage reaches an optimum value of 76 kPa/V when the central electrode radius is 35 µm. The electromechanical transformer ratio drops to zero as the central electrode radius covers the whole plate. As such, the central electrode radius is designed as 35 µm (or 70% plate radius coverage) to maximize the electromechanical transformer ratio between the input voltage and the output pressure. The first-mode circuit parameters with a central electrode radius of 35 µm as described in Fig. 4 are summarized in Table II. The average plate displacement versus frequency is calculated using the circuit model for an input ac voltage of 1 V and under both air and water loading. As shown in Fig. 6, the static average plate displacement is the same under both loading mediums and calculated as 6.54 nm/V. The first-mode mechanical resonant frequency is 2.25 MHz with air loading and drops to 1.70 MHz under water loading. Fig. 7 shows the mechanical impedance versus frequency. For a large pMUT array, the loading medium acoustic impedance becomes resistive. The 3-dB

TABLE I. The Geometric Dimensions and Material Properties of the Simulated pMUT Plate. Parameter

Description

h a ρ Y0 ν δ31 ε

Layer thickness Plate radius Plate density Young’s modulus Poisson’s ratio Piezoelectric charge constant Dielectric constant

Value 2 50 7750 61 0.33 171 1700

Unit μm μm kg/m3 GPa pm/V

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TABLE II. Theoretical First Mode Circuit Parameters of the Clamped Circular pMUT for the Three Considered Cases. Unit a C0 n Cs Cm Lm

µm pF kPa/V m3/Pa m3/Pa kg/m4

Case I 35 37.365 76.998 6.29881 × 10−21 0.6123 × 10−21 7.4021 × 106

mechanical bandwidth can be calculated using the method outlined in [25] as 3.65 MHz or 214% and 1.13 kHz or 0.05% under water and air loading, respectively. The predicted real and imaginary parts of the electrical input impedance are plotted in Figs. 8 and 9, respectively. The resonant frequency decreases as the fluid loading on one side of the plate changes from air to water in agreement with the plate loading predictions [16]. Because the immersion acoustic impedance in water is higher than that in air, the pMUT resonant behavior becomes more heavily damped. The mechanical quality factor, Q, is calculated from the resonance, fr, and antiresonance, fa, frequencies as follows [26]:

1 f − fr . (53) = a fr Q

Case II 35 78.87 0 ∞ 0.67823 × 10−21 7.4021 × 106

Case II 35 78.87 153.996 6.29881 × 10−21 0.6123 × 10−21 7.4021 × 106

rized in Table II. The electromechanical transformer ratio n is zero as expected, and the capacitance is a parallel combination of the central and annular capacitances. The plate does not vibrate in the flexural node with the application of a bias voltage along an electrode covering the whole plate, but rather develops an alternating stress in the radial direction. It is worth mentioning that the model is capable of accurately predicting the mechanical impedance, because the plate can still vibrate under the influence of an acoustic pressure without generating an electric displacement current. C. Case III: Active Central and Annular Electrodes With Opposite Polarities

The first-mode circuit parameters for a clamped circular pMUT with active central and annular electrodes excited with voltages of the same polarity are summa-

The first-mode circuit parameters for a clamped circular pMUT plate with active central and annular electrodes excited with voltages of the same magnitude but opposite polarity are shown in Table II. The electromechanical transformer ratio n is 154 kPa/V, which is double the value for the case in which only the central electrode is active. The central electrode applies a tensile stress to the inner piezoelectric layer, while the annular electrode causes a compressive stress. As such, the plate deflection is expected to increase. Fig. 10 compares the average plate deflection for cases I and II, in which the static plate deflection for inner and annular electrodes excited with voltages of reverse polarity is calculated as 13.1 nm/V. The plate resonant frequencies remain unchanged under both

Fig. 5. Model-predicted transformer ratio or maximum output pressure per unit input voltage versus central electrode radius. There is an optimum central electrode radius of 35 µm to obtain maximum electromechanical transformer ratio of the clamped circular bimorph pMUT. 

Fig. 6. Model-predicted average plate displacement per unit input voltage for a clamped circular bimorph pMUT, with the geometric and material properties shown in Table I, under both air and water loading. Both the first and second resonance modes are displayed. 

