Design and modelling of MEMS DC–DC converter H. Samaali, B. Ouni✉ and F. Najar A new microelectromechanical systems (MEMS) design of an integrated DC–DC converter electrostatically driven is proposed. The converter has one output based on the tunable capacitor principle. The output voltage has been achieved by designing a capacitor with different gaps. Compared to previous research works, the complete mathematical model of the converter is proposed. To examine the dynamic behaviour of the converter two different methods are used: the differential quadratic method and the Galerkin method. The obtained results show good agreement between the methods. In addition, two mechanical switches are added in the structure to replace the conventional diodes that have the advantage of zero leakage current, low power consumption and synchronous operations. Results show that the converter takes about 22.73 ms to convert a 5 V voltage to a range of voltages that are varied from 10 to 50 V.
Introduction: The integrated microelectromechanical systems (MEMS) converter converts DC voltage to another DC voltage, that is, step-up or step-down conversion using a variable capacitor [1, 2]. The electrostatically driven microstructure is suitable to be used in the monolithic resonant DC–DC converter [3]. In [3], the mechanical resonant structure is used instead of an LC circuit and a transformer. DC–DC converter may be used to supply micro-resonator drivers [4], scientific instruments [5] and micro-propulsion units [6]. Previous researches [1, 7–9] have used both interdigitated comb fingers and parallel plates to design and to model MEMS DC–DC converters. All these works have presented a lumped-model (mass-spring) to model the converter and they have used diodes or active contact circuits to store the electric energy. In this Letter, a new design of a DC– DC converter as well as its associated complete mathematical model is introduced. In fact, we propose a complete mathematical model of the converter and we replace the traditional diode by mechanical switches that have the advantage of zero leakage current, low power consumption and synchronous operations. Later, the proposed mathematical model is solved with two different methods [the differential quadratic method (DQM) and the Galerkin method]. L fixed plate substrate d1
b
K1 substrate
Cvar1 mobile plate
x y
Cstor
Vout
K2
h
VDC
V0
VAC
d2
Cvar2
hc
substrate
fixed plate
Lc
bc
a
Vvar1
Cvar1
V0
injected into it. Owing to the drive signal (VDC + VAC), the cantilever system (cantilever beam and mobile plate) reverses its direction and both switches K1 and K2 are Off, keeping a constant charge Q. At the lower position (ymin) that corresponds to the minimum value Cmin of Cvar1, K2 turns On (stiffness K is added also to the system) and the charge Q is injected into the storage capacitor Cstor. The output voltage Vout provided by the device can be expressed as Vout = (CMax/ Cmin)V0 [10]. The driving electrical signal of the MEMS converter is composed by a DC voltage VDC = 5 V and a bipolar square waveform voltage VAC = ±5 V, at a frequency f = 11 728 kHz that equals the natural frequency of the MEMS converter structure. The electric schema shown in Fig. 1b gives an explicit description of the system. The variable capacitors Cvar1 and Cvar2 are expressed as: Cvar1 = ((εbcLc)/(d1+y(L, t))), Cvar2 = ((εbcLc)/(d2−y(L, t))). The geometric parameters of the MEMS converter are summarised in Table 1.
Table 1: Geometrical parameters of MEMS converter L = 250 μm Lc = 50 μm h = 1.5 μm
Length of the cantilever Length of the three plates Thickness of the cantilever, upper and lower plates
Thickness of mobile plate hc = 10 μm Width of the three plates bc = 40 μm Width of the cantilever beam b = 5 μm Initial gap distance between mobile and lower plates d2 = 15 μm Initial gap distance between mobile and upper plates d1 depends on the design Modulus of elasticity Density Permittivity of free space Mass of the mobile plate Stiffness of switches K1 and K2 Mass moment of inertia of the mobile plate
E = 166 GPa ρ = 2332 kg/m3 ε = 8.851 × 10−12 F/m M = 1.165 × 10−14 kg K = 191 N/m 1 J = ML2c 3
Equation (1) and its boundary conditions (2)–(4) are solved using the DQM or the Galirkin method (for more details see works published by Samaali et al. [11, 12]). Legend: Position of the lower switch ymin = −5 μm Position of the upper switch ymax = 5 μm Displacement of the mobile plate in the y-direction y(x, t) Derivative with respect to space variable x ((∂y(x, t))/∂x) Derivative with respect to time t ((∂y(x, t))/∂t) Derivative with respect to space variable x at point L ((∂y(L, t))/∂x) Derivative with respect to time t at point L ((∂y(L, t))/∂t) Derivative with respect to space variable x at point 0 ((∂y(0, t))/∂x) Displacement of the mobile plate at point 0 and L y(0, t), y(L, t)
Vout
K1
5 K2
y, µm
VDC Vvar2 Cvar2
VAC
b
0
–5
Fig. 1 Schematic of MEMS DC–DC converter and its electric schema a Schematic of MEMS DC–DC converter b Its electric schema
Design and modelling: The proposed design is presented in Fig. 1a. It is based on two variable capacitors Cvar1 and Cvar2. To a cantilever beam, that is fixed to the substrate at the left end on the right end , is attached a rigid plate of mass M. When an electrical voltage is applied between the lower and the mobile plate, a non linear electrostatic force is generated causing the movement of the mobile plate according to (1) and its boundary conditions (2)–(4). At its high position (ymax) that corresponds to the maximum value CMax of Cvar1 (capacitance created by the upper and mobile plates) the switch K1 turns On. Knowing that the switch K1 is a cantilever with stiffness K; this should be added to (1). At this step, the capacitor Cvar1 is connected to the initial voltage V0 and a charge Q is
0
1000
2000
3000 4000 t, µs
5000
6000
Fig. 2 Effect of switches cantilever on response of DC–DC converter
Results and discussion: Fig. 2 resumes the effect of the switches cantilever (here the switches serve also as a stopper to limit the displacement of the central plate and prevent an eventual short circuit). As shown in Fig. 2, without switches (red curve), the mobile plate oscillates between –8.135 and 8.135 µm, whereas with switches (blue curve) placed at ymin=−5 μm and ymax=5 μm, its oscillation varies between −5.3 and 5.3 µm. In addition, the mobile plate takes 34.83 µs before it stops osillating. This time is enough to charge and discharge the variable capacitor Cvar1.
