A Variable-Capacitance Vibration-to-Electric Energy Harvester

Transcription

1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS 1 A Variable-Capacitance Vibration-to-Electric Energy Harvester Bernard C. Yen, Student Member, IEEE, and Jeffrey H. Lang, Fellow, IEEE Abstract Past research on vibration energy harvesting has focused primarily on the use of magnets or piezoelectric materials as the basis of energy transduction, with few experimental studies implementing variable-capacitance-based scavenging. In contrast, this paper presents the design and demonstration of a variable-capacitance vibration energy harvester that combines an asynchronous diode-based charge pump with an inductive energy flyback circuit to deliver 1.8 µw to a resistive load. A cantilever beam variable capacitor with a 65-pF DC capacitance and a 348- pf zero-to-peak AC capacitance, formed by a cm 2 spring steel top plate attached to an aluminum base, drives the charge pump at its out-of-plane resonant frequency of 1.56 khz. The entire harvester requires only one gated MOSFET for energy flyback control, greatly simplifying the clocking scheme and avoiding synchronization issues. Furthermore, the system exhibits a startup voltage requirement below 89 mv, indicating that it can potentially be turned on using just a piezoelectric film. Index Terms Vibration energy harvesting, vibration energy scavenging, variable-capacitance-based energy conversion. I. INTRODUCTION IN RESPONSE to a growing interest in autonomous systems, such as RF sensor nodes that must operate for prolonged periods of time without human intervention, vibrationto-electric energy harvesters have attracted wide research interest. Three main strategies for energy transduction dominate: piezoelectric, magnetic, and electric. Magnetic harvesting can be further categorized into systems with a time-varying inductor and systems that employ moving permanent magnets. Likewise, electric harvesting employs either a time-varying capacitor or a moving permanent electret. Piezoelectric materials, such as quartz and barium titanate, contain permanently-polarized structures that produce an electric field when the materials deform due to imposed mechanical forces [1]. Such a mechanically excited element can be modeled as a current source with a capacitive source impedance [2] where the current amplitude depends on the applied force. Therefore, if the material is connected to a vibration source, it can harvest the vibration energy and generate electric power [3], [4]. Magnetic energy harvesters, on the other hand, convert vibration energy into an induced voltage across wire coils, which can then deliver power to a load. This is typically done by attaching either a permanent magnet, such as that made from Manuscript received March 25, 25; revised June 1, 25. This paper was recommended by Associate Editor M. K. Kazimierczuk. The authors are with the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 2139, USA ( bernardy@mit.edu; lang@mit.edu). /$. c 25 IEEE Neodymium Iron Boron, or a coil of wire onto a cantilever beam that is vibrationally actuated; the other element remains fixed [5], [6]. In either scenario, the coil will cut through magnetic flux as the cantilever beam vibrates, creating an induced voltage at the terminals of the coil. Vibration energy can also be transduced using a variable inductor, although no such studies have been reported to date, presumably due to the inherent advantages of permanent magnets. Finally, electric energy harvesting transduces vibration energy through the electric fields between a parallel plate capacitor. Typically, charge is injected into the capacitor at maximum capacitance and pulled off at minimum capacitance. Between these points, vibration separates the plates against their attractive force, performing work on the injected charge, which is then harvested [7], [8], [9], [1]. Besides the variable capacitor, one can also employ a moving layer of permanently embedded charge, or electret, to carry out electric energy harvesting [11], although such systems currently have power densities inferior to those using variable capacitors. Roundy et al. prototyped a simple variable-capacitancebased charge pump and showed that mechanical-to-electrical energy transduction was possible, but they did not explore the regime where the harvester saturates due to the lack of an energy flyback path [12]. Mur-Miranda conducted extensive studies on a synchronously-excited capacitive energy harvester involving two active switches [8]. Although energy conversion was demonstrated, difficulties in gate clocking and inefficiencies of the power electronics prevented net energy conversion to a load. Miyazaki et al. improved the timing scheme of this topology and achieved 12 nw of converted power from a 45 Hz vibration [9]. However, they did not analyze whether energy injection from the clock signal contributed to the harvested energy. As this paper will show, such energy injection can be significant, and can be mistaken for converted energy. Here, we present an optimized asynchronous capacitive energy harvester that requires only one active switch, thereby greatly simplifying clocking. For this paper, emphasis is placed on the circuitry, not on the implementation of the variable capacitor. The circuit employs a charge pump in its forward harvesting path and an inductive energy flyback to return net energy to a central reservoir. The harvester is demonstrated experimentally using a spring steel variable capacitor with capacitance variation between 32 pf and 998 pf and an outof-plane resonant frequency of 1.56 khz. It delivers 1.8 µw of power to a resistive load, translating to an efficiency of 19.1%. Experimental data prove that net energy conversion does not result from clock energy injection.

