Rectifier-free piezoelectric energy harverster and battery charger

[0001] This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/345,770, filed May 18, 2010, the entirety of which is hereby incorporated herein by reference.

[0002] 1. Field of the Invention

[0003] The present invention relates to electrical energy harvesting systems and, more specifically, to an electrical energy harvesting system that is rectifier free.

[0004] 2. Description of the Related Art

[0005] Microscale and nanoscale integrated devices are used in a variety of applications, including in biomedical implants and sensors. Many such devices are powered by electrical energy. Conventional power sources, such as electrochemical batteries, are often used to power such integrated devices. However, in certain applications, such as biomedical implants, replacing or recharging batteries may be difficult and costly. One approach to recharging such batteries is to harvest electrical energy from piezoelectric structures subjected to ambient motion (such as muscle flexing in the case of biomedical implants or vibration in the case of environmental sensors, etc.). Typical piezoelectric structures include cantilevers made from a piezoelectric material, such as certain crystals, organic materials and ceramic structures, generate electrical charge when subject to mechanical force. For example, a zinc oxide crystal will generate a charge imbalance when subjected to a bending force. Other piezoelectric structures generate charge when subjected to stress or strain.

[0006] Given that many piezoelectric structures are subject to periodic movements that have both a positive cycle and a negative cycle (such as a vibrating cantilever), the electricity generated by a piezoelectric structure is typically in the alternating current form. However, most batteries require direct current to be recharged. Therefore, recharging a battery from a piezoelectric structure usually requires a device to convert the alternating current to direct current.

[0007] A typical microscale alternating current to direct current converter includes a diode bride rectifier. Such a rectifier employs an arrangement of diodes that channels current during both the positive cycle and the negative cycle onto an output in the form of a series of positive half-waves. In such a rectifier, current from the piezoelectric structure usually must pass through two diodes and each diode typically has a voltage drop of between 0.2 V to 0.7 V (depending on the semiconductor technology employed). This voltage drop can be greater than the voltage output from the piezoelectric structure and, therefore, usually requires that a voltage boost be applied to the output from the piezoelectric structure so that the voltage being applied to the battery is greater than the voltage of the battery, otherwise no current would flow into the battery. Such a system tends to waste a considerable amount of energy.

[0008] The problem is rectifying unpredictably small ac signals (which are prevalent in small volumes and with weak vibrations whose peak voltages fall below the rectified output level targeted, requires low-loss, no-threshold rectifiers. Quasi-lossless LC energy-transfer networks that precede or follow the rectifier can extract all the energy stored in the piezoelectric capacitance and therefore overcome the basic threshold-voltage limitation, except the rectifier and its controller's headroom and quiescent current nonetheless limit the input voltage range of the system and dissipate power.

[0009] Miniaturized mobile electronic systems, such as biomedical drug-delivery implants, acceleration-monitoring and pressure-monitoring sensors, and micro-sensor nodes in wireless networks, have so little space for energy storage that they suffer from short operational lives. Unfortunately, replacing easily exhaustible onboard batteries is prohibitive because the systems often conduct in situ measurements in unreachable places and operate in concert with numerous other devices, where the personnel and logistical costs of maintaining all batteries charged are unacceptably high. Harvesting energy, however, from light, heat, radiation, and motion is an attractive, though not easy alternative for replenishing small batteries and capacitors.

[0010] Although the application ultimately determines what energy source suits best, harvesting kinetic energy is promising because motion and vibrations are abundant and produce moderate power levels. For context, solar light generates more power, but only when exposed directly to the sun, and power densities derived from indoor lighting, thermal gradients, and radio-frequency (RF) waves fall well below their kinetic counterparts, although not all motion-based transducers perform equally well. Piezoelectric devices, in fact, when constrained to small platforms, generate more power than variable (electrostatic) capacitors and moving (electromagnetic) coils.

[0011] When considering a piezoelectric source, the internal charge configuration of the material changes (much like an ac current source) to generate an alternating voltage across the equivalent capacitance that its opposing surfaces present. The harvester circuit must therefore extract energy from the changing voltages of the piezoelectric capacitor and deposit charge into an energy-storage device. Because harvested power is low and uncorrelated to the load, a small battery or capacitor serves as the reservoir from which electrical functions in the system draw power on demand.

