ISSN 1292-862

TIMA Lab. Research Reports

CNRS INPG UJF

TIMA Laboratory, 46 avenue Félix Viallet, 38000 Grenoble France

Dynamic simulation of an electrostatic power micro-generator

Wei Ma1.2, Man Wong3, Libor Rufer1 1TIMA Laboratory, 46 Av. Félix Viallet, 38 031 Grenoble, France

2Dept of Mech. Engg., 3Dept of EE Engg., the Hong Kong University of Science and Technology, Clear Water Bay,

Kowloon, Hong Kong

Abstract Micro-fabricated electric generators, scavenging ambient mechanical energy, are the potential power sources for autonomous systems. The analysis of such a micro-generator including an energy coupling field shows a high degree of complexity. Nonlinear behavior of the system makes the analysis even more difficult. In this paper, we present the modeling of an integrated floating gate electrostatic power micro-generator. Nonlinear state equations describing its dynamic behavior are integrated numerically using SIMULINK. A linear equivalent circuit network based on an electro-mechanical analogy was also built. Both models show the consistency in the small-signal regime. Power generation up to 1µW is predicted, at a driving frequency around 4 kHz and assuming an input displacement of 5µm. Keywords: Energy scavenging, electrostatic MEMS device, transducer modeling.

1. Introduction

Vibration-to-electricity conversion offers the potential for autonomous systems to be self-sustaining in many environments. Recent advances in MEMS technology enable the creation of a self-powered system with a MEMS device acting as an electromechanical transducer. There are three physical principles typically used to generate electrical power from mechanical motion: electromagnetic [1], electrostatic [2] and piezoelectric [3]. Electrostatic effect was shown as a promising solution for the most significant advantage of its potential for integration with microelectronics.

A single-wafer integrated floating-gate electrostatic power micro-generator has been implemented in our recent work. Comparing with the structure assembling a mechanical resonator to another substrate containing an electret, described in [4, 5], our generator uses an insulated floating gate made of polycrystalline silicon (poly-Si) to provide a bias voltage necessary for its operation [6]. Such a solution eliminates the non-trivial alignment of the substrates that hinders optimal operation and scaling of the generators.

The dynamic behavior of the generator can be described using a set of differential

electro-dynamical equations. Closed-form analytical solution to the complete system

is difficult to obtain, due to its nonlinearity. For small-signals where the system can be considered as linear, a linearized model, based on the equivalent circuit network, can be used. Such a linear equivalent circuit network was built. When the input displacement attains large variations, it is necessary to use the differential equations in their general form. We have also built a system model based on the state equations that can be used for the simulation of both linear and nonlinear behavior. The results obtained from these two models are presented and discussed. The paper is structured as follows. In Section 2, the working principle of the floating gate electrostatic power micro-generator is discussed. In Section 3, the background of the device modeling is provided. In Section 4, the simulation results are shown and in Section 5, the implementation of the device is described. Finally, Section 6 is the conclusion.

2. Working principle

The present power generator (Figure 1) includes a mechanical resonator and an insulated floating gate made of poly-Si. Charged by electron tunneling, the floating gate works like one in a non-volatile memory device. Power is generated using a variable capacitor (Ccf), formed between a movable electrode (also the proof mass of the resonator) and a fixed counter electrode (also the floating gate). Mechanical

motion in x-direction induces changes in ( ) ( )yxbha

yxC ocf −−=−

ε, hence also a

current through the external load (RL). eo is the electric permittivity, h is the gap between the electrodes, a is the width of the capacitor electrodes normal to the direction of motion and b is the length of the electrodes along the direction of motion.

The resonator is realized using a photoresist molded low temperature electroplating process [7] that is compatible with the floating gate process. The movable electrode with its suspension elements are made of gold, thus bringing an additional advantage of reduced internal power loss.

y(t)

Floating gate with net charge Qf

M

k

ß

x(t)

Ccf(x-y)RL

Cfg

Arbitrary reference

iL

Figure 1. Lumped electromechanical model of the floating-gate power generator.

3. Dynamic model and linearization

The power generator is an electromechanical system, including a purely electrical part, a purely mechanical part, and a coupling mechanism linking the two parts. An electrostatic coupling field between the electrodes of the device provides energy exchange of the power generator. Supposing there is no loss in the coupling field, the energy put into the field by the electrical and mechanical source is stored and can be recovered completely due to its conservation. Energy exchange in the coupling field can be described by the terminal pairs of voltage Vm and current i in the electrical part and force Fe and velocity x& in the mechanical part. Both voltage Vm and force Fe can

be derived from the total differential of the energy cf

e CQ

W2

21

≡ stored in the field:

( )yxbaQh

QW

Vo

em −−

=∂∂

≡ε

(1)

( ) 2

2

2

2

2)sgn(

21

sgncf

ocf

cf

ee

CQa

yxyx

C

CQ

yxyx

WF

δε

−=−∂

∂−=

−∂∂

≡ (2)

where ( ) 1sgn =− yx if yx ≥ and 1− if yx < .

