This paper has presented the design, fabrication and testing of both types of the
ECAs, i.e. rectangular and trapezoidal. In comparison with the classical rectangular ECAs,
the trapezoidal ones have several advantages and also minor disadvantage as below:
1. With the same teeth number, counted in the same length unit, the forces generated
in the trapezoidal ECA is larger than those in the rectangular one. The attracting force
in the movement direction increases 1.1 times initially, and 1.55 times at displacement of
10 µm.
2. The obtained results show that displacement of the trapezoidal ECA is larger
the those of the rectangular ECA. The largest deviation (62%) was observed at voltage
of 50 V. On the other hand, the measured values of displacement are smaller than the
theoretically calculated and simulated ones, which can be explained by measurement and
fabrication errors.
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Vietnam Journal of Mechanics, VAST, Vol. 34, No. 4 (2012), pp. 261 – 269
DESIGN AND FABRICATION OF THE TRAPEZOIDAL
ELECTROSTATIC COMB-DRIVE ACTUATOR
Pham Hong Phuc1, Dinh Khac Toan1, Dang Bao Lam1,
Nguyen Tuan Khoa1, Nguyen Tien Dung2
1Hanoi University of Science and Technology, Vietnam
2Thai Nguyen University of Technology, Vietnam
Abstract. This paper reports the design, fabrication and characterization process of
the trapezoidal Electrostatic Comb-drive Actuator (ECA) with the slope angle α = 2◦.
Together with the trapezoidal ones, the rectangular ECA with identical dimension was
also designed and fabricated for comparison purpose. In order to reduce calculating de-
viation, the fringing effect was also taken into consider while carrying out theoretical
analysis. The obtained results pointed out the fact, that the trapezoidal ECA excels the
rectangular ones with the same numbers of teeth in electrostatic force and displacement
generation, while requires relatively low driving voltage. But it is also observed that with
higher driving voltage (larger than 50 V), the trapezoidal ECA starts to lose its stability
(the lateral pull-in phenomenon occurs).
Keyword: electrostatic comb-drive actuator.
1. INTRODUCTION
A rectangular electrostatic comb-drive actuator (ECA) was firstly introduced by
Tang et al. [1]. While working, the fabrication errors can cause imbalance of the forces
acting on the electrodes, which can lead to destabilization of the rectangular ECA [2].
Impact of destabilization increases with a thinner finger thickness and lower stiffness of
the beam of the actuator. On the other hand, for overcoming this problem, choosing larger
size of the electrodes could also lead to bulky dimension of the whole system. In [3], Johan
B. C. Engelen et al.mentioned about the optimized shape of the ECAs. After optimization,
the ECA can generate larger and constant electrostatic force. With the same number of
fingers in a length unit, the ECA with optimized teeth can work with relatively high
stability, good vibration damping ability, and can produce the forces, which is 1.8 larger
than those generated by the normal rectangular ECA.
Since the first research of an ECA by Tang et al [1], there have been a large number
of applications of this type of actuators [4]. One of them was a linear motor [5] using
the linear ECAs, the other one was a rotational motor [6, 7] with the curved ECA. The
ECAs can also be found in the micro transportation and assembling systems [8, 9, 10]. In
those applications, the actuating members in the ECA are tangential electrostatic forces,
which are relatively small. The normal electrostatic forces, which are much larger, can also
262 Pham Hong Phuc, Dinh Khac Toan, Dang Bao Lam, Nguyen Tuan Khoa, Nguyen Tien Dung
be used e.g. in the gap-closing actuator (GCA) [5], etc. However, those actuators using
the normal electrostatic force have to deal with a disadvantage that the normal force
could rapidly decrease when the distance between the fingers of ECA becomes larger. The
rectangular ECAs using tangential forces have the simple structure, simple fabrication
process, and high stability, but they are still accompanied with low overall performance
characteristics such as small force at relatively high driving voltage and large dimension.
To overcome those disadvantages, the authors present the ECA with the trapezoidal finger,
in which the combination of the tangential and normal electrostatic forces will be used as
a driving force.
2. THE TRAPEZOIDAL ECA
2.1. ECA design
Fig. 1 shows the configuration of the trapezoidal ECA with following parameters:
slope angle α = 2◦, initial gap between two opposite finger d, overlap a, tooth thickness
b and tooth length lc. We know that when the slope angle α increases, the electrostatic
b d
a
lcα
Fig. 1. Configuration of the trapezoidal ECA
forces acting on the fingers also increase, but the gained displacement will decrease, and
vice versa. While the trapezoidal ECA is working, the overlap becomes larger, the gap d
becomes smaller, and therefore, the acting forces in the ECA increase simultaneously.
2.2. Edge effect (fringing effect)
When two plates are applied by V -voltage between (one plate is positive and the
other is negative) as shown in Fig. 2, an electric field is created between them. The electric
field lines seem linear in the center and non-linear toward the edge of the capacitor. This
phenomenon changes its capacitance, therefore, it varies forces applying on two plates.