[0012] Conventional approaches first rectify the incoming ac voltage with a diode bridge. Some techniques reduce the voltage (and therefore power losses) across the pn-junction diodes by using MOS switches and driving them with a comparator that senses and ensures only small positive terminal voltages allow the switches to close. Unfortunately, input voltages must nonetheless exceed their rectified outputs for the MOS switches to conduct, which means rectifiers place a threshold limit on the mechanical input. In other words, rectified harvesters only harvest energy above a minimum input level, so they cannot extract all the energy the piezoelectric material offers. Although some systems extract more energy from the environment by boosting the transducer's electrical damping force with higher (LC-induced) piezoelectric voltages, the subsequent rectifier still suffers from a threshold minimum below which the harvester cannot harness energy. Furthermore, drawing maximum power requires an optimal rectified output voltage, so some approaches, at the cost of power, employ a correcting feedback loop that senses the harvester's output current to set the optimal rectified level.

[0013] Therefore, there is a need for an alternating current to direct current converter that does not incur the losses associated with a bridge rectifier.

[0014] The disadvantages of the prior art are overcome by the present invention which, in one aspect, is a circuit for harvesting electrical energy from a piezoelectric source and for storing the electrical energy in a battery that includes an inductor that is configured to store electrical energy. A diode bridge-free switching network is configured to: direct electrical energy from the piezoelectric source to the inductor during a first portion of a piezoelectric charge generating cycle; and direct electrical energy from the inductor to the battery during a second portion of the piezoelectric charge generating cycle.

[0015] In another aspect, the invention is a circuit for harvesting electrical energy that includes an alternating current source that is coupled between a first note and a second node. A load is coupled between a fifth node and the second node. An inductor, coupled between a third node and a fourth node, is configured to store electrical energy. A first switch is coupled between the first node and the third node. A second switch is coupled between the fourth node and the second node. A first unidirectional current directing device is coupled between the third node and the fifth node and is configured to allow current to flow only from the third node to the fifth node. A second unidirectional current directing device is coupled between the fourth node and the fifth node and is configured to allow current to flow only from the fourth node to the fifth node. A switch state controller is configured to control the first switch and the second switch. The switch state controller directs electrical energy from the alternating current source to the inductor during a first portion of an alternating current charge generating cycle and directs electrical energy from the inductor to the load during a second portion of the alternating current charge generating cycle.

[0016] In yet another aspect, the invention is a method of harvesting energy from a piezoelectric current generating source, in which an inductor is charged with a first bias during a first positive portion of a piezoelectric charge generating cycle from the piezoelectric current generating source. Current from the inductor is directed to a positive terminal of a load during a second positive portion of the piezoelectric charge generating cycle by coupling a first terminal of the inductor to the positive terminal of the load along a first circuit path. The inductor is charged with a second bias, opposite the first bias, during a first negative portion of the piezoelectric charge generating cycle from the piezoelectric current generating source. Current is directed from the inductor to the positive terminal of the load during a second negative portion of the piezoelectric charge generating cycle by coupling a second terminal of the inductor, different from the first terminal of the inductor, to the positive terminal of the load along a second circuit path, different from the first circuit path.

[0017] These and other aspects of the invention will become apparent from the following description of the preferred embodiments taken in conjunction with the following drawings. As would be obvious to one skilled in the art, many variations and modifications of the invention may be effected without departing from the spirit and scope of the novel concepts of the disclosure.

[0024] A preferred embodiment of the invention is now described in detail. Referring to the drawings, like numbers indicate like parts throughout the views. Unless otherwise specifically indicated in the disclosure that follows, the drawings are not necessarily drawn to scale. As used in the description herein and throughout the claims, the following terms take the meanings explicitly associated herein, unless the context clearly dictates otherwise: the meaning of “a,” “an,” and “the” includes plural reference, the meaning of “in” includes “in” and “on.”

[0025] As shown in FIG. 1, one embodiment of a circuit 100 for harvesting electrical energy from a piezoelectric source 10 (which can include a piezoelectric structure 12 and an inherent capacitance 14), or other type of alternating current source, and for storing the electrical energy in a battery 20, or other type of load, includes an inductor 118 that stores electrical energy from the piezoelectric source 10. A diode bridge-free switching network 110 directs electrical energy from the piezoelectric source 12 to the inductor 118 during a first portion of a piezoelectric charge generating cycle (such a as a period between when the piezoelectric source 10 begins either a positive (or negative) voltage cycle and when it reaches a maximum (or minimum) voltage peak) and directs electrical energy from the inductor 118 to the battery 20 during a second portion (e.g., after a voltage peak) of the piezoelectric charge generating cycle.

