In this chapter, we will discuss the fundamentals of micro electrostatic capacitive energy converters. The energy generation depends on the change of capacitance of a variable capacitor caused by vibration. Kinetic energy is converted into electrical energy during this process. The variable capacitor is initially charged by an external power supply such as a battery. The charge-discharge cycles are controlled by mechanical switches. The optimal design and analysis are presented in this chapter. A theoretical model was established to analyze the device characteristics; the results are compared with simulation in our previous work [39].
2.1 Characteristics of vibration sources
Vibration sources are generally more ubiquitous. The output power of a vibration driven converter depends on the nature of the vibration source, which should be known in order to estimate the power generating capability. There are various kinds of vibration sources in the environment. Measurement of different vibration sources was conducted by Roundy [9], as shown in Fig. 2.1. From the low level vibration sources, two characteristics can be observed. First, fundamental peaks occur at a common low frequency. Second, the high frequency vibration modes are lower in acceleration magnitude than the low frequency fundamental modes. Low level vibration in typical households, offices, and manufacturing environments is considered also as a possible power source for wireless sensor nodes.
Our measurement of the vibration spectrum of an air purifier is shown in Fig. 2.2.
A piezoelectric accelerometer (PCB Piezotronics model 353B17) was put on the air purifier and the data was collected by an oscilloscope. The fundamental vibration mode was at about 120 Hz, as observed by Roundy. The peak acceleration of the air
purifier is about 2.2m/s2 at about 120 Hz, as shown in Fig. 2.3. These results will be used as our targeted input vibration source due to its common existence.
Fig. 2.1 Vibration spectra by Roundy [9]
Fig. 2.2 Measurement of air purifier vibration Air purifier
To oscilloscope
Accelerometer
Fig. 2.3 Vibration spectrum of an air purifier
2.2 Operation principle
The main component of the electrostatic energy converter is a variable capacitor Cv [36]. A schematic circuit of the energy converter is shown in Fig. 2.4. It is composed of an auxiliary battery supply Vin, a vibration driven variable capacitor Cv
and an output storage capacitor Cstor, which is connected to the load RL. Two switches, SW1 and SW2, are used to connect these components and control the charge-discharge conversion timing [39].
Fig. 2.4 Operation circuit of the electrostatic energy converter
A B C
D
Vin Cv Cstor RL VL
SW1 SW2
20 70 120 170 220 Frequency (Hz)
1 10
2 Acceleration (m/s) 0.1
0.01 0.001
100 2.2 m/s2at 120 Hz
A more detailed schematic of the energy converter is shown in Fig. 2.5. The change of the capacitance is driven by the external vibration source. SW1 is implemented by a contact mechanism between nodes A and B. SW2 is actuated by the electrostatic pull-in force between nodes B and D. When node B reaches the pull-in voltage, it will be attracted by node D and touch node C.
Fig. 2.5 Variable capacitor schematic
In the energy conversion cycle, the variable capacitor Cv is first charged by the auxiliary voltage supply Vin through the switch SW1 when Cv is at its maximum Cmax, as shown in Fig. 2.6. After Cv is charged to Vin, SW1 is opened and the capacitance changes from Cmax to Cmin due to the electrode displacement by vibration. In the process, the charge Q on the capacitor remains constant (SW1 and SW2 both open).
Therefore, the terminal voltage across the capacitor Cv is increased and the mechanical energy is converted to the electrical energy stored in the capacitor.
Fig. 2.6 Capacitor charging and capacitance change by vibration VL
SW2 Cstor RL Pull-in electrode (GND)
B A
Cmin Cstor RL VL
When the capacitance reaches Cmin and terminal voltage reaches Vmax, SW2 is closed and the charge redistributes between Cv andCstor with balanced voltage Vo, as shown in Fig. 2.7 [36]. The energy stored in the variable capacitor Cv is transferred to the the storage capacitor Cstor. SW2 is then opened and Cv varies back to Cmax, preparing for the next conversion cycle. We notice that there are two conversion cycles in one oscillation cycle since the period of oscillation contains two headings.
The charge on Cstor is dissapated through the load resistance RL with a time constant τ
= RLCstor before it is charged again by Cv. The output voltage VL will eventually reach the steady state when the initial and final voltages of the charge-discharge process become equal.
Fig. 2.7 Charge transfer and discharge process [36]
2.3 Device design
The variable capacitor is the main component of the converter. In this study, capacitors with and without an external mass are both considered. To meet the 120Hz vibration source, we used an external mass mounted on the device to adjust its resonance frequency. In the case without external mass, the 120Hz requirement results in a very small mechanical spring constant. Because of the small mass, the kinetic energy in the system is inevitably small, causing very low output power compared to the case with an external mass. However, the vertical force caused by the weight of the mass is also reduced so that damage caused by the external mass can be alleviated.