ELECTRONICS LETTERS 28th May 2015 Vol. 51 No. 11 pp. 860–861
Fig. 3 shows the variation of Vout with the initial gap distance d1 between the plates of capacitor Cvar1. The design result shows the Vout range from 10 to 50 V for an initial input voltage Vout=5 V. Consequently, the gain in voltage varies from 100 to 900%, which is very significant.
2 ∂3 y(L, t) ∂2 y(L, t) Lc ∂ ∂ y(L, t) 1bc (VDC + VAC )2 EI =M +M −
2 3 2 2 ∂∂t 2 ∂x ∂t 2 ∂y(L, t) /∂x
1 1
− × d2 − y(L, t) d2 − y(L, t) − Lc ∂y(L, t) /∂x (4)
50
© The Institution of Engineering and Technology 2015 16 December 2014 doi: 10.1049/el.2014.4378 One or more of the Figures in this Letter are available in colour online.
Vout, V
40
30
H. Samaali and F. Najar (Applied Mechanics and Systems Research Laboratory, Tunisia Polytechnic School, University of Carthage, BP 743, La Marsa 2078, La Marsa, Tunisia)
20
10 6
7
8 d1, µm
9
10
B. Ouni (Laboratory of Electronic and Microelectronic, Faculty of Science, Monastir, Tunisia)
Fig. 3 Variation of Vout with initial gap distance d1
✉ E-mail:
[email protected]
We define the conversion time Tconv as the time required to convert V0 to Vout. On the basis of Fig. 4, we can show that Tconv of our design is 22.73 ms.
References
4
5 µm
y, µm
2
0
–2
–4 0
200
400
600
800
t 0.000017418 s
1000
1200 22.73 ms
Fig. 4 Estimation of conversion time Tconv
Conclusion: This Letter demonstrates a synchronous capacitor DC–DC converter circuit. The mathematical model of the proposed design has been presented and solved using the DQM and the Galarkin method. Mechanical switches are proposed in the model with the aim of replacing the conventional diodes. A parametric study has been done and has shown that the 5 V voltage can be converted from 10 to 50 V by varying the gap distance from 6 to 10 µm. The design takes only 22.73 ms to convert the input voltage ⎧ ∂4 y(x, t) ∂y(x, t) ∂2 y(x, t) ⎪ ⎪ + rA +C =0 EI ⎪ 4 ⎪ ∂x ∂t ∂t 2 ⎪ ⎪ ⎪ ⎪ when ymin ≤ y ≤ ymax ⎪ ⎪ ⎪ ⎪ ⎨ ∂4 y(x, t) ∂y(x, t) ∂2 y(x, t) + rA EI + Ky(L, t) + C =0 4 (1) ∂x ∂t ∂t 2 ⎪ ⎪ ⎪ when y ≤ ymax ⎪ ⎪ ⎪ ⎪ ∂4 y(x, t) ∂y(x, t) ∂2 y(x, t) ⎪ ⎪ ⎪ + rA EI + Ky(L, t) + C =0 ⎪ 4 ⎪ ∂x ∂t ∂t 2 ⎩ when y ≥ ymin y(0, t) =
EI
∂y(0, t) ∂y(L, t) = y(L, t) = =0 ∂x ∂x
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(2)
∂2 y(L, t) Lc ∂2 y(L, t) = −M 2 2 ∂t 2 ∂x
∂ ∂2 y(L, t) Lc 2 1bc (VDC + VAC )2 − M +J +
2 2 2 ∂x∂t 2 ∂y(L, t) /∂x
Lc ∂y(L, t) /∂x
× d2 − y(L, t) − Lc ∂y(L, t) /∂x d2 − y(L, t)
− ln d2 − y(L, t) − Lc ∂y(L, t) /∂x (3)
ELECTRONICS LETTERS 28th May 2015 Vol. 51 No. 11 pp. 860–861