The behavior of the electrostatic power generator can be described by a set of

differential equations. The differential equation of the mechanical part of the system is obtained by the application of Newton’s 2nd law:

0)( =+−++ eFyxkxxM &&& β , (3)

where M is the mass, ß is the damping coefficient, k is the spring constant and Fe is the electrical force acting on M by the coupling field. The values x and y are the displacements of the upper and lower (frame) electrodes of Ccf, as shown in Figure 1. The differential equation of the electrical part of the system is based on Kirchoff’s voltage law, taking into account voltage drops on the components Cfg, Ccf, and RL, also shown in Figure 1. The important part of all electrostatic transducers is a bias voltage supplying the energy to the system. Unlike traditional solutions using electrets with fixed charges, a poly-Si floating gate is presently used. The differential equation governing the electrical part of the generator is given by

0=−+

+Lfg

f

Lcffg

fgcf

RC

QQ

RCC

CC

dtdQ

, (4)

where Cfg is the floating gate to ground capacitance, Q is the charge moving through

the loop. Qf is the net charge stored in the floating gate and fgf CQ denotes the

effective bias voltage Vf set up by Qf.

It is clear that the force Fe displays a quadratic dependence on the charge Q, which makes the system nonlinear.

3.1 General model

The dynamic behavior of the electrostatic power generator in its general form can be described by the set of differential state equations [8]. Such a description can be used for the simulation of both linear and nonlinear behavior of the system. State equations can be solved either numerically or using a system diagram that represents by blocks each part of a mathematical description. We have built a system model based on the state equations using SIMULINK, a software package for multi-domain modeling and design of dynamic systems. The model is based on following state equations of the system

12

21 )(

)(X

RCyXC

CyXC

RC

QX

Lfgcf

fgcf

Lfg

f

−

+−−=& (5)

32 XX =& (6)

−−+−+−=

)(2)sgn()(

1

22

210

2233 yXhCXa

yXyXkXM

Xcf

εβ& (7)

where X1, X2 and X3 are the state variables. They denote the charge Q, displacement x and velocity x& , respectively, and are determined by the present state and the

instantaneous values of all inputs. Ccf is a function of yX −2 . The complete block

diagram used for the analysis is shown in Figure 2.

The charge Q is obtained on the output of the integration unit shown in the upper part of the block diagram, the mechanical components, velocity x& and displacement x are evaluated in the lower part of the same diagram. In the diagram is also defined the electrostatic force making the coupling between the mechanical and electrical parts.

7

Charge

6Displacement5Velocity4Acceleration3Force

2

Current1

Voltage

B

k

h/2/eps/a/N

eps*a*N/h

R

R

R

u2

Q0

u2

1s

1s

1s

b

Electrode Width

Cfg

Cfg

Cfg

X0

Balance Point

|u|

Abs

1/m

1

Input Displacement

Electrical part

Mechanical part

Figure 2. Dynamic model of the power generator built in SIMULINK.

3.2 Linearized model

A linear model has a great practical importance in the cases when a small-signal operation near an equilibrium point can be supposed. The system can be thus described in a simple way by means of a linearized lumped-parameter model based on an electromechanical analogy. Linear behavior is achieved for incremental variations around an equilibrium state. The linearized voltage Vml and force Fel can be obtained through Taylor series expansion about a static equilibrium point, the state with zero

displacement 0=− oo yx in our design, as

( )[ ] ( ) ( ) ( )yxb

VQ

Cyx

yxV

QQV

yxQV o

oo

m

o

mml −+=−

−∂∂

+∂∂

=− δδδδδδδ1

, (8)

( )[ ] ( ) ( ) ( )

−+=−

−∂∂

+∂∂

=− yxb

VQ

CbQ

yxyx

FQ

QF

yxQF o

o

o

o

e

o

eel δδδδδδδ

1, (9)

where Co is the capacitance of Ccf at the equilibrium point, fgo

ofo CC

CQQ

+= . Then

the linearized equations can be written as

( )[ ]

=Γ+−+−+

=−Γ+++

0)()(d

dd

d

01

dd

22

2

VxxbVQ

ktx

tx

M

xxQCC

QtQ

R

inoo

inofg

L

δδδ

βδ

δδδδ

(10)

where b

QVF o

m

e =≡Γδδ

. It is obvious that second item of Fel in equation (9) acts like a

positive spring in the mechanical part 2bVQ

k ooe = . Figure 3 shows the linearized

equivalent circuit network that has been set up based on the equation (10) for the case of electrostatic power generator. The networks of electrical and mechanical parts are linked by an ideal transformer with a transduction factor Γ.