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS 2, - $,. / ( ) *+ ( *3 Fig. 1. Block diagram of asynchronous capacitive energy harvester. % % %, *$ " &! " &! " #$ & #$ ) *+ ' ' ' Buck-converter flyback is presented here as a highefficiency standard baseline implementation against which other schemes, such as switched-capacitor flyback, can be compared. Thus, even though the flyback circuitry involves an inductor, it is worthwhile examining. When employed as part of an integrated system, the required inductor in the flyback path could be implemented as an external component. Alternatively, planar inductors could be deposited or electroplated as part of the fabrication process [13]. II. POWER ELECTRONICS Fig. 1 shows a block diagram of the capacitor-based energy harvester. The charge pump in the forward path transduces vibration energy into electric energy, which is then delivered to a temporary storage. Periodically, the inductive flyback circuit sends the harvested energy back to a reservoir that powers the attached load, priming the charge pump. A circuit implementation of the block diagram is shown in Fig. 2, with R LOAD representing the load. The charge pump consists of two diodes, D 1 and D 2, that sandwich a variable capacitor C VAR. The flyback section includes a gated MOSFET, freewheeling diode D FLY, and inductor L FLY ; R WIRE and R CORE model the winding and core loss of the inductor respectively. Finally, the reservoir C RES and temporary storage C STORE are each formed by a single capacitor under the requirement that C RES C STORE. In this paper, the MOSFET gate drive is powered from an external power supply for simplicity. It is recognized that the gate drive, along with other required control logic, must ultimately be powered directly from the harvested energy. This appears to be feasible for an integrated system because power generation on the order of microwatts is demonstrated below. Based on a typical MOSFET gate-source capacitance of C GS = 1 pf, a drive voltage of v GS = 1 V for an IC implementation, and a drive frequency of f CLK = 475 Hz, the power dissipated in driving the MOSFET gate is P CLK = C GS v 2 GSf CLK = 5 nw. (1) Thus, the harvested energy should be adequate. A. Charge Pump Section To understand the transduction of vibration energy to electric energy, assume initially that v VAR = v RES = v STORE and C VAR = C MAX, the maximum capacitance of the variable capacitor. At this point, both D 1 and D 2 are off, meaning that Q VAR, the charge on the variable capacitor, cannot change. As ambient vibration pulls the capacitor plates apart, C VAR decreases, which causes v VAR to rise given constant Q VAR. This voltage rise eventually turns on D 2, resulting in the partial : ; < = >? 4@ AB C 7 D E 5 F8D < FG8B? Fig. 2. Capacitive energy harvester with source-referenced clock controlling the flyback switch. discharge of C VAR into C STORE with v VAR = v STORE. When the vibration reverses direction, C VAR increases, causing v VAR to drop, thereby turning off D 2. As v VAR drops further, D 1 turns on, which results in charge injection from C RES to C VAR. This charging causes v VAR to be held at v RES. The cycle repeats itself as charge and energy are pumped into C STORE. Without the flyback section, v STORE will eventually saturate. Because this saturation voltage impacts the overall energy harvester performance, it is now computed analytically. Let the variable capacitor exhibit C MIN C VAR C MAX, and define a complete energy harvesting cycle as one variation of C VAR from C MAX to C MIN and back to C MAX. Let v STORE,n, where n is an integer index starting from, represent the voltage on C STORE at the end of n cycles. Based on this definition, v STORE, = v RES. Furthermore, since C RES C STORE, approximate v RES as a constant voltage source. Finally, assume the diodes are ideal. Fig. 3(a) shows an equivalent circuit diagram at the start of cycle n. At this point, the total charge, Q TOTAL, stored in C VAR and C STORE is Q TOTAL,n 1 = C MAX V RES + C STORE v STORE,n 1. (2) Next, as C