For air loading, fr and fa are measured from the input impedance curves as 2.24 and 2.34 MHz, respectively. As such, the mechanical quality factor is calculated as 21.11. For water loading, fr and fa are 1.67 and 1.85 MHz, respectively, and Q reduces to 10.06. B. Case II: Active Central and Annular Electrodes With the Same Polarity

sammoura et al.: an equivalent network representation of a clamped bimorph pmut

Fig. 7. Model-predicted plate mechanical impedance versus frequency for a clamped circular bimorph pMUT with the geometric and material properties shown in Table I. Both the first and second resonance modes are displayed. 

excitation schemes because this is governed by plate geometry and material composition. D. Model Verification Using FEM To verify the developed circuit, Comsol Multiphysics (Comsol Inc., Burlington, MA) was used as the FEM tool [27]. Fig. 11 shows a schematic diagram of a Comsol model for an axisymmetric pMUT cell exposed from one side to an acoustic medium, which mimics a piston in a rigid baffle. A piezoelectric/acoustic boundary condition is enforced at the pMUT/acoustic medium interface and a sound-hard boundary is imposed at the bottom plane outside the vibrating pMUT. The model is terminated with a radiating boundary condition few wavelengths away from the pMUT cell. Fig. 12 compares the simulated and theoretically predicted dynamic average plate displacement under both air and water loading. The simulation predicts a static dis-

Fig. 8. Model-predicted real part of the electrical input impedance for a clamped circular bimorph pMUT with the geometric and material properties shown in Table I under air and water loading. Both the first and second resonance modes are displayed. 

1997

Fig. 9. Model-predicted imaginary part of the electrical input impedance for a clamped circular bimorph pMUT with the geometric and material properties shown in Table I under air and water loading. Both the first and second resonance modes are displayed. 

placement of 6.5 nm/V under both air and water loading, which is within 0.6% of the theoretically reported value. The plate resonant frequencies as calculated using Comsol Multiphysics were 2.32 and 1.62 MHz for air and water loading, which correspond to absolute errors of 3% and 4.7% from the theoretically predicted values, respectively. The peak average plate displacements at resonance were 77 and 87 nm as calculated from the theoretical model and the Comsol simulation, respectively, indicating an agreement of 11.5% in displacement at resonance under water loading. Fig. 13 compares the magnitude of the simulated input electrical impedance of a pMUT cell under water loading vs. frequency to the theoretically predicted one. The simulated resonance and antiresonance frequencies for a central-electrode-actuated pMUT are reported as 1.595 and 1.725 MHz, respectively. These values agree very well with the theoretically predicted values presented in an earlier section, and the absolute errors are 4.5% and 6.7%, respectively. The electrical input impedance is important to

Fig. 10. Model-predicted average plate displacement per unit input voltage for a clamped circular bimorph pMUT with 1 and 2 electrode configuration under both air and water loading. The pMUT has the geometric and material properties shown in Table I. 

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2013

Fig. 12. Comparison of the model-predicted and simulated average plate displacement per unit input voltage for a clamped circular bimorph pMUT with the geometric and material properties shown in Table I under both air and water loading. Both the first and second resonance modes are displayed. 

Fig. 11. A Comsol model for an axisymmetric clamped circular bimorph pMUT cell in an acoustic medium. The pMUT cell is modeled as a vibrating plate in a rigid baffle, where a sound hard boundary is imposed outside the vibrating device and a piezoelectric/acoustic boundary condition is enforced at the interface between the pMUT and the acoustic medium. Radiating boundary conditions are specified a few wavelengths from the pMUT cell. 

2 calculate the effective coupling factor of the pMUT, k eff , which is extracted from fr and fa as [28]: 2



f  2 k eff = 1 −  r  . (54)  fa 

Using (54), the effective coupling factor for the pMUT is 14.5% as simulated and 18.5% as theoretically predicted. The open-circuit receive sensitivity of the clamped pMUT with central electrode was also characterized. As shown in Fig. 14, the maximum receive sensitivities under water loading are −107.1 dB re 1 V/µPa at 1.73 MHz and −107.9 dB re 1 V/µPa at 1.71 MHz as simulated and theoretically predicted, respectively. Finally, the total acoustic pressure at the plate/water interface was simulated and compared with the average value predicted by the developed model for an array of pMUT cells with size of 1, 7, and 19. As shown in Fig. 15, the model predicted pressures versus frequency track very well the simulated values for all array sizes. The small differences are attributed to the slight mismatches in the simulated and model-predicted resonant frequency and displacement. For a pMUT cell operated at 1 MHz, for instance, the total acoustic pressures predicted by the developed model and the simulation results are 16.56 and 17.47 kPa, respectively, which correspond to an error of less than 5.2%. As the number of array elements increases,