RL

Cfg

Ccfo k1

ek1

ß

M

x&inx&

( )xx in && −δ

VF δδ Γ=

+

-

( )xxin && −Γδ

iL

i=

Vδ

+

-

Γ:1

*

1k

Figure 3. Equivalent circuit of the linear model.

The linearized equivalent network can be easily analyzed using the Laplace transform and impedance method. The resulting output current iL and output power P dissipated on the load RL are given by following expressions

+−+

−−

−Γ

=

m

e

memem

mmeoLL

j

jCRY

ji

ωω

ζωω

ωω

ωωω

ζωω

ωω

ωω

ζωω

2121

)2(

2

22

2

2

2

2

(11)

( ) 2

2

2

2

222

2

2

2

22

4

4

2

22

2

2

2121

)4(

+−+

−−

−Γ

=

m

e

memem

mmoLe CRY

P

ωω

ζωω

ωω

ωωω

ζωω

ωω

ζωω

ωω

ω (12)

In these expressionsMk

m

*

=ω , is the natural mechanical angular frequency,

ofg

ofgL

e

CC

CCR

RC+

==11

ω is the natural electrical angular frequency, and mMω

βζ

2=

is the damping ratio of the system.

4. Simulation results

Simulation was first done using the linear model shown in Figure 3. The values of

elements used in this model were defined based on the technology parameters and device dimensions. We have thus considered mass of movable electrode M = 2x10-7 kg, spring constant was evaluated based on FEA as k = 138 kg s-², and the damping coefficient was estimated as β = 0.002 kg s-1. The capacitance Co = 1.9 pF and Cfg = 1 pF. The transduction factor was Γ = 1.4x10-6 NV-1.

Analysis was undertaken by an input displacement ranging from 1µm to 5µm.

The results of power and current output show obviously positive dependence on the increasing of the input. Figure 4 shows the 3D plot of the output power P depending on the load RL and vibrating frequency f for the input displacement of y = 5µm. The output power P clearly shows the maximum response at the natural vibrating frequency 4.3 kHz and the optimum value of the load resistance RL=58 MO. When the input displacement is 1µm, the frequency response of the output current for the optimal load resistance is shown in Figure 5.

2

4

6

8

0

50

100

150

2000

0.5

1

1.5

Frequency [kHz]

Output Power P = f(f,RL)

RL [MOhm]

Out

put P

ower

[µW

]

Figure 4. Power output from the linear model dependent on the vibration frequency and resistance

load (input displacement y=5µm).

2 3 4 5 6 7 80

5

10

15

20

25

30

Output Current I = f(f), RL = 58MOhm

Frequency [kHz]

Out

put C

urre

nt [n

A]

Figure 5. Frequency response of the output current obtained from the linear model (input displacement y=1µm).

Using the same component values as for the linear model and optimum conditions obtained from the linear analysis, we have performed the simulation based on the SIMULINK model. Analysis was done at the resonant frequency fr=4.3 kHz, with the input displacement ranging from 1 µm to 5 µm.

In the device described here, there is an effect causing a strong nonlinearity of the

second order that is based on the geometrical disposition of the structure. Let us suppose that both electrodes are in the equilibrium placed so that their planes of symmetry perpendicular to the main surfaces are identical. If the movable electrode is under the sinusoidal movement perpendicular to the plane of symmetry, both the positive and the negative halves of the displacement period lead to a decreasing of the capacity Ccf. The absolute value of the displacement must be thus considered which shows a strong presence of the second harmonic frequency in the output signal. This effect is illustrated in Figure 6 where the main frequency component has a value equal to the 2fr.

(a) (b)

1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-80

-60

-40

-20

0

20

40

60

time [ms]

Out

put C

urre

nt [n

A]

Output Current I = f(t), Displacement = 5µm

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250

300

Frequency [kHz]

Pow

er S

pect

ral D

ensi

ty [n

A²]

/1e3

Frequency Content of Output Current, Displacement = 5µm

Figure 6. Waveform and spectrum of the output current for the input displacement of y = 5 µm.