According to Guo Zhanshe et al. [11], the capacitance C can be expressed as:
C =
abεε0
d
+
bεε0
pi
{
1 + ln
[
1 +
pia
d
+ ln
(
1 +
pia
d
)]}
+
aεε0
pi
{
1 + ln
[
1 +
pib
d
+ ln
(
1 +
pib
d
)]}
(1)
Design and fabrication of the trapezoidal electrostatic comb-drive actuator 263
where d is the gap between two fingers, a is the overlap of those and b is the thickness of
a comb finger.
Fig. 2. Electric field between two parallel plates
Fig. 3a presents the force analysis in the well-known rectangular ECA, in which the
acting tangential force Fy and normal force Fx can be expressed as below:
Fx =
1
2
∂C (d)
∂d
V 2, Fy =
1
2
∂C (a)
∂a
V 2 (2)
2.3. Force analysis in the trapezoidal ECA
In Fig. 3b, the initial position of a point on the surface of trapezoidal tooth is at
A0. After applying a driving voltage, the movable tooth is pulled in y-direction, thus, it
Fixed combs
Movable combs
α
b) Forces in the trapezoidal ECA
Fig. 3. Force analysis in the comb actuator
is moved a distance of ∆y, and the point A0 travels to the point A. Therefore, the gap d
and the overlap a are calculated as:
d = d0 −∆y sinα, a = a0 +∆y cosα (3)
264 Pham Hong Phuc, Dinh Khac Toan, Dang Bao Lam, Nguyen Tuan Khoa, Nguyen Tien Dung
As the rectangular ECA, the trapezoidal ECA is acted by the tangential force Ft
and the normal force Fn. Those are calculated in Eq. (2), where the gap d and the overlap
a are defined by Eq. (3). Fig. 3b shows that the force Ft is separated into Ftx in x-direction
and Fty in y-direction, and similarly, Fn is separated into Fnx and Fny .
Total forces acting on the one side of the finger in x and y direction can be calculated
as below:
∑
Fy = Fty + Fny,
∑
Fx = Fnx − Ftx (4)
Or ∑
Fy = Ft cosα+ Fn sinα and
∑
Fx = Fn cosα− Ft sinα (5)
Where parameters of the trapezoidal ECA were chosen as follows: the initial overlap
a0=5 µm, the initial gap d0=2 µm, the angle α=2
◦. The dimensions of the rectangular
ECA are intentionally chosen as the same as those of the trapezoidal ECA.
From Eqs. (1), (2) and (4), the relation between the forces acting in the ECA and the
moved distance ∆y can be established in Fig. 4a. The ratio of the force in the trapezoidal
ECA to the force in the rectangular ECA is shown in Fig. 4b.
y(µm)
F (N)
(a) Force - displacement relation
y(µm)
Ftrapezoidal/Frectangular
(b) The ratio of forces
Fig. 4. Effects of the ratio R/h (m = 1, n = 6, k = k1 = 1, T = 300 K,
ξ∗ = 0.1, L/R = 1)
It is observed that when ∆y increases, the force in the rectangular ECA decreases,
but on the contrary, the force in the trapezoidal ECA increases rapidly. The ratio of two
forces also increases as presented in Fig. 4b and this ratio approximately equals 1.55 when
∆y = 10 µm.
Design and fabrication of the trapezoidal electrostatic comb-drive actuator 265
2.4. Displacement in the trapezoidal ECA
Fig. 5a shows the designed trapezoidal ECA which following parameters: the length
of the beam l = 700 µm, the rectangular section bw×h = 4×30 µm
2, number of movable
fingers n = 134, E = 169 GPa is Young’s modulus of single crystalline silicon. Fig. 5b
presents the calculation diagram, where the forces acting on beam nF t are replaced by a
(a) Configuration of the trapezoidal ECA (b) Calculation diagram
Fig. 5. Force calculation in the trapezoidal ECA
spring with a stiffness of k. The stiffness k is defined as:
k =
4Ehb3w
l3
(6)
Replacing parameters in Eq. (6) by their values, we get the value of beam stiffness
k = 3.7845 µN/µm. Because the viscous drag force and friction force are negligible while
the ECA is working, the equation of force balance of the actuator can be presented as
below:
nFt (∆y)− k∆y = 0 (7)
Table 1. Calculated displacements of the trapezoidal ECA
Voltage(V) 20 30 40 50
Displacement (µm) 1,25 2,95 5,69 10,57
We use the numerical method to solve Eq. (7) with two steps: Firstly estimating
initial roots by geometry method, and then using Newton-Raphson method to solve Eq.
(7) to get value of displacement of the trapezoidal ECA. The results are shown in Tab. 1.