[0026] The piezoelectric source 10 is coupled between a first node 111 and a second node 112, the inductor 118 is coupled between a third node 113 and a fourth node 114, and the battery 20 is coupled between a fifth node 115 and the second node 112. The switching network includes: a first switch 120 coupled between the first node 111 and the third node 113; a second switch 122 coupled between the fourth node 114 and the second node 112. A first unidirectional current directing device 124 coupled between the third node 113 and the fifth node 115. The first unidirectional current directing device 124 allows current to flow only from the third node 113 to the fifth node 115. Similarly, a second unidirectional current directing device 126 is coupled between the fourth node 114 and the fifth node 115. The second unidirectional current directing device 126 allows current to flow only from the fourth node 114 to the fifth node 115.

[0027] In one embodiment, both the first unidirectional current directing device 124 and the second unidirectional current directing device 126 each include a transistor 128 having a source, a drain and a gate. A comparator 130 compares the voltage at the source to the voltage at the drain and drives the gate so as to put the transistor 128 in a conducting state when current is flowing into the source side of the transistor 128. A driver element (such as a plurality of cascaded inverters, shown in FIG. 6) may be disposed between the output of the comparator and the gate of the transistor.

[0028] A switch state controller 140, which is coupled to the piezoelectric source 10 via a resistor 142 and a series capacitor 144 (also, the addition of a parallel capacitor might be necessary if the inherent capacitance 14 of the piezoelectric source is insufficient), controls the states of the first switch 120 and the second switch 122, based on a voltage sensed at the piezoelectric source 10.

[0029] Both the first switch 120 and the second switch 122 are closed during a first portion of a positive current generating portion of the piezoelectric charge generating cycle, as shown in FIG. 2A, during which time the inductor 118 is charged. Then the first switch 120 is maintained in a closed state and the second switch 122 is opened during a second portion of the positive current generating portion of the piezoelectric charge generating cycle. At the beginning of a first portion of a negative current generating portion of the piezoelectric charge generating cycle, as shown in FIG. 2C, both the first switch 120 and the second switch 122 are again closed, thereby charging the inductor 118 with a polarity opposite of the polarity it was charged in FIG. 2A. Then, as shown in FIG. 2D, the first switch 120 is opened and the second switch 122 is maintained in a closed state during a second portion of the negative current generating portion of the piezoelectric charge generating cycle, thereby allowing current to flow from the inductor 118 into the battery. As shown in the graph 300 in FIG. 3A, both switches 120 and 122 are closed in τ_L⁺ during which the inductor 118 is charged from the piezoelectric source 10. Then the first switch 120 is maintained in a closed state and the second switch 122 is opened during τ_B⁺ to allow the inductor to charge the battery. As shown in the graph 310 in FIG. 3B, both switches 120 and 122 are closed during τ_L⁻, thereby charging the inductor 118 with a polarity opposite of the polarity it was charged in the positive cycle. Then, the first switch 120 is opened and the second switch 122 is maintained in a closed state during τ_B⁻ to allow the current to flow from the inductor 118 into the battery 20.

[0030] As shown in FIG. 4, one embodiment of the switch state controller 140 includes a comparator 410 coupled across resistor 142 and decoupled from a ground via capacitor 412. The comparator 410 detects the positive and negative voltage events across resistor 142 and generates an output 414 representative thereof, which is fed into a delay circuit 420 (one embodiment of which is shown in FIG. 5). The comparator output 414 and the delay output 416 are fed into both an AND gate 422 and a NOR gate 432. A first driver element 424, which includes a plurality of cascaded inverters, repowers the output from the AND gate 422 to generate a first switch control signal 425 that drives the first switch 120. Similarly, a second driver element 434, which includes a second plurality of cascaded inverters, repowers the output from NOR gate 432 to generate a second switch control signal 435 that drives the second switch 122. The first switch 120 and the second switch 122 each include a first transistor 426 and a second transistor 428, whose gates are coupled to each other. The first transistor 426 has a leakage current that flows in a first direction and the second transistor 428 has a leakage current that flows in a second direction that is opposite of the first direction.

[0031] One experimental embodiment 500 is shown in FIG. 6. In this embodiment 500, most of the components of the diode bridge-free switching network 510, except for the inductor 518, can be embodied on a single chip.

[0032] In this experimental embodiment, when vibrations move the piezoelectric cantilever in the positive cycle, the system waits until the voltage across piezoelectric capacitance C_PZT peaks, which happens when the transducer's cantilever reaches its maximum displacement. At this point, the system energizes harvesting inductor L_H from C_PZT in energizing time τ_L⁺ and then de-energizes L_H into the battery in de-energizing time τ_B⁺. Transferring energy this way only requires a few microseconds, which represents a negligible fraction of the millisecond vibration period, so the position of the transducer is, for all practical purposes, static during the transfer. Similarly, after depositing positive-cycle charge into the battery, the system waits until the piezoelectric device charges C_PZT in the negative direction to its negative peak. The switched-inductor circuit then quickly energizes and de-energizes L_H in τ_L⁻ and τ_B⁻ from C_PZT to the battery. This negative-cycle transfer concludes the full cycle, and the sequence repeats as long as vibrations persist.