VAR decreases, v VAR rises so that D 1 immediately turns off and Q TOTAL remains constant. Eventually, v VAR increases enough so that D 2 turns on and the equivalent circuit diagram shown in Fig. 3(b) results. During this part of the cycle, charge is transferred from C VAR to C STORE, giving C STORE v STORE = v STORE,n 1 C MIN + C STORE C MAX + V RES (3) C MIN + C STORE at the end of the half cycle. Then, as C VAR increases and v VAR decreases, D 2 immediately turns off, keeping the charge on C STORE, and hence v STORE, constant throughout the remainder of the cycle. Therefore, v STORE,n = v STORE (4) where v STORE is taken from (3). Because charge is transferred from C VAR to C STORE during the cycle, v VAR will head towards a voltage that is less than V RES as C VAR continues to increase. Therefore, D 1 is guaranteed to turn on by the point when C VAR = C MAX, so Fig. 3(a) is again the equivalent

3 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS 3 H I H J [ \ [ ] Z R S O K L M N T U R S O K O P Q R S T U O P Q R S WX Y I l e f b ^ _ `a g h b c d e f ij ^ b c d e f g h b c d e f ij V V k k (a) Start of first half of cycle (b) Start of second half of cycle Fig. 3. Equivalent circuit diagrams for different parts of the harvesting cycle. Fig. 4. u v w x ƒ u v yz ˆ ƒ Š m nop ~ Œ Œ Ž Ž u v w x { { } ƒ ˆ ƒ Š Ž Ž q r s t u v yz Q-V plane contour representing one energy harvesting cycle. circuit diagram for the charge pump at the start of the next cycle. Define C STORE α = C MIN + C STORE (5) C MAX β = V RES C MIN + C STORE (6) so that (3) and (4) can be rewritten as v STORE,n = αv STORE,n 1 + β, (7) which is a recurrence relation in the variable v STORE,i. From [14], the solution of this recurrence relation is [( v STORE,n = V RES 1 C ) MAX C MIN ( ) n C STORE + C ] MAX. (8) C MIN + C STORE C MIN To determine the saturation value of v STORE without energy flyback, substitute n = to obtain v STORE, = C MAX C MIN V RES. (9) (9) indicates that the saturation value is related to the ratio C MAX /C MIN. This equation may be rewritten as v STORE, = C DC + C AC C DC C AC V RES (1) where C DC represents the average value of C VAR and C AC represents its zero-to-peak capacitance variation. From (1), it is apparent that DC parasitic capacitances in parallel with C VAR must be minimized to prevent early saturation. The energy transduction process just described can also be viewed graphically as a Q-V plane contour shown in Fig. 4. In this diagram, Point 1 corresponds to the moment when both D 1 and D 2 are off and the capacitor plates are just starting to pull apart for the nth energy conversion cycle. At Point 2, D 1 turns on, allowing charge to transfer from C VAR to C STORE ; correspondingly Q VAR falls from Point 2 to Point 3. During this part of the cycle, v VAR = v STORE. At Point 3, vibration has pulled the capacitor plates to their maximum separation, decreasing C VAR to C MIN. Next, C VAR increases and v VAR falls, turning off D 2. Correspondingly, Q VAR remains constant until Point 4, at which time D 1 turns on and v VAR is held at V RES. The area within the closed curve in Fig. 4 equals the mechanical vibration energy converted to electrical energy and delivered to C STORE. As the cycles advance, Point 2 moves to the right and Point 3 rises to meet it. At the same time, Point 4 rises to meet Point 1, and the converted energy shrinks to zero as v STORE,n saturates. Because of this, it is important to send the energy stored in C STORE back to C RES quickly enough to permit continuous energy conversion. This is the purpose of the flyback circuitry shown in Fig. 1 and Fig. 2. B. Inductive Flyback Section To power the attached load and prevent v STORE from saturating, an inductive flyback circuit, modeled after a DC/DC buck converter and shown in Fig. 2, complements the charge pump. Nominally in its off state, the MOSFET switch turns on every few energy harvesting cycles to energize L FLY. The current i FLY ramps up according to di FLY dt = v FLY L FLY (11) where v FLY = v STORE v RES. After DT CLK, where D represents