The output power strongly depends on the maximum capacitance Cmax. But the electrostatic force also increases with the Cmax, indicating that it may influence the movement of the mass for large Cmax. Therefore, the variable capacitor must be designed by considering the 1 cm device area constraint, the maximum 2 capacitanceCmaxand the associated electrostatic force.
2.3.1 Auxiliary battery supply
The auxiliary battery supply is used to pre-charge the variable capacitor through SW1. Typical energy storage devices include capacitors, inductors and batteries.
Capacitors and inductors have lower energy density. They often serve as short-term energy storage cells. Batteries, such as NiZn, NiMH, NiCd, and Lithium-ion (Li-ion), store energy chemically and are rechargeable. Among these types, Li-ion batteries (Fig. 2.8) offer the best performance with high energy density, high discharge rate, high cell voltage, long life span, and no memory effect. In this study, LIR1620 (3.6 V, Φ 16 mm, H 2.2 mm, 1.2 g) and LIR2016 (3.6 V, Φ 20 mm, H 1.8 mm, 1.6 g) Li-ion cells can be used as the auxiliary battery supply. Moreover, the battery can act as part of the external mass if it is well bonded on the device.
Fig. 2.8 Lithium-ion rechargeable battery
2.3.2 Variable capacitor design
SOI wafers with highly doped thick device layers and deep silicon etching technology were used to fabricate the devices. An in-plane gap-closing comb structure is used for the variable capacitor, as shown in Fig. 2.9. Compared with the in-plane overlap type comb structures, this topology has the advantage of larger capacitance change for smaller displacement. Compared with the out-of-plane gap closing capacitors, this topology has the advantage of lower mechanical damping loss and possibility to incorporate minimum gap control designs.
Fig. 2.9 Top view of the in-plane gap closing variable capacitor topology
The symbols used in the following discussion are listed below:
d: initial gap between comb fingers dmin: minimum air gap between fingers t: comb finger width
Lf: overlap length of comb fingers h: thickness of device layer
z : relative displacement between the movable and fixed electrodes Ng: number of variable capacitor finger cells
Relative motion
S S’
⇒
ε : permittivity of free space (0 ε0 = 8.842 10× -12 F/m) μ: viscosity of air at 1 atm (μ =1.82 10 × -5 Pa-sec)
α: damping coefficient depending on effective region (α ≈ 1.74)
The total variable capacitance between the comb fingers is [36]
v 2Ng 02 L hd2f C (z)=
d -z
ε (2.1)
From this equation, the variable capacitance strongly depends on the comb finger structure. Layout design under restricted area directly affects the variable capacitance.
A general model of the comb finger layout is shown in Fig. 2.10. The total layout area is limited to 1cm . A number of free parameters can be adjusted to obtain the optimal 2 design. In Fig. 2.10, the layout can be divided into three parts. First is the gray portion for the support of the fixed comb fingers. The second part is the rest of the layout area which is defined as S. The third part is the area occupied by the finger cells in S, which is defined as S', the area ratio is defined as
r S'
S = S (2.2)
Fig 2.10 A generalized layout design
For the movable part of the variable capacitor, the area S' occupied by fingers can be divided into cells, as marked by the rectangle in Fig. 2.11. The air gaps around the fingers are equal to maintain consistent etching rate in the ICP deep RIE process.
Fig 2.11 Single cell schematic
The number of fingers is the finger area S'in layout divided by the finger cell area g r In these two equations, two free parameters, Sr and d, are utilized to optimize the layout design.
The device will be fabricated on SOI wafers. The thickness h is chosen as 200 μm to have large capacitance. The finger width t of 10 μm is restricted by the aspect ratio of up to 20:1 in the deep reactive ion etching process. A maximum device area of 1cm is set as the device size constraint. 2
2.3.3 Dynamic analysis
The dynamic analysis is performed to decide the mechanical spring constant k under certain proof mass m in order to achieve the theoretical maximum displacement under the targeted input vibration. The electro-mechanical dynamics of the variable capacitor can be modeled as a spring-damper-mass system, as shown in Fig. 2.12.
Fig. 2.12 Schematic of the dynamic model
The equation of motion is
mz + b z + (k +k )z = -my m 0 e (2.5) where y (=Y0sinωt) is the displacement of device frame caused by the vibration, z is the relative displacement between movable and fixed plates, k is the mechanical 0 spring coefficient, b zm is the mechanical damping force, and k z is the e electrostatic force caused by the charge on the capacitor, which acts as a negative spring force with ke as the electrostatic spring constant.