[0033] C_PZT stores the energy that the piezoelectric material produces each half cycle (E_IN⁺) and the switched-inductor converter extracts E_IN⁺ from C_PZT to reset C_PZT's voltage v_PZT to 0 V. Assuming piezoelectric current i_PZT is sinusoidal at I_PZT sin(ω_VIBt), v_PZT rises in positive-cycle time τ_C⁺ to

[0034] $v_{\mathrm{PZT}}\left(t\right)=\frac{1}{C_{\mathrm{PZT}}}{\int }_0^t{i}_{\mathrm{PZT}}\left(\tau \right)d\tau =\left(\frac{I_{\mathrm{PZT}}}{{\omega }_{\mathrm{VIB}}{C}_{\mathrm{PZT}}}\right)\left[1-\cos \left({\omega }_{\mathrm{VIB}}t\right)\right]$
After τ_C⁺, C_PZT accumulates

[0035] $E_{\mathrm{IN}}^{+}\approx {\int }_0^{{\tau }_C^{+}\approx T/2}{i}_{\mathrm{PZT}}\left(t\right)v_{\mathrm{PZT}}\left(t\right)dt=\frac{2I_{\mathrm{PZT}}^2}{{\omega }_{\mathrm{VIB}}^2{C}_{\mathrm{PZT}}}$
Since the negative cycle generates an equivalent amount (i.e., E_IN⁺≈E_IN⁻), the system harnesses 4I_PZT²/ω_VIB²C_PZT from each vibration period, which is four times (4×) more energy than the ideal diode-rectified case can possibly induce.

[0036] In the positive cycle, with switch S_I open, the switched-inductor power stage waits for i_PZT to charge C_PZT to peak positive voltage V_PIEZOP⁺. V_PIEZOP⁺ is not the actual peak because the implemented circuit includes delay and offset, the causes and implications of which subsequent subsections will address. The controller then closes S_I and S_N to discharge C_PZT into L_H for energizing time τ_L⁺, allowing L_H's current i_L to reach positive peak current I_LP⁺. Because a capacitor fully discharges into an inductor in a quarter of its LC resonance period, the system sets τ_L⁺ to one-fourth of 2π√(L_HC_PZT). After τ_L⁺, S_N opens and i_L charges the parasitic capacitance at v_SW⁺ until non-inverting diode D_N forward biases and steers i_L's de-energizing (battery) current i_BAT⁺ into the battery. Once D_N completely drains L_H, after de-energizing time τ_B⁺, D_N opens automatically (i.e., asynchronously).

[0037] The negative cycle operates similarly, except the power stage inverts its functionality to realize an inverting buck-boost translation. As such, the system waits for i_PZT to charge C_PZT in the negative direction until v_PZT reaches negative peak voltage V_PIEZOP⁻. (Like before, V_PIEZOP⁻ is not accurate because of non-idealities in the circuit.) S_I and S_N then drain C_PZT into L_H for one-fourth of L_H-C_PZT's resonance period, after which point S_I opens and i_L charges v_SW⁻'s parasitic capacitance until inverting diode D_I forward biases and conducts i_L's de-energizing (battery) current i_BAT⁻ into the battery.

[0038] Unlike many rectified harvesters, this embodiment of a switched-inductor circuit can de-energize (and therefore harness) all piezoelectric energy in C_PZT, which is to say the circuit does not suffer from a minimum input threshold. What is more, L_H automatically raises v_SW⁺ and v_SW⁻ to whatever voltage the battery demands, which means the circuit does not require the additional dc-dc converter stage that use to drive charge into the battery. Unlike some other converters, which include a lossy feedback loop to adjust the rectifier output voltage to draw the maximum power from C_PZT, embodiments of the present invention can derive four times (4×) more energy than rectified harvesters without a correcting loop.

[0039] An integrated circuit version of this embodiment integrates the switches, controller, and bias generator into a single 2-μm BiCMOS IC. Bias resistor R_PTAT and delay filter elements R_D and C_D are off chip for testing flexibility. For the same reasons, external voltages V_ADJ⁺ and v_ADJ⁻ adjust energizing times τ_L⁺ and τ_L⁻ externally. The piezoelectric cantilever, the battery, and L_H are also off chip.