the clock duty ratio and T CLK represents the clock period, the MOSFET turns off, forcing i FLY to commutate to the freewheeling diode D FLY. This makes v FLY = v RES, so di FLY dt = v RES L FLY (12) until i FLY = A, at which point the diode turns off. In this way, energy is transferred from C STORE back to C RES where it can power the load, represented here by R LOAD. Note that it is not necessary for this process to be synchronized with the energy conversion cycles described in Section II-A. Indeed, it might be best to initiate the energy flyback whenever v STORE rises to a threshold value. This simplifies the control of the

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS 4 TABLE I HARVESTED POWER AS A FUNCTION OF v G USING GROUND-REFERENCED CLOCK. v G (V) v RES, (V) v RES, (V) P CONV (W) à ÂÄ ¹Å Â Æ ÀÇ» ¹È É À Ê Ì Í À È ± ² Î œ ž Ÿ ³ ² µ Ï Ð Ñ Ò ¹º» ¼ ½ ¹¾ ¼ ½ À Á  š ª «Ó» Ë Ë Â Â ÈÀ Ç ¹ ¾ (a) Charge injection (b) Charge leakage Fig. 6. Spring steel variable capacitor side view (not to scale). Fig. 5. Cycle of circuit operation that results in energy injection from CLK. MOSFET switch by making it independent of the motion of the variable capacitor. III. CLOCK ENERGY INJECTION As the previous section shows, the capacitive energy harvester uses a source-referenced clock to drive the MOSFET. Typically, this gate drive is undesirable because the floating reference voltage makes implementation difficult. Here, SPICE simulation explains the necessity of this choice. Table I shows simulated data when the clock is referenced to ground. Circuit parameters used in the simulation are C RES = 1 µf, C STORE = 3.3 nf, R LOAD = 2 MΩ, L FLY = 2.5 mh, R CORE = 36 kω, R WIRE = 8 Ω, v th = 2.5 V, C DC = 1.22 nf, C AC = 3 pf, v RES = 6 V, f CLK = 475 Hz, and D =.19. The simulation uses a SPICE Level-3 2N72 n- channel MOSFET model and a realistic diode model based on the characteristics of a 1N6263 low-leakage Schottky barrier diode. In the table, v RES, and v RES, represent the original and final voltages on C RES respectively. Furthermore, v G is the magnitude of the gate drive voltage. In all cases, v G exceeds the MOSFET threshold voltage, so the on-resistance of the MOSFET does not change much between simulations. The important point to observe is that the converted power P CONV rises significantly as v G increases. Given that all other parameters remain fixed, the additional harvested energy must come from the clock. This is important for two reasons. First, in an experimental system in which the MOSFET is not powered by C RES, energy injection from the clock through the MOSFET gate can be confused as net energy conversion when in fact net energy is not converted. Second, the energy injected into the harvester diminishes the energy that can actually be harvested during a cycle. To understand the source of clock energy injection, consider the simplified circuit diagram shown in Fig. 5. In this figure, C GS is the MOSFET gate-source capacitance, D 3 is the MOSFET body diode, and C 1 is the parasitic junction capacitance of D FLY. When a rising clock places Q GS onto C GS, the voltage at node X rises according to the capacitive divider involving C 1 and C GS. If the voltage rises enough, D 3 turns on and leaks a fraction of Q GS onto C STORE, which cannot be recovered when the clock goes low. Using a source-referenced clock prevents node X from being pulled up, thereby preventing the accidental turn-on of D 3 and the corresponding energy injection. Table II shows the SPICE simulation when a sourcereferenced clock is used. Here, P CONV does not increase with v GS, indicating that clock energy injection has been eliminated. The first row in the table proves that if v GS is less than the MOSFET threshold voltage, the capacitive energy harvester cannot sustain itself. IV. VARIABLE CAPACITOR To test the asynchronous capacitive energy harvesting circuit, a macro scale spring steel variable capacitor supported by eight cantilever beams was designed and fabricated. As shown Fig. 7. Spring steel variable capacitor prototype.