In order to eliminate the nonlinearity due to the device symmetry, a non-zero mechanical bias was introduced to the model. This bias is represented by a balance point X0 in Figure 2. In the following, a mechanically biased device will be supposed. Such a condition will be provided by setting the mechanical bias to a value at least equal to the peak electrode displacement.

When the input displacement of 1 µm is considered, the output current waveform presents perfect linearity after the first oscillation (Figure 7a). This is confirmed by the power spectral density of the current signal that contains principally one component corresponding to the frequency of the input signal (Figure 7b). The steady current value for this case corresponds to the output from the linear analysis that is shown in Figure 5.

(a) (b)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 10-3

-30

-20

-10

0

10

20

30

40

time [ms]

Out

put C

urre

nt [n

A]

Output Current I = f(t), Displacement = 1µm

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40

Frequency [kHz]

Pow

er S

pect

ral D

ensi

ty [n

A²]

/1e3

Frequency Content of Output Current, Displacement = 1µm

Figure 7. Waveform and spectrum of current response in linear range

(input displacement y = 1 µm).

Nonlinearity appears gradually when the input displacement increases. When the

input reaches 3 µm, there is an obvious distortion in the waveform of the current

output (Figure 8a); harmonic distortion effects due to the nonlinearities are visible in the frequency spectrum (Figure 8b) as multiples of the frequency of the input signal.

(a) (b)

1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-200

-150

-100

-50

0

50

100

time [ms]

Out

put C

urre

nt [n

A]

Output Current I = f(t), Displacement = 3µm

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

Frequency [kHz]

Pow

er S

pect

ral D

ensi

ty [n

A²]

/1e3

Frequency Content of Output Current, Displacement = 3µm

Figure 8. Waveform and spectrum of the output current for the input displacement of y = 3 µm.

5. Device implementation

The micro power generator was designed with the dimension based on the described analysis and implemented using a CMOS front-end electronic device fabrication and the post-CMOS back-end MEMS device process. The heavily doped poly-Si floating gate was sandwiched between two layers of low-stress silicon-rich nitride. Using aluminum as the sacrificial layer, the resonator was formed by electroplated gold. The resonator structures were totally suspended after removing the sacrificial layer. A micrograph of a fabricated generator with a 5x4 resonator array is shown in Figure 9. Device testing is undertaking now.

Figure 9. Micrograph of the implemented 5x4 array micro power generator.

6. Conclusions

A general model that describes the dynamic behavior of the implemented floating-gate electrostatic micro power generator was presented based on the nonlinear state equations that govern the electrical, mechanical and the coupling field respectively. Numerical simulation done in SIMULINK discloses the system response clearly and rapidly comparing the tedious analytical solutions. A linearized equivalent circuit network based on the electro-mechanical analogy was also built to make the small signal analysis straightforward. Both models show the consistency in the obtained results at linear range and predict the power of 1µW generated by the device around 4kHz assuming an input displacement of 5µm. Testing on the devices is undertaking and further analysis will be done by comparing the testing and simulation results.

7. References

1. H. Kulah and K. Najafi, “An electromagnetic micro power generator for low-frequency environmental vibrations”, MEMS 2004, pp. 237-240.

2. S. Meninger, J. O. Mur-Miranda, R. Amirtharajah, A. P. Chandrakasan and J. Lang, “Vibration-to-electric conversion”, IEEE Trans. VLSI Syst., Vol. 9, No. 1, 2001, pp. 64-76.

3. P. Gynne-Jones, S. P. Beeby, N. W. White, “Toward a piezoelectric vibration-powered microgenerator”, IEEE proc. Sci. Meas. Technology, Vol. 148, No. 2, 2001, pp. 68-72.

4. J. Boland, Y. H. Chao, Y. Suzuki and Y. C. Tai, “Micro electret power generator”, MEMS’03, Kyoto, 19-23, Jan., 2003, pp. 538-541.

5. T. Sterken, P. Fiorini, K. Baert, R. Puers and G. Borghs, “An electret-based electrostatic µ-generator”, Transducer’03, 8-12 June, 2003, pp. 1291-1294.

6. T. Ma, T. Y. Man, Y. C. Chan, Y. Zohar and M. Wong, “Design and fabrication of an integrated programmable floating-gate microphone”, MEMS 2002, pp. 288-291.

7. W. Ma, G. Li, Y. Zohar and M. Wong, “Fabrication and packaging of inertia micro-switch using low-temperature photo-resist molded metal-electroplating technology,” Sensors & Actuators A, vol. 111, pp. 63-70, 2004.

8. S. D. Senturia, “Microsystem design”, Kluwer Academic Publishers, Boston, 2001.