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Devices that harvest electrical energy from mechanical vibrations have the problem that the frequency of the source vibration is often not matched to the resonant frequency of the energy harvesting device. Manufacturing tolerances make it difficult to match the Energy Harvesting Device (EHD) resonant frequency to the source vibration frequency, and the source vibration frequency may vary with time. Previous work has recognized that it is possible to tune the resonant frequency of an EHD using a tunable, reactive impedance at the output of the device. The present paper develops the theory of electrical tuning, and proposes the Bias-Flip (BF) technique, to implement this tunable, reactive impedance.

Schematic of a Piezo-Electric (PZ) Energy Harvesting Device (EHD) based on the Cantilever Beam structure.

In the interest of simplicity, we will analyze the structure in

Schematic of the simplified EHD that is analyzed in this paper. A_{p} is the area of the PZ capacitor, and t_{p} is the thickness. Z is the complex amplitude of the source vibration, and X is the complex amplitude of the mechanical displacement of the mass M. This simplified model illustrates the concepts of electronic tuning that apply to the cantilever structure of

This paper describes three concepts for electrically tuning of PZ EHDs.

Use of voltage amplitude to tune the mechanical stiffness of the EHD;

Coupling of the mechanical resonator to an electrical RLC tank circuit;

Bias-Flip (BF) technique to emulate the large tunable inductor that is required for the RLC tank circuit.

These three concepts were introduced in summary form in [

In this paper, we will analyze the PZ EHD. However, many of the results and conclusions are equally applicable to electromagnetic and electrostatic EHDs. Cammarano

The material equations for PZ material can be written as follows [

The parameters are defined below.

^{2})

_{m}

Refer to the device of _{oc}_{m}

Similarly, we define the electrical capacitance

The equations for the PZ EHD shown in

These equations can be solved for

When the source vibration frequency ω equals the mechanical resonant frequency _{m}

This results in the familiar circuit model for the PZ EHD, shown at the left of

The circuit within the dashed box is the equivalent circuit that applies to a PZ EHD when _{m}_{p}_{m}_{p}

If a purely resistive load is connected to the EHD, the device capacitance _{mc}

In succeeding sections, simulations of voltage and output power are shown as a function of frequency. In [_{m}_{m}

Voltage magnitude for the case of no inductor. Voltage is normalized to

_{mc}_{m}

_{m}

It is tempting to assume from the above that frequency tuning is possible only in the narrow range _{oc}

In the simulations in the previous section, we did not attempt to cancel the reactive admittance of the PZ capacitor, and we observed the degradation in output voltage. Since _{m}_{mc}_{m}_{m}_{m}_{m}_{m}

However, for _{m}

Solution to the pole-splitting equation above, for the case _{mc}_{m}_{+} and _{−} above. Note that, when _{mc}_{−} = 0 and

Voltage magnitude for the case when an inductor of value

Roots of the Pole-splitting Equation (9). The normalized pole frequencies

The roots of the pole-splitting equation are shown in _{mc}_{m}_{m}_{+} and _{−}. (We will show in the next section that output power is also optimized at these frequencies). This analysis suggests that we can tune the EHD resonant frequency by varying _{mc}

We can gain further insight into the pole-splitting by returning to Equations (5) and (6). For small

Using Equation (10) the force of the spring can be written as

The above equation shows that, by tuning L, we can vary _{mc}_{mc}_{mc}_{mc}_{mc}

Equation (12) shows that the effective spring constant can be tuned over a wide range.

It is also somewhat surprising that the peak voltages at _{−} and _{+} are 3.5× to 5.5× higher than ^{2}, and

In the previous section, we showed that voltage can be made to peak at frequencies _{−} and _{+}, which are different from _{m}_{m}_{m}_{−} and _{+} that have output power comparable to _{−} and _{+} decreases.

Normalized average output power for the case when an inductor of value

_{m}_{mc}

When we use Equation (13) to determine the value of _{mc}

Equation (13) suggests that we need two strategies for optimizing power, depending on the source frequency _{m}

Note that, when _{m}_{mc}_{m}_{m}_{m}_{m}

Normalized voltage magnitude for the case when the reactive admittance is optimized to give maximum output power using Equation (13). Plots of Voltage _{mc}

Normalized average power for the case when the reactive admittance is optimized to give maximum output power using Equation (13). Plots of power

Within the region _{m}

Cammarano _{m}_{m}

For a typical, discrete EHD, _{mc}_{m}

Operation of a Bias Flip (BF) Inductor. (

The BF technique is illustrated in

Refer to _{L}_{in}

Normalized Average Power delivered to the load at _{m}_{L}_{in}

In the worst case of very large _{mc}^{2} = 81% of the max power obtained using an ideal inductor. This illustrates the effectiveness of Bias-Flip circuits to achieve high output power when _{mc}

So far in this paper, we have discussed the case in which AC power is delivered to a resistive load. We have done this because the analysis can be performed in closed form. However, in many energy harvesting applications, it is necessary to rectify the AC power and store it in a battery or super-capacitor. The Bias-Flip technique is especially applicable to this case, as shown in [

The rectification circuit analyzed in [_{RECT}_{RECT}

Circuit to rectify and store the AC power being generated by the EHD. It is assumed that the EHD is operating at the mechanical resonance frequency.

Operation of the Bias-Flip rectifier is described with reference to

(_{mc}_{mc}_{mc}_{RECT}

When the capacitance is zero, as shown in _{on}_{RECT}_{on}_{off}_{off}_{mc}_{mc}_{RECT}_{mc}

Power transferred to the storage capacitor _{RECT}_{RECT}_{mc}_{mc}_{mc}

The energy transferred per cycle depends on _{RECT}_{RECT}_{off}_{on}_{RECT}_{RECT}_{RECT}_{mc}

Shows the rectified power as a function of _{mc}_{RECT}

_{mc}

In _{m}_{m}_{m}

In order to maximize output power at any frequency, we need to maximize the input power delivered from the mechanical source to the EHD. In other words, we need to align the phase of the force with the phase of the source velocity.

In the following analysis, we assume the phase of ^{o}. The force acting on the EHD is given by Equation (3). Our goal is to maximize.

Where _{I}

Phase of mechanical displacement _{L}

Additional insight into maximization of output power is seen in

No inductor. The phase is −90° only at _{m}_{m}_{oc}

Inductor, optimized using Equation (13). Note that the phase approaches −90° above and below _{m}

Inductor optimized using the pole-splitting Equation (9). The phase is −90° for all frequencies.

The improvement in power at frequencies above and below _{m}

No inductor:

Inductor optimized using Equation (13):

Inductor optimized using Equation (13):

Normalized Average Output Power for three cases. (1) No inductor &

Case #2 illustrates the case where the reactive admittance is chosen to optimize output power, but the load conductance _{m}_{m}

Additional insight into the mechanism of frequency tuning can be obtained by transforming the mechanical equations of motion to an equivalent circuit [

(_{e}.

The equations for the mechanical portion of the equivalent circuit are shown below.

_{e}

Define _{m}_{F}_{m}

The last term in the above equation can be used to tune the resonance frequency above or below the mechanical resonance frequency. The last term takes the form
_{eff}_{m}_{m}_{mc}_{m}_{−} must be reduced. This can be achieved by increasing the effective inductance, which can be achieved by setting _{mc}_{m}_{+} must be increased. This