5 - + IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS 5 TABLE II HARVESTED POWER AS A FUNCTION OF v GS USING SOURCE-REFERENCED CLOCK. v GS (V) v RES, (V) v RES, (V) P CONV (W) C AC (pf) 35 1 pf 3 1 K V+ 25 C VAR V - + V BIAS 2 v TEST Frequency (Hz) C AC (pf) Vamp,p-p (mv) Fig. 8. Frequency sweep for spring steel variable capacitor with v AMP,p p = 5 mv. Fig. 9. C AC as a function of shaking strength at f = 156 Hz. in Fig. 6 and Fig. 7, a square sheet of.5 thick 195 bluetempered spring steel, with a mass density of 785 kg/m 3 and a Young s Modulus of 25 GPa, measuring 6.6 cm on each side formed the flexible top plate of C VAR. A solid block of aluminum formed the base. Mylar tape acted as a spacer that defined the nominal gap between the plates. To prevent the top and bottom plates from shorting, nylon screws were used to hold the structure together. FEA optimization was used to keep the out-of-plane resonant mode close to 15 Hz while pushing the torsion and rotation modes towards higher frequencies above 6 Hz. The resulting springs have lengths of.8 cm and widths of.7 cm. Finally, fillets with 2 mm radii were inserted at all perpendicular intersections to help reduce stress levels in the bending capacitor plate. The prototype spring steel variable capacitor gave C DC = 65 pf when measured on a standard bridge, which is close to the calculated value of 566 pf. The capacitor was vibrated with a Ling Dynamic System V456 shaker table and its corresponding PA 1L amplifier. Using the circuit shown in Fig. 8, in which v TEST is proportional to C AC, a frequency sweep was performed. The result in Fig. 8 shows the first resonant mode to be at f = 156 Hz with a Q of approximately 3.5. Next, by applying 156 Hz vibrations of varying strengths to the variable capacitor, created by changing the peak-to-peak voltage input v AMP,p p to the PA 1L amplifier, a plot of C AC as a function of v AMP,p p was generated. Fig. 9 shows the collected data. V. EXPERIMENTAL RESULTS The circuit in Fig. 2 was fabricated using surface mount components to test the validity of SPICE simulation results and to demonstrate its operation. The components include a 2N72 n-channel MOSFET, three 1N6263 low-leakage Schottky barrier diodes, and a CD447 low power monostable multivibrator gate drive. Before every experiment, a shorting jumper connects a battery to precharge C RES up to v RES = 1.5 V, the starting steady-state voltage. The shaker table is then ramped up from no shaking to the desired test level and the jumper is disconnected. In all cases, the flyback circuit is activated after every 4 energy conversion cycles. Because a resistive load R LOAD formed by a 1 MΩ scope probe in series with a 1 MΩ resistor is attached to the reservoir, a rising v RES indicates positive energy conversion. Fig. 1 plots the evolution of v RES for various shaker table amplifier inputs v AMP,p p. When v AMP,p p = 12 mv (C AC = 84 pf), the energy harvester just barely sustains v RES while powering the 2 MΩ load. For v AMP,p p = 36 mv (C AC = 26 pf), v RES charges up to 6 V, which means that the system delivers 1.8 µw to R LOAD. Although the energy harvester does not have an integrated acceleration sensor to determine the applied acceleration directly, an approximate value can be obtained at the resonant frequency. Assuming a sinusoidal travel of the upper variable capacitor plate and using Q 3.5 from Fig. 8, and the nominal dimensions of the variable capacitor, the applied acceleration per 1 mv of v AMP,p p is approximately 8.4g where g =