the water loading impedance becomes more resistive. As such, the peak pressure decreases and the bandwidth increases. Table III summarizes the bandwidth of a pMUT array with varying element numbers as predicted theoretically and simulated by Comsol. Although the theoretical and simulated BWs agree fairly well, the former results in a slightly larger BW than the latter. The error increases with the element number and is attributed to the reduction in the acoustic radiation impedance resulting from the area not covered by the vibrating plate [15]. V. Conclusion In this work, a novel electric circuit representation for a clamped circular bimorph pMUT with central and annular electrodes was demonstrated for the first time using simple matrix manipulation techniques. The electrode radii and excitation scheme were optimized to maximize the pMUT acoustic pressure output per unit input voltage.

Fig. 13. Comparison of the model-predicted and simulated magnitudes of the electrical input impedance for a clamped circular bimorph pMUT with the geometric and material properties shown in Table I under water loading. Both the first and second resonance modes are displayed. 

sammoura et al.: an equivalent network representation of a clamped bimorph pmut

Fig. 14. Comparison of the model-predicted and simulated open circuit receive sensitivity for a clamped circular bimorph pMUT with the geometric and material properties shown in Table I under water loading. 

The developed circuit was used to characterize the pMUT under air and water acoustic loading. FEM was used to validate the pMUT circuit parameters. The pMUT’s independent electrodes were excited by two different voltage sources. The constitutive equations governing the vibration of circular plates were derived using Hamilton’s principle, and the corresponding forcing and velocity vectors were determined. The electromechanical transformation matrix for the clamped circular plate was built using simple matrix subtraction and inversion techniques of the matrices of circular and annular subplates by invoking the necessary boundary conditions. The equivalent network was subsequently extracted, as the pMUT was reciprocal. The central electrode coverage was theoretically predicted as 70% of the plate radius for ideal electromechanical transformer ratio, which doubles when exciting the central and annular electrodes with voltages of the same magnitude and opposite polarity versus central electrode actuation only. This unprecedented electrode coverage optimization and biasing scheme would enable pMUTs for high-acoustic-pressure applications at low voltages, while operating at the same resonant frequency and small piezoelectric stress. The average plate displacement for a pMUT cell as described in Table I was extracted under both air and water loading conditions. For a pMUT cell actuated by a central electrode, the static average plate displacement was theoretically reported as 6.54 nm/V, with an absolute error

TABLE III. Theoretical and Simulated pMUT Array Bandwidth (BW) as Predicted Theoretically and Simulated Using Finite Element Modeling. Number of array elements, n Theoretical BW (%) Simulated BW (%)

1

2

3

16.2 15.2

76.9 66.4

263 204

1999

Fig. 15. Comparison of the model-predicted and simulated output pressure per unit input voltage for a 1, 7, and 19-element array of clamped circular bimorph pMUT cells with the geometric and material properties shown in Table I under water loading. As the number of array elements increases, the absolute pressure peak decreases and the bandwidth increases. 

of 0.6% from the simulated data. The model-predicted mechanical resonant frequency dropped from 2.25 MHz under air loading to 1.70 MHz under water loading, which is within 4.7% of the simulated data. An excellent effective coupling factor under water loading was calculated from the derived electrical input impedance as 18.5%, which is in good agreement with the simulated value of 14.5%. A high open-circuit receive sensitivity of −107.9 dB re 1 V/µPa was predicted theoretically at resonance under water loading and was supported by FEM data. A wide pMUT array bandwidth of 263% with a maximum output pressure of 120.7 kPa was theoretically determined under water loading and confirmed by FEM. The simulated and theoretical data showed a decrease in output pressure and increase in bandwidth as the number of array elements increased, because the acoustic load became resistive as the number of array elements increased. The error in bandwidth prediction increased with number of array elements and was attributed to the reduction in the radiation real impedance resulting from the nonvibrating area of the array. It is envisioned that the circuit model introduced in this work will provide improved understanding of pMUTs for better design, easier characterization, and performance comparison to their counterpart cMUTs. Appendix A Plate Vibration Equation Using Hamilton’s Principle Hamilton’s principle states that the variation in the internal energy of a system is equal to the variation in the work done by the nonconservative forces acting on the system [29], [30]. The internal energy, or the Lagrangian L, is defined as the difference between the kinetic energy and the potential energy:

2000

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

L = KE − PE, (A1)



where KE and PE are the kinetic and the potential energies of the system, respectively. The kinetic energy for the annular plate shown in Fig. 3(a) can be defined as

KE n =

1 2

h

2π

b

∫−h ∫0 ∫a

(ρω 2w 02)r dr dθ dz , (A2)

where ρ is the mass density of the piezoelectric material and ω is the frequency of operation. The potential energy of the annular piezoelectric plate is defined as



δKE n =



∂ w0 (A4) ∂r 2 1 ∂w 0 = −z . (A5) r ∂r

εrr = −z εθθ

2

The stress–strain relationships for an isotropic piezoelectric material are

Trr = Y 0m[εrr + v εθθ − d 31mE z ] (A6) Tθθ = Y 0m[v εrr + εθθ − d 31mE z ],

2013

h

b

2π

∫−h ∫0 ∫a (ρω 2w 0δw 0)r dr dθ dz . (A10)

Upon performing the integral in (A10), the variation in the kinetic energy can be rewritten as δKE n = 2πρ s



b

∫a (ω 2w 0δw 0)r dr, (A11)

where ρs is the surface density or the mass per unit area, defined as

∫ ∫ ∫

where Trr and Tθθ are the stresses in the r- and θ-directions, respectively; εrr and εθθ are the strains in the r- and θ-directions, respectively; Dz is the electric displacement current is the z-direction; and Ez is the electric field in the z-direction. The strains εrr and εθθ can be related to the out-of-plane displacement w0 and the distance z from the neutral axis using classic plate theory as

September

Using (A2), the variation in the kinetic energy can be expressed as

1 h 2π b (T ε + Tθθεθθ − D zE z )r dr dθ dz , 2 −h 0 a rr rr (A3) PE n =

vol. 60, no. 9,

ρs =

h

∫−h ρ dz . (A12)

Using (A3), (A6), and (A8), the variation in potential energy can be expressed as δ PE n =

h

b

2π

∫−h ∫0 ∫a (Trrδεrr + Yθθδεθθ − DzδE z )r dr dθ dz .

(A13)

The variation in potential energy can be rewritten using (A4), (A5), and (A13) as



δ PE n = −2π

b

2   M ∂ δw 0  r dr  rr ∂r 2 

∫a

− 2π

1 ∂δw 0   M θθ r dr dθ − Q nδV n,  r ∂r 

b

∫a

(A14)

where the moment in the r-direction Mrr, the moment in the θ-direction Mθθ, and the total charge Qn separated in the piezoelectric layers of the annular plate are defined as h

where ν is Poisson’s ratio. The modified Young’s modulus, Y0m, and the modified piezoelectric charge coefficient, d31m, are defined as



Y0 (A7) 1−v2 = d 31(1 + v),





Y 0m = d 31m

where Y0 is the Young’s modulus and d31 is the piezoelectric charge coefficient. The electric displacement current can be related to the stresses and electric field as

D z = d 31(Trr + Tθθ) + εzE z , (A8)

where εz is the dielectric constant of the piezoelectric material. The electric field developed along the thickness of each piezoelectric layer is linearly related to the voltage applied as

Ez =

V . (A9) h

M rr =

∫ zTrr dz (A15)

−h h

M θθ =

∫ zTθθ dz (A16)

−h



h

bD

2π

∫0 ∫0 ∫a

Qn = 2

z

h

r dr dθ dz. (A17)

Upon integrating (A14) by parts, the variation in the potential energy simplifies to b



b

δ PE n = − (2πrM rrδφ0) a + (2πrQ rδw 0) a − 2π

b ∂(rQ

∫a

∂r

r)

δw 0 dr − Q nδV n,

(A18)

where ϕ0 and Qr are the angular rotation and the shear stress in the radial direction, respectively, defined as

φ0 =

∂w 0 (A19) ∂r

sammoura et al.: an equivalent network representation of a clamped bimorph pmut

Qr =



1  ∂(rM rr )  − M θθ  . (A20)  r  ∂r 

The work done by the uniform pressure q applied on the top surface of the annular pMUT plate is defined as 2π

b

∫0 ∫a (qw 0)r dr dθ. (A21)

Wn = −



The variation in the work done by the nonconservative forces is inferred from (A21) as δW n = −2π



b

∫a (qδw 0)r dr. (A22)