2.5. Simulation of displacements of the trapezoidal ECA
The rectangular and trapezoidal ECA are simulated in multi-field (electric and
mechanical fields) using FEA method (Finite Element Analysis). Fig. 6 shows a result
from simulation of the trapezoidal ECA working at voltage of 50 V and with the number
266 Pham Hong Phuc, Dinh Khac Toan, Dang Bao Lam, Nguyen Tuan Khoa, Nguyen Tien Dung
Fig. 6. Displacement of the trapezoidal ECA at 50 V (11 comb fingers)
Table 2. Simulated displacements of the trapezoidal ECA
Voltage(V) 20 30 40 50
Displacement (µm) 1,32 3,16 6,15 10,81
of comb fingers n =11. Simulation results with the driving voltages ranged from 20 V to
50 V are presented in Tab. 2.
3. FABRICATION AND CHARACTERIZATION
3.1. Fabrication
The ECAs has been fabricated by using SOI wafer with the thicknesses of the device
layer, buried SiO2 layer, and silicon substrate were 30 µm, 4 µm and 450 µm, respectively.
(a) Rectangular teeth (b) Trapezoidal teeth
Fig. 7. SEM images of the rectangular and trapezoidal ECAs
The rectangular and trapezoidal ECAs after fabrication are shown in Fig. 7a and 7b,
respectively. Fig. 8 shows the video-captured of the rectangular (8a) and trapezoidal (8b)
Design and fabrication of the trapezoidal electrostatic comb-drive actuator 267
(a) Rectangular ECA (b) Trapezoidal ECA
Fig. 8. The video-captured image when the ECAs working at 30V
ECA. It is clearly observed that at the same driving voltage (Vpp=50 V), the displacements
of the trapezoidal one is larger than those of the rectangular ECA.
To perform the operational test, the ECAs were driven by square wave periodic
voltage ranged from 20 V to 50 V. Fig. 9 shows the video-captured photograph taken
(a) Displacement - Driving voltage (b) Deviation between calculated and
measured data
Fig. 9. Displacement of trapezoidal ECA
when measuring displacements of the rectangular and trapezoidal ECAs at the voltage
of 30 V. Tab. 3 shows the measured displacements at 20, 30, 40 and 50 V, respectively.
Table 3. Measured displacements (µm) of the rectangular and trapezoidal ECAs
Finger/Voltage (V) 20 30 40 50
Rectangular 1 3 4 6
Trapezoidal 1.5 3.5 7 10.5
It is also observed that when driving voltage was increased to 60 V, the lateral pull-in
phenomenon occurs in the trapezoidal ECA.
268 Pham Hong Phuc, Dinh Khac Toan, Dang Bao Lam, Nguyen Tuan Khoa, Nguyen Tien Dung
3.2. Characterization
Fig. 9a shows the relation between the calculated, simulated and measured displace-
ments and the driving voltage of the trapezoidal ECA. With increased value of the driving
voltage, the value of displacement also increases and a kind of quadratic relation is created.
The largest deviation between the calculated data and the measured ones is close to 8%
at the voltage of 50 V as shown in Fig. 9b.
(a)
T
ra
p
e
z
o
id
a
l
/
R
e
c
ta
n
g
u
la
r
(b)
Fig. 10. Relation between measured displacements and driving voltage (a); the
displacement ratio (b) of the rectangular and trapezoidal ECAs
The graph comparing measured displacements in the rectangular and trapezoidal
ECAs is shown in Fig. 10. At low voltages, there is almost no deviation between both types
of ECAs. When driving voltage increases (from 40 V to 50 V), the measured displacements
of the trapezoidal ECA are much larger than those of the rectangular ones. At voltage of
50 V, this ratio reaches the value of 1.62.
4. CONCLUSION
This paper has presented the design, fabrication and testing of both types of the
ECAs, i.e. rectangular and trapezoidal. In comparison with the classical rectangular ECAs,
the trapezoidal ones have several advantages and also minor disadvantage as below:
1. With the same teeth number, counted in the same length unit, the forces generated
in the trapezoidal ECA is larger than those in the rectangular one. The attracting force
in the movement direction increases 1.1 times initially, and 1.55 times at displacement of
10 µm.
2. The obtained results show that displacement of the trapezoidal ECA is larger
the those of the rectangular ECA. The largest deviation (62%) was observed at voltage
of 50 V. On the other hand, the measured values of displacement are smaller than the
theoretically calculated and simulated ones, which can be explained by measurement and
fabrication errors.
Design and fabrication of the trapezoidal electrostatic comb-drive actuator 269
3. When driving voltage increased to 60 V, the lateral pull-in phenomenon occured
because of the unbalanced state of the force components in x-direction.
ACKNOWLEDGEMENT
This work is supported by the project number: B2012.01.32, Ministry of Education
and Training of Vietnam.
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Received July 31, 2012
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