6 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS Vres (V) mv 28 mv 24 mv 2 mv 16 mv 12 mv Time (sec) Fig. 1. Evolution of v RES as a function of time parameterized by v AMP,p p. The factor of.5 in the vertical axis results from the 1 MΩ resistor in series with the scope probe. Saturation Vres (V) Frequency (Hz) Fig. 13. Plot of saturation v RES as a function of frequency with v AMP,p p = 32 mv..5vres (V) Fig Vgs = 4.5 V Vgs = 6. V Time (sec) v RES as a function of source-referenced clocking voltage. 9.8 m/s 2. This translates to 4g when C AC = 35 pf. Notice that the plots in Fig. 1 all flatten out and saturate. This is due to the nonlinear core loss present in L FLY that increases as v FLY goes up. In Fig. 2, this nonlinearity is modeled as a nonlinear resistor R CORE whose resistance depends on v FLY. As v FLY varies between V and 3 V, R CORE drops from its nominal value of 4 kω down to approximately 1 kω. To prove that the harvested energy does not include clock energy injection, two additional experiments are performed. First, the shaker table is stopped while the clock signal continued running. The scope probe attached to C RES shows that v RES decays to below 8 mv, much lower than the voltages sustained in Fig. 1. This strongly suggests that even if there is energy injection from the source-referenced clock, it is small compared to the transduced energy. Next,.5Vres (V) System startup Time (sec) Fig. 12. Plot of v RES as circuit starts up from v RES = 2 mv when v AMP,p p = 32 mv. the MOSFET clock amplitude, or equivalently v GS, is varied to determined its effect on the harvested energy. The result, shown in Fig. 11, indicates that the amount of harvested energy is independent of v GS, consistent with SPICE simulations presented in Table II. Thus, it is believed that no energy injected through the MOSFET gate is present in Fig. 1. If the capacitive energy harvester is to be integrated onto an IC with a MEMS capacitor, the minimum v RES required for system startup is critical. A high requirement would necessitate electrochemical cells, defeating the purpose of using an energy harvester in the first place. Startup tests were conducted for the prototype circuit by lowering v RES using a voltage divider and observing whether v RES could still rise to the levels shown in Fig. 1. Fig. 12 plots a sample result with v AMP,p p = 32 mv and v RES = 2 mv at the start. An iteration process shows that even at the noise floor of the oscilloscope, around v RES = 89 mv, the system can still startup, allowing for a piezoelectric film or an electret to jump start the harvester. In real world applications, the vibration frequency will have some spread, so the sensitivity of the capacitive energy harvester to frequency variations is also important. Fig. 13 shows the saturation point of v RES as a function of the drive frequency at v AMP,p p = 32 mv. From this plot, a deviation of 15 Hz from the resonant frequency causes a 37% drop in the saturation voltage, which may be unacceptable. A reduction in the mechanical Q will lower this sensitivity at the cost of C AC, so a trade-off exists. If the frequency spectrum of the ambient vibration is not known a priori, the variable capacitor can be designed with a tunable resonant frequency, perhaps through an adaptive scheme employing electric spring stiffening. VI. ENERGY CONVERSION EFFICIENCY Having demonstrated the behavior of the capacitive energy harvester, the energy conversion efficiency of the circuit is now computed. As a figure of merit, the efficiency η will be defined as Power delivered to load η Theoretical power harvested from Q-V cycle. (13) The numerator represents the amount of useful energy the harvester can provide to a load while the denominator represents the theoretical maximum that the circuit can deliver if the process was 1% efficient.