[0040] A series combination of two NMOS transistors in an isolated p-well implements both S_I and S_N. The purpose of the back-to-back body diodes is to block the undesired diode current that would otherwise result through the body diode of a single transistor when v_PZT swings below ground (in the negative cycle) or above battery voltage V_BAT (in the positive cycle). Note that the series combination of two switches increases channel resistance and gate capacitance, which raise conduction and switching losses, respectively, but not to the extent body-diode conduction would dissipate power.

[0041] Allowing v_PZT to swing below ground demands that S_N's gate driver GD_SN outputs a negative voltage. The reason for this is that, while S_N is open during the negative cycle, when v_PZT falls below ground, S_I remains closed and, as a result, switch-node voltages v_SW⁺ and v_SW⁻ follow v_PZT below ground. During this time, driving S_N's gate to zero would not disengage S_N. To avoid this problem, GD_SN's last inverter stage connects to v_PZT instead of ground. Accordingly, GD_SN's output v_GDSN—OUT follows v_PZT during the negative cycle and nears V_BAT otherwise. Because S_N's gate capacitance is orders of magnitude below C_PZT's, GD_SN hardly drains C_PZT, that is, has negligible impact on v_PZT.

[0042] To reduce the diode-voltage conduction loss across battery-charging diodes D_N and D_I in the power stage, the circuit replaces them with low-drop synchronous MOS switches M_PDN and M_PDT. Here, comparators CP_DN and CP_DI sense when D_N and D_P's anode (switching) voltages v_SW⁺ and v_SW⁻ just surpass their cathode counterpart V_BAT to engage M_PDN and M_PDI, and switch them off otherwise. CP_DN and CP_DI must therefore (i) respond quickly (to reduce the time the lossy body-diodes conduct charging currents and keep M_PDN and M_PDI from draining the battery with negative current) and (ii) and dissipate little energy.

[0043] A fast response implies considerable quiescent current and low energy means the comparators must operate only when needed (asynchronously): when L_H drains i_L into the battery (during τ_B⁺ and τ_B⁻). Fortunately, τ_B⁺ and τ_B⁻ are a small fraction of the vibration period so energy remains low (at 0.5 nJ per cycle in the prototyped case). Considering that, the comparator powers only when i_L is non-zero and the corresponding switching (anode) voltage v_SW is well above zero (above M_PB and M_NB's combined gate-source voltages), which only happens when L_H is ready to charge the battery. During that time, some of i_L (as limited by R_BIAS) flows into bias mirror M_NB-M_NO to establish a threshold current about which M_PO determines the state of the comparator. That way, since M_PB and M_PO match, M_PO conducts less current and engages M_PDN (or M_PDI) when V_BAT is below v_SW, and vice versa otherwise.

[0044] Because the comparator does not draw current until v_SW rises well above ground, i_L charges v_SW's parasitic capacitance C_SW—PAR quickly and M_NO sinks sufficient current to ensure V_CP—OUT transitions low quickly (to engage M_PDN or M_PDI). Note M_PDN's (or M_PDI's) body diode conducts current while the comparator reacts to close M_PDN (or M_PDI). Once engaged, v_SW remains above V_BAT by the Ohmic voltage i_L produces across M_PDN's (or M_PDI's) series resistance. As a result, v_SW decreases with L_H's falling i_L, as L_H de-energizes. When i_L reaches zero, which happens when the system exhausts L_H's energy, v_SW falls to V_BAT and the comparator transitions to disengage M_PDN (or M_PDI). Because i_L no longer carries sufficient current to power the comparator, the circuit shuts down automatically.

[0045] During the turn-off transition, the comparator receives a small overdrive voltage so its shut-off time is longer than its counterpart is. As a result, while still closed, M_PDN's (or M_PDI's) reverse current can not only discharge the battery but also dissipate power. To prevent this reverse current, the comparator shifts the shut-off trip-point up to transition earlier (when v_SW is slightly above V_BAT) by leaking current away from M_NB through M_NH1. The positive feedback that M_NH1 creates engages only after V_CP—OUT rises above ground by more than M_NH1's V_TN, which happens after the comparator decides to raise V_CP—OUT. In other words, M_NH1 accelerates the circuit only after it begins to trip. M_NH1 also imbalances gate-coupled pair M_PB-M_PO with M_NH2's current to offset v_SW's rising trip-point (and create a hysteresis) that helps in preventing transient and ringing events in v_SW from asserting inadvertent transitions in the comparator.