The mechanical damping can be determined from experimental data. From our previous MMA (MEMS motion analyzer) measurements, the quality factor is approximately 6 to 8. The corresponding mechanical damping is 0.38 to 0.52 based on
m
bm
ke
k z(t)
y(t)
2
m
Q = km
b (2.6) The electrostatic force induced by constant charge Q on Cv is [36]
e e 2 The force can be viewed as a negative spring force with a negative electrostatic spring constantk . The electrostatic spring constant is determined by the charge Q on the e variable capacitor which alternates betweenQmaxand Qminin the charge-discharge process. The corresponding electrostatic spring constants are defined as ke,max and
e,min
k . A newly introduced parameterQ is defined as the charge ratio between r
QmintoQmax
r min
max
Q Q
=Q (2.8) The charge ratio is related to the output characteristics. Further discussion of this parameter will be presented in the next section.
The relationship between the electrostatic spring constant and the dynamic activities of the system is
1 0 e,max The system becomes a piecewise linear system described by
1 m 1 1 1 where z is the relative displacement between fingers after charging at ends and 1 before discharging at center with Q = Qmax , z is relative displacement after 2 discharging at center and before charging at ends withQ = Qmin. In Fig. 2.13, the conditions z(t)z(t) < 0 andz(t)z(t) 0 > are satisfied at z = z or z ' and 1 1 z = z or z ' 2 2
respectively. The solutions to Eq. 2.10 are composed of homogeneous solution and
z and p1 z are the particular solutions of the equations as shown below p2
Fig.2.13 Displacement of shuttle mass versus time
Due to the difference in total spring constants, the time needed for the mass to move from ends to center is not equal to the time needed from center to ends.
Therefore a parameterα is defined as the portion of time occupied by maximum charge onC . V
wheret1 andt represent the time occupied by the 2 k and1 k equations in Eq. 2.10, 2 respectively. Duringt , the charge on1 C isV Qmax; duringt , the charge on2 C isV Qmin.
Fig. 2.14 Time shift for t and T scale
A new time scale is also defined to describe the system behavior. At T = 0, the mass is at the maximum displacement, as shown in Fig. 2.14. The delay between T and t is TΔ . If T t - T= Δ is substitute into Eq. 2.11, the solution or the displacement function can be rewritten as
( )
The velocity is the derivative of the displacement functions
-1 1 -1 2
The boundary conditions between z and 1 z can be found from the stable 2 oscillating dynamics. At the ends of its travel range indicated in Fig.2.13, the velocity of the moving mass is equal to zero; at the center of travel, the velocity of z and 1 z 2 are continues and the displacement is equal to zero. The next conversion cycle has the same dynamic characteristics; however, the direction is opposite to the previous one.
Therefore, there are two conversion cycles in one oscillation cycle. The boundary conditions used in this analysis can be summarized as
z (0) = 01 , (2.16a) Eqs. 2.16a and b mean the velocity at both ends are equal to zero; Eqs. 2.16c and d show the amplitude equals to A; Eq. 2.16e and f show that the displacement at center is zero; Eq. 2.16g is the continuity of velocity at center. The next conversion cycle has identical behavior as the previous one except for the opposite direction. The boundary conditions are depicted in Fig. 2.15.
Fig. 2.15 Boundary conditions between z1 and z2.
The amplitude A solved from above equations is directly related toke,max. The amplitude A and the initial gap d of the comb fingers are depicted in Fig. 2.16. The capacitance is maximum as the shuttle mass reaches its maximum displacement A.
The electrostatic spring constant ke,max is related to A as
e,max 2max 2max in2 20 f2 2 in2
0 f 0 f
-Q -C V -2Nε L hdV
k = = =
2Nε L hd 2Nε L hd (d -A ) (2.17) The electrostatic constant ke,min is also a function of amplitude because
e,min min 2 e,max 2r e,max
max
k = (Q ) k = Q k
Q (2.18) Therefore, the parameters related tok such as the nature frequencies and damping e ratios can also be expressed as functions of the amplitude A. The seven unknowns
1 2 3 4
2.16. Since they are complicated set of nonlinear equations, a MATLAB solver is used to find the numerical solution.
Fig. 2.16 Fingers at maximum displacement
2.3.4 Static analysis
Static analysis can be used to obtain mathematical guidelines for overall system design including layout and circuit. Once the amplitude is decided from the dynamic analysis, parameters such as initial gap and mechanical spring constant can be determined directly. At the output node of the variable capacitor, the charge ratio before and after SW2 is switched and can be expressed as
r min min o min o o
max max min min max max
C V C V V
Q = Q = = =
Q C V C V V , (2.19) where Vmax oand V are the capacitor voltage before and after discharge by SW2, as discussed in section 2.2. Assume the storage capacitorCstoris large and the output ripple can be ignored, the charge flow through output load is
out o
L
Q = V
2fR (2.20) The charge flow into the variable capacitor is
Q = C V (1- Q ) (2.21) in max in r m
d A
Eq. 2.22 can be rearranged to the following form L r
min r
R = Q
2fC (1- Q ) (2.23) From Eq. 2.22, we know that the charge ratio is determined by the frequency of vibration, the minimum capacitance and the output load. If the voltage ripple of
Cstor needs to be considered, the charge ratio is
The approximation of Eq. 2.24 for largeCstoris the same as Eq. 2.22.