7 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS 7 As mentioned in Section V, the prototype system presents 6 V to a 2 MΩ resistive load under heavy shaking. Hence, the amount of power delivered to the load for that experiment is 1.8 µw. To determine the denominator, we can compute the total enclosed area A 1 + A 2 in Fig. 4. For simplicity, approximate A 2 as a triangle. Also, because the flyback switch is enabled every 4 harvesting cycles in this paper, take n = 2 in (8) to find the average energy harvested per cycle. Begin by considering Point 1, which corresponds to the moment when both D 1 and D 2 are off and the plates of the variable capacitor are beginning to pull apart. At this point, and therefore v 1 = V RES = 6 V (14) Q 1 = Q 2 = C MAX V RES = C. (15) Point 2 occurs when D 2 turns on, and this happens when v VAR is equal to the previous cycle s v STORE, which is just v STORE,1. From (8) with n = 1, v STORE,1 = 7.2 V, so v 2 = 7.2 V. The trace between Points 2 and 3 represents the charge transfer between C VAR and C STORE when D 2 turns on. By setting n = 2 in (8), it is obtained that v 3 = 8.2 V. From this, one can also calculate that Q 3 = Q 4 = C MIN v 3 = C. (16) Finally, the capacitor plates move toward each other until D 1 turns on, from which point on v VAR is held at V RES until the end of the cycle. Therefore, v 4 = V RES = 6 V. (17) Having determined all the necessary values, areas A 1 and A 2 can be computed as A 1 = (v 2 v 1 ) (Q 1 Q 4 ) = J (18) A 2 = 1 2 (v 3 v 2 ) (Q 1 Q 4 ) = J. (19) Therefore, the average converted energy W CONV per cycle is which means that W CONV = A 1 + A 2 = 6 nj, (2) P CONV = W CONV f = 9.4 µw. (21) Finally, referring back to (13), η = 1.8 µw = 19.1%. (22) 9.4 µw An efficiency that removes the losses of the MOSFET and diodes can also be calculated. This value corresponds to the deliverable power given lossless power electronics. Because the MOSFET on-resistance, R DS,ON, is approximately a constant when operated as a switch, the average conduction loss, P FET,COND, is related to the root-mean-square current, i RMS. Using typical parameter values from the 2N72 n- channel MOSFET datasheet, P FET,COND i 2 RMS R DS,ON =.36 nw. (23) The value of i RMS = 14 µa comes from HSPICE simulation [1]. On the other hand, diodes have a constant voltage drop, v D, across them when turned on, so their average conduction loss is related to the average current, i D. Using typical parameter values from the 1N6263 Schottky diode datasheet, P D,COND i D v D = 2 µw. (24) Again, the value of i D = 1 µa comes from HSPICE simulation [1]. Adding these losses to the numerator, one obtains η 3.8 µw = = 4.4%. (25) 9.4 µw VII. SUMMARY AND CONCLUSION This paper demonstrates an asynchronous capacitive energy harvesting circuit employing a charge pump and inductive flyback. Coupled to an experimental variable capacitor, it delivers 1.8 µw of power to a 2 MΩ resistive load at a steady-state voltage of 6 V while exhibiting an efficiency of 19.1%. The spring steel variable capacitor used to drive the charge pump achieved a C MAX /C MIN ratio of 3.3 with a 1.56 khz out-of-plane resonant frequency. Because the harvester employs only one asynchronous MOSFET switch, clocking is greatly simplified. However, this paper shows that in order to obtain accurate data, a source-referenced gate drive should be used to prevent spurious clock energy injection, which can artificially inflate the conversion efficiency and diminish actual conversion. Experiments conducted using a prototype circuit not only demonstrated close match between theory and actual data, but they also revealed an extremely low startup requirement of v RES 89 mv for the capacitive harvester. Finally, the system can tolerate a frequency variation of 15 Hz before the steady-state voltage drops to 2/3 of its peak value. ACKNOWLEDGMENT This paper is based in part on [1]. The authors would like to thank both Professor Alex Slocum and Alexis Weber for their assistance in designing and fabricating the spring steel variable capacitor. We are also indebted to Professor Charles Sodini, who helped us work out the clock energy injection issues associated with ground-referenced gate drives. Finally, Professor Dave Perreault provided us with invaluable suggestions on the diode selection as well as alternative energy flyback techniques. REFERENCES [1] Materials by Design: Piezoelectric Materials, [2] G. K. Ottman, H. F. Hofmann, A. C. Bhatt, and G. A. Lesieutre, Adaptive Piezoelectric Energy Harvesting Circuit for Wireless Remote Power Supply, IEEE Transactions on Power Electronics, vol. 17, no. 5, pp , September 22. [3] S. Roundy, B. Otis, Y.-H. Chee, J. M. Rabaey, and P. Wright, A 1.9GHz RF Transmit Beacon using Environmentally Scavenged Energy, in International Symposium on Low Power Electronics and Design, 23. [4] G. K. Ottman, H. F. Hofmann, and G. A. Lesieutre, Optimized Piezoelectric Energy Harvesting Circuit Using Step-Down Converter in Discontinuous Conduction Mode, IEEE Transactions on Power Electronics, vol. 18, no. 2, pp , March 23.