[0046] The purpose of the control block is to determine when to start and stop energizing L_H. First, S_I and S_N should start discharging C_PZT into L_H when v_PZT peaks. To this end, comparator CP_PK compares v_PZT with a delayed version of itself (v_D) to capture the moments when v_PZT stops leading v_D. That is to say, v_PZT drops below v_D when v_PZT just begins falling from its positive peak and v_PZT rises above v_D when v_PZT just begins increasing from its negative peak.

[0047] CP_PK incorporates about ±10 mV of hysteresis to deglitch noise in v_PZT and v_D. The cross-coupled, positive-feedback p-type mirror load unbalances the n-type differential pair in both directions for that purpose. The circuit operates in sub-threshold to minimize power (with nano-amps) so its delay is on the microsecond range. Such a slow response is tolerable because microseconds constitute a negligible fraction of the millisecond vibration period, meaning v_PZT does not change appreciably in microseconds. To mitigate risk in evaluating the power stage and its basic control scheme, external negative source V_SS extends CP_PK's input common-mode range (ICMR) below ground so it can accommodate negative v_PZT voltages.

[0048] Ideally, the harvester should stop energizing L_H after one-fourth of L_H-C_PZT's resonance period, after energizing periods τ_L⁺ and τ_L⁻. For that, v_ADJ⁺-R_ADJ⁺ and v_ADJ⁻-R_ADJ⁻ set the source and sink currents that charge and discharge C_ADJ so that C_ADJ's voltage reaches the threshold of the ensuing inverter in ½π√(L_HC_PZT). Since tolerance in R_ADJ and C_ADJ offset this time, and other parasitic effects similarly shift how long C_PZT should discharge into L_H, v_ADJ⁺ and v_ADJ⁻ are adjustable (and off chip for testing flexibility).

[0049] After discounting losses, the system harnesses up to 30 μW (P_H) from 72 μW of transducer-generated power (P_IN(LOADED)). Without the harvester (i.e., unloaded) and under equal mechanical stimulation, energy per cycle in C_PZT (i.e., C_PZTΔv_{PZT(UNLOADED)}²f_VIB) can generate less power (P_IN(UNLOADED)) than when loaded (e.g., 40 versus 72 μW). This results, as expected, because extracting all C_PZT'S positive- and negative-cycle energy resets (i.e., conditions) v_PZT to zero, further increasing the transducer's electrical damping force and thereby inducing it to derive more energy from the environment. Note P_IN(UNLOADED) also represents the maximum input power a rectifier-based circuit can produce because a rectifier draws charge from C_PZT without conditioning its voltage. In other words, the best possible (and ideal) rectifier-based system would draw roughly 56% of what this embodiment can (e.g., 40 of 72 μW).

[0050] Although increasing the transducer's output power by up to 78% or 1.78× (when Δv_{PZT(UNLOADED)} is 1.2 V) by loading it with the prototyped harvester corroborates theory, which predicts a 4× (i.e., 400%) improvement. One reason for this discrepancy is CP_PK's 36-mV input-referred offset causes the system to discharge C_PZT past v_PZT's peak after C_PZT loses some energy to vibrations. What is more, since the offset voltage does not scale with the input, the fraction of energy lost to vibrations is larger for smaller inputs, so loaded-to-unloaded input-power ratio P_IN(LOADED)/P_IN(UNLOADED) decreases from 1.78 rapidly (i.e., nonlinearly) with respect to Δv_{PZT(UNLOADED)}. Parasitic resistances in switches S_N and S_I and across L_H also reduce the transducer's output power because, instead of fully draining C_PZT into L_H by lowering v_PZT to zero, the converter pulls v_PZT to the voltage drop across S_N, L_H, and S_I's combined series resistance R_SN+R_ESR R_SI. This non-zero voltage not only represents remnant energy that the system does not harvest but also lowers the peak amplitude (i.e., electrical damping force) of the ensuing half cycle.