The variation in the work done by the external pressure q can be also expressed as the product of the applied pressure and the variation in the volumetric displacement w 0n as δW n = −qδw 0n, (A23)



where the volumetric displacement of the annular plate is defined as w 0n =



2π

2001 b

b

δL n = (2πrM rrδφ0) a − (2πrQ rδw 0) a + Q nδV n + qδw 0n . (A27) As such, the force vector should include the moments and shear forces applied at the inner and outer radii, the acoustic pressure, and the applied electric potential. The velocity vector should include the angular velocities and translational velocities at the inner and outer radii, the volumetric velocity, and the electric current. A similar solution approach can be undertaken for the circular plate, in which the same plate vibration equation is derived. The internal energy for the circular plate can be expressed as δL c = (2πrM rrδφ0) a − (2πrQ rδw 0) a + Q cδV c + qδw 0c, (A28) where the total charge Qc separated in the piezoelectric layers and the volumetric displacement w 0c of the circular plate are defined as

b

∫0 ∫a (w 0)r dr dθ. (A24)

h

a

2π

∫0 ∫0 ∫0

Qc = 2

w 0c =

2π

Dz r dr dθ dz (A29) h

a

∫0 ∫0 (w 0)r dr dθ. (A30)

Appendix B Shape Function of an Annular Plate

Using Hamilton’s principle along with (A11), (A18), and (A22), the pMUT plate vibration is governed by

The moment Mrr and the shear stress Qr in the radial 1 ∂ ( rQ r )  b direction, as defined in (A15) and (A20), can be written 2π  ρ sω w 0 + + q  δw 0r dr + (2πrM rrδφ0) a  r ∂r a  in terms of the out-of-plane displacement, material propb erties, plate thickness, and the applied voltage V using − (2πrQ rδw 0) a + Q nδVn = 0. (A25) (A4)–(A6), (A9), (A15), and (A16) as

∫

b

2

The integrand in (A25) must be zero, yielding the axisymmetric plate vibration equation:

−

2

1 ∂(rQ r ) ∂ w0 + ρs = q. (A26) r ∂r ∂t 2

Eq. (A25) also yields information regarding boundary condition. For instance, either the angular rotations must be zero at the inner or outer radii or the applied moments must be equal in magnitude and opposite in sign to the internally developed ones, as in (A15). Similarly, either the vertical plate displacements at the inner or outer radii must be zero or the applied shear stresses must be equal in magnitude or opposite in sign to the internally defined ones, as in (A20). Using (A1), (A23), (A25), and (A26), the variation of the internal energy is equal to the work done by the externally applied moments and shear forces at the inner and outer radii, the work done by acoustic pressure, and the electrostatic energy as



 ∂ 2w 0 v ∂w 0   + M PV (B1) M rr = −D  +  ∂r 2 r ∂r 



 ∂ 3w 0 1 ∂ 2w 0 1 ∂w 0   , (B2) Q r = −D  + − 2 2  ∂r 3 r ∂r r ∂r 

where the flexural rigidity of the bimorph plate D and the piezoelectric moment coefficient MP are defined as

D=

2Y 0mh 3 (B3) 3

M P = −Y 0md 31mh . (B4)

The general displacement shape function for the first vibration mode of an annular plate governed by the vibration equation described in (A26) is presented as

w 0(r ) = A1I 0(βr ) + A2J 0(βr ) + A3K 0(βr ) (B5) q + A4Y 0(βr ) − 2 , ω ρs

2002

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

where I0(r), J0(r), K0(r), and Y0(r) are the modified Bessel function of the first order, the Bessel function of the first order, the modified Bessel function of the second order, and the Bessel function of the second order, respectively, A1, A2, A3, and A4 are constants that depend on the boundary conditions, and β is a constant that depends on the surface density, flexural rigidity, and the frequency of operation as β4 =



ρs ω 2 . (B6) D

The translational velocity U(r) and the rotational velocity Ω(r) are deduced using (B5) as



 U (r ) = j ω  A1I 0(βr ) + A2J 0(βr ) + A3K 0(βr )  (B7) q  + A4Y 0(βr ) − 2  ω ρs 

Ω(r ) = j ωβ[A1I 1(βr ) − A2J 1(βr ) − A3K 1(βr ) − A4Y1(βr )]. (B8) Appendix C Derived Functions Definition The functions II(x), JJ(x), KK(x), YY(x), III(x), JJJ(x), KKK(x), and YYY(x) that relate mechanical force vector, Fn, to the mode shape vector, An, are defined as:



II (x ) = JJ (x ) = KK (x ) = YY (x ) =

+0.5xI 0(x ) + vI 1(x ) + 0.5xI 2(x ) −0.5xJ 0(x ) − vJ 1(x ) + 0.5xJ 2(x ) (C1) +0.5xK 0(x ) − vK 1(x ) + 0.5xK 2(x ) −0.5Y 0x(x ) − vY1(x ) + 0.5xY 2(x )

III (x ) = [2xI 0(x ) + (−4 + 3x 2)I 1(x ) + 2xI 2(x ) + x 2I 3(x )]/(4x ) JJJ (x ) = [−2xJ 0(x ) + (4 + 3x 2)J 1(x ) + 2xJ 2(x )

− x 2J 3(x )]/(4x ) KKK (x ) = [2xK 0(x ) + (4 − 3x 2)K 1(x ) + 2xK 2(x )

(C2)

− x 2K 3(x )]/(4x ) YYY (x ) = [−2xY 0(x ) + (4 + 3x 2)Y1(x ) + 2xY 2(x ) − x 2Y 3(x )]/(4x ). References [1] T. Ikeda, Fundamentals of Piezoelectricity. New York, NY: Oxford University Press, 1990. [2] G. S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1987. [3] K. K. Shung, “Ultrasonic transducers and arrays,” IEEE Mag. Eng. Med. Biol., vol. 15, no. 6, pp. 20–30, Dec. 1996. [4] B. T. Khuri-Yakub and O. Oralkan, “Capacitive micromachined ultrasonic transducers for medical imaging and therapy,” J. Micromech. Microeng., vol. 21, no. 5, pp. 1–11, 2011.

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Firas Sammoura is currently an Assistant Professor at the Microsystems Division of the Masdar Institute of Science and Technology, Abu Dhabi. Firas Sammoura received his B.E. degree in mechanical engineering from the American University of Beirut in June 2001 with high distinction. From 2001 until 2006, he was a graduate student researcher at the University of California at Berkeley. In May 2006, he earned his Ph.D. degree in the field of microelectromechanical systems (MEMS). His dissertation focused on building plastic millimeter-wave systems for radar applications at 95 GHz. Prior to joining the Masdar Institute, Dr. Sammoura was a visiting scientist in the MI/MIT cooperative program at the Massachusetts Institute of Technology, where he started a new research program in the field of piezoelectric micromachined ultrasonic transducers (pMUTs) in collaboration with Prof. Sang-Gook Kim in the mechanical engineering department. From January 2007 until January 2011, he was senior device characterization engineer in the Advanced Development Group at the MEMS/Sensors division of Analog Devices in Wilmington, MA. From March 2004 until May 2006, he also worked as a student researcher with Hitachi Global Storage Technologies at the IBM Almaden Research Center, where he did proprietary research in MEMS applications for the hard drive disk industry. Dr. Sammoura won the Spot Award for solving the stiction problem that plagued consumer inertial MEMS accelerometers. He has several pending and issued patents in the field of microwave engineering and MEMS fabrication, design, and characterization.

2003

Katherine Smyth is a Ph.D. candidate in the Department of Mechanical Engineering at MIT. She received her S.B. and S.M. degrees in mechanical engineering from MIT in 2010 and 2012, respectively. Before graduate school, Katherine worked in the medical device industry as a summer intern in research and development in the 3M Drug Delivery Systems Division in 2008 and component engineering at Medtronic in 2009. Her current research interests include medical applications, and modeling and fabrication of thin film piezoelectric MEMS and nano-scale devices.

Sang-Gook Kim received his B.S. degree from Seoul National University, Korea, his M.S. degree from KAIST, and his Ph.D. degree from the Massachusetts Institute of Technology (MIT). He held positions at Axiomatics Co. and the Korea Institute of Science and Technology from 1986 to 1991. He joined Daewoo Corporation, Korea, in 1991 as a General Manager and then directed the Central Research Institute of Daewoo Electronics Co. as a corporate executive director before he joined MIT as a professor in 2000. His research and teaching at MIT has addressed the issues in bridging the gap between scientific findings and engineering innovations, developing novel manufacturing processes for newly-developed materials, and designing and realizing new products at micro- and nano-scales. They include carbon nanotube assembly, muscle-inspired micro-actuators, nano-engineered solar cells, MEMS energy harvesters, and piezoelectric transducers. He is a fellow of CIRP and ASME. He currently serves as the Director of the Park Center for Complex Systems at MIT.