8 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS 8 [5] P. Glynne-Jones, M. Tudor, S. Beeby, and N. White, An Electromagnetic, Vibration-Powered Generator for Intelligent Sensor Systems, Sensors and Actuators A, vol. 11, no. 1-3, pp , February 24. [6] C. Williams and R. Yates, Analysis of a Micro-electric Generator for Microsystems, in 8th International Conference on Solid-State Sensors and Actuators, vol. 1, June 1995, pp [7] P. Miao, A. S. Holmes, E. M. Yeatman, and T. C. Green, Micro- Machined Variable Capacitors for Power Generation, March 23, unpublished. [8] J. O. Mur-Miranda, MEMS-Enabled Electrostatic Vibration-to-Electric Energy Conversion, Ph.D. dissertation, Massachusetts Institute of Technology, 23. [9] M. Miyazaki, H. Tanaka, G. Ono, T. Nagano, N. Ohkubo, T. Kawahara, and K. Yano, Electric-Energy Generation Using Variable-Capacitive Resonator for Power-Free LSI: Efficiency Analysis and Fundamental Experiment, in ISPLED 23, August 23, pp [1] B. C. Yen, Vibration-to-Electric Energy Conversion Using a Mechanically-Varied Capacitor, Master s thesis, Massachusetts Institute of Technology, 25. [11] T. Sterken, K. Baert, R. Puers, G. Borghs, and R. Mertens, A New Power MEMS Component with Variable Capacitance, in Pan Pacific Microelectronics Symposium, February 23, pp [12] S. Roundy, P. K. Wright, and J. Rabaey, A Study of Low Level Vibrations as a Power Source for Wireless Sensor Nodes, Computer Communications, vol. 26, no. 11, pp , July 23. [13] D. P. Arnold, F. Cros, I. Zana, D. R. Veazie, and M. G. Allen, Electroplated Metal Microstructures Embedded in Fusion-Bonded Silicon: Conductors and Magnetic Materials, Journal of Microelectromechanical Systems, vol. 13, no. 5, October 24. [14] K. H. Rosen, Discrete Mathematics and Its Applications, 4th ed. McGraw-Hill, New York, Bernard C. Yen (S 5) received the B.S. degree in electrical engineering from the University of California, Berkeley, in 23, and the S.M. degree in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge, in 25. He is currently working toward the Ph.D. degree in electrical engineering at MIT. His research interests include vibration-to-electric energy harvesting, power electronics, micro-electromechanical systems, and analog circuit design. Jeffrey H. Lang (S 78 M 79 SM 95 F 98) received the S.B., S.M., and Ph.D. degrees in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge, in 1975, 1977, and 198, respectively. Currently, he is a Professor of Electrical Engineering at MIT and has been a faculty member since 198. His research and teaching interests focus on the analysis, design, and control of electromechanical systems with an emphasis on rotating machinery, microsensors and actuators, and flexible structures. He has written over 18 papers and holds 11 patents in the areas of electromechanics, power electronics and applied control, and has been awarded four best paper prizes from various IEEE societies. Prof. Lang is a former Hertz Foundation Fellowand a former Associate Editor of Sensors and Actuators.