The output power can also be determined as we know the amplitude of the shuttle mass displacement. For largeCstor, the output power is
Output power can also be expressed in following form from Eq. 2.22 and Eq. 2.23 P = 2fC V Q (1- Q ) (2.26) out min max2 r r
It is worthy to mention that an optimum load based on a similar system analysis was derived in [41] by finding the derivative of output power with respect to the output load. The optimal load is [41]
L,opt
min
R = 1
2fC (2.27) However; the result in Eq. 2.27 was based on an assumption of fixed amplitude. The same result can also be derived by finding the derivative of output power respect to
charge ratio in Eq. 2.26 if we regard other parameters as constants. The maximum output power occurred atQr =0.5. As we substituteQr =0.5 into Eq. 2.23, we can obtain an identical result as Eq. 2.27.
However; in this research, the charge ratio is related to the electrostatic force and thus the amplitude. Therefore, we should not assume a fixed amplitude arbitrarily.
The assumption ignored the electromechanical coupling effect in the mass-spring-damper system. We treat Q as an operation parameter instead of a r constant.
2.4 Optimizing process
Because Eqs. 2.16 are highly nonlinear, we use “fsolve” in Matlab to find numerical solution. The “fsolve” solver needs an initial guess to find the solution of the set of equations. At low amplitude, the system can be regarded as linear. The starting guess was therefore found by the approximated linear results. For the next calculation the previous solution was used as new starting guess. After repeating these steps, the maximum amplitude can be found.
The dimensions of the device are partially fixed as shown in Table 2.1. Four free parameters are used to find the optimal condition under restricted area, as listed in Table 2.2. The detailed optimization flow chart is shown in Fig. 2.17.
Table 2.1 Fixed parameters of optimizing process
Parameter Description of constants Value
h Device thickness 200μm
t Finger width 10μm
Table 2.2 Free parameters of optimizing process
Variable Description of variables Range
k 0 Mechanical spring constant 1~3000N/m
Q r Charge ratio 0.1~0.9
S r Surface ratio 0.05~0.95
d Initial gap 1μm~70μm
Fig. 2.17 Optimization flow chart
Qr Sr
Mechanical spring constant k0 (N/m)
The oscillation amplitude A was calculated by the “fsolve” solver for each set of free parameter values in the ranges in Table 2.2. For example, Fig.2.18 shows the calculation results for the initial gap of 40 mμ , charge ratio of 0.5 and area ratio of 0.8 as a function of k0. At k0= 2392N / mthe amplitude reaches the maximum of 39.35 mμ . The corresponding output power is 45.8μW and output voltage is 68.62V.
Similar results can also be obtained in frequency domain. Fig.2.19 shows the amplitude and power versus frequency for k0=2392N/m, which corresponds to the maximum power in Fig 2.18.
Fig 2.18 Amplitude and power vs. spring constant
It is interesting to notice that there is a range of design or operation parameters where no solution of the oscillation amplitude A exists in Figs. 2.18 and 2.19. This phenomenon is discussed in the next section from an energy point of view.
50
Fig. 2.19 Amplitude and power vs. Frequency
2.4.1 Conditions for normal oscillation discussion
At maximum displacement, the energy in the system is the sum of the mechanical spring potential energy and the capacitor electrostatic energy. The capacitor energy increases after charging by the battery. At center, the system energy is the sum of the kinetic energy and the capacitor energy. The capacitor energy decreases after discharging to the load. The system energy at different instances and positions is listed in Table 2.3. Fig. 2.20 shows these calculated system energies at different frequencies. The interaction between the external force and the restoring force in the system cause two conditions for which the energy can not be balanced and no normal oscillation can be found, as discussed next.
at center before discharging
Table 2.3 System Energy at different instances and positions
Position
Fig. 2.20 System energy versus driven frequency
In Fig 2.20, we can see the energy at center after discharging crosses the energy at ends after charging at 120Hz. As the system approaches resonance at f2, the output reaches maximum and the energy left in the system after discharging at center becomes insufficient for the mass to move to maximum displacement. Similar observation can be found at f1. As the system approaches the frequency at f1 the energy
the mass to maintain stable operation.
2. 4 .2 Optimum design
The output voltage is limited to 40V for further integration with power management circuits. The calculated power can be plotted versus area ratio Sr and
The output voltage is limited to 40V for further integration with power management circuits. The calculated power can be plotted versus area ratio Sr and