[0051] Ultimately, the harvesting conversion efficiency of the system with respect to P_IN(LOADED) across vibration strength falls between 40% and 50%. The system loses conduction energy E_C to the switches' finite resistances and L_H's equivalent-series resistance (ESR) when i_L flows through L_H during L_H's positive- and negative-cycle energizing and de-energizing times τ_L⁺, τ_B⁺, τ_L⁻, and τ_B⁻. However, L_H de-energizes in considerably less time than it energizes (i.e., τ_L⁺ and τ_L⁻ are considerably longer than τ_B⁺ and τ_B⁻) so E_C mostly consists of energizing losses:

[0052] $E_C\approx E_{C\left(\mathrm{LE}\right)}\approx \left[{\left(\frac{I_{\mathrm{LP}}^{+}}{\sqrt{3}}\right)}^2+{\left(\frac{I_{\mathrm{LP}}^{-}}{\sqrt{3}}\right)}^2\right]\left(R_{\mathrm{SI}}+R_{\mathrm{SN}}+R_{\mathrm{ESR}}\right){\tau }_{\mathrm{LE}}$
where energizing times τ_L⁺ and τ_L⁻ equal (to τ_LE) and I_LP⁺/√3 and I_LP⁻/√3 are the root-mean-squared (RMS) currents flowing through the resistors across τ_L⁺ and τ_L⁻, respectively. E_C falls with increasing L_H values because, even though τ_LE increases with L_H^0.5 (from ½π√(L_HC_PZT)), (I_LP⁺)² and (I_LP⁻)² decrease with L_H¹. Wider MOS transistors (i.e., lower R_SI and R_SN) also decrease E_C, but at the expense of silicon real estate and higher switching gate-drive losses E_SW(GD), as the following discussion will elucidate.

[0053] The system loses E_SW(GD) in inverter drivers when they charge and discharge parasitic gate capacitances in S_I, S_N, and D_N and D_I's M_PDN and M_PDI. Since all the switches in this embodiment engage and disengage once per cycle, gate-drive losses amount to
E_SW(GD)≈(C_SI+C_DI+C_DN)V_DD²+C_SN(V_DD+|V_PIEZOP⁻|)²
where C_SI, C_DI, C_DN, and C_SN refer to the parasitic capacitances that S_I, D_I's M_PDI, D_N's M_PDN, and S_N introduce, respectively, which scale with transistor channel widths. Note S_N causes more energy loss than other switches because its gate swings between V_DD and V_PIEZOP⁻.

[0054] The system also loses overlap losses E_SW(IV) when MOS drain currents and drain-source voltages shortly overlap every time they transition. This overlap occurs when S_N and S_I open because v_SW⁺ and v_SW⁻ transition with a non-zero i_L:
E_SW(IV)≈(0.5I_LP⁺τ_SN—OFF+0.5I_LP⁻τ_SI—OFF)(V_BAT+V_D)
where V_D refers to M_PDN and M_PDI's as body-diode voltage. Here, because i_L is triangular and τ_SN—OFF and τ_SI—OFF are S_N and S_I's corresponding off times, 0.5I_LP⁺τ_SN—OFF and 0.5I_LP⁻τ_SI—OFF represent S_N and S_I's total conducted charge during transitions.

[0055] Peak-detector comparator CP_PK, the nanoamp-bias generator, adjustable delay block τ_ADJ, the logic gates, and D_N and D_I's comparators CP_DN and CP_DI also dissipate quiescent energy (E_Q). The losses in CP_PK and the bias circuit, however, dominate E_Q because they operate through the entire vibration cycle, whereas the other circuits only engage for a substantially smaller fraction. Overall, although higher input power increases |V_PIEZOP⁻|, I_LP⁺, and I_LP⁻ and their associated switching losses E_SW(GD) and E_SW(IV), conduction losses E_C scales more rapidly with power (i.e., proportional to (I_LP⁺)² and (I_LP⁻)²) so E_C ultimately dominates and limits η_H when mechanical vibrations are strong. Conversely, when vibrations are weak, E_Q may dominate because E_C, E_SW(GD), and E_SW(IV) scale with power.

[0056] Vibrations in practical operating environments occur, for the most part, at relatively low frequencies, between maybe 1 Hz for a person walking to 167 Hz for an engine cycling at 10,000 revolutions per minute. What is more, the rate is often inconsistent and maybe even non-recurring, as in the case of human motion, wind-propelled movements, and vibrations generated from impact. What all this means is that matching the narrow band of a transducer to vibrations, which is the recipe for high conversion efficiency, is oftentimes impractical and unrealistic.

[0057] Because the prototyped harvester automatically detects when to draw energy from C_PZT, the system is capable of harnessing energy from non-periodic stimuli. When dropping a 145-g official major-league baseball from 40 cm above the experimental setup table, for example, the impact of the ball bouncing once off the table induces the piezoelectric cantilever to vibrate and produce the pulse train. In this case, the prototyped system charged 500 nF (from 3.04 V) from a single bounce for three separate trials by roughly 200 mV.

[0058] Each drop produces the decaying vibrations where the system harvests energy by quickly draining C_PZT into L_H and then L_H into the 500-nF capacitor each time v_PZT peaks. As v_PZT's peaks (and mechanical strength) decrease, however, peak-detector comparator CP_PK's offset has a larger impact on how much input power P_IN(LOADED) the system harnesses so output power P_H decreases more than basic theory predicts. Under these weak vibrations, in fact, R_D-C_D and CP_PK's delays increase CP_PK's offset to degrade performance. In contrast, just as CP_PK's offset adds to the effects of CP_PK's delay in the positive half-cycle, the offset cancels delay in the negative half-cycle which means alternating the polarity of the offset can improve overall performance.

[0059] Although the immediate aim of this research is to raise P_H by increasing η_H and P_IN(LOADED), the ultimate metric is end-to-end harvesting conversion efficiency η_TOTAL, which refers to how much mechanical energy E_ME reaches the output as harvested energy E_H. A means of applying a known value for E_ME (and approximating η_TOTAL) is to tie an object of known mass m_W to the tip of the piezoelectric cantilever with a light string and subsequently cut the link. Since the cantilever's elastic force F_C balances the object's gravitational pull F_G, E_ME relates to F_G and the cantilever's tip displacement distance d_C:

[0060] $E_{\mathrm{ME}}=\frac{K_C{d}_C^2}{2}=\frac{\left(K_C{d}_C\right)d_C}{2}=\frac{F_C{d}_C}{2}=\frac{F_G{d}_C}{2}=\frac{\left(m_{W}g\right)d_C}{2}$
where K_C and g refer to the spring constant and gravity's acceleration, respectively.

[0061] This way, neglecting parasitic weights and assuming the string severs instantaneously, the prototyped harvester charged (experimentally) 500 nF by roughly 400 mV from a 3-g weight. As a result, subtracting the external supply's quiescent energy E_Q (to the chip for CP_PK, bias generator, adjustable delay, and logic gates) and gate-driving losses E_SW(GD) from the energy deposited in the 500-nF capacitor reduces E_H to
E_H=0.5C_BAT(V_BAT(F)²−V_BAT(I)²)−E_Q−E_SW(GD)
which ranged between 100 and 660 nJ when stimulated with 1.2 to 10.8 μJ of E_ME from 1-, 2-, and 3-g weights to yield 6.1±1.5% to 8.8±6.9% of η_TOTAL or E_H/E_ME.

[0062] η_TOTAL incorporates the collective performance of the transducer (η_PZT(LOADED)) and the harvester (η_H), which means η_PZT(LOADED is η_TOTAL/η_H or roughly 14%-20% when assuming η_H is 45% on average. Since η_PZT(LOADED) includes a factor improvement the harvester induces, the unloaded counterpart (η_{PZT(UNLOADED)}) is less than 14%-20% by up to 1.78×. Although this simple analysis is by no means accurate or complete, it shows that mechanical losses are significant at around 80% and conditioning the transducer to increase P_IN(LOADED) is as important as reducing the losses across the switched-inductor converter (i.e., increasing η_H).

[0063] The embodiment of a 2-μm BiCMOS switched-inductor piezoelectric harvester generates and steers up to 30 μW from a periodic 72-μW piezoelectric source into a capacitor or battery directly. In doing so, it increases the piezoelectric cantilever's electrical damping force to raise the transducer's (and therefore system's end-to-end) mechanical-electrical efficiency by up to 78%. The system also harnesses up to 659 nJ from non-periodic vibrations with 6.1±1.5% to 8.8±6.9% end-to-end mechanical-electrical efficiencies. One key feature of the presented harvester is that it eliminates the need for a rectifier. As a result, the system no longer (i) places a threshold limit imposed by diodes and/or the output voltage on mechanical vibrations, (ii) loses power across an otherwise additional stage (i.e., across a rectifier), and (iii) limits how much the circuit dampens the transducer (to produce more power). Conditioning the piezoelectric device to generate more power is an important attribute, as is the relatively simple and low-loss control strategy the system adopts to energize and de-energize the inductor directly into the energy-storage device. From a practical standpoint, the harvester is also able to harness energy from short, non-periodic mechanical vibrations, which are more prevalent in real-life applications, such as in human motion, wind-propelled movements, vibrations that objects crashing produce, and so on.

[0064] The above described embodiments, while including the preferred embodiment and the best mode of the invention known to the inventor at the time of filing, are given as illustrative examples only. It will be readily appreciated that many deviations may be made from the specific embodiments disclosed in this specification without departing from the spirit and scope of the invention. Accordingly, the scope of the invention is to be determined by the claims below rather than being limited to the specifically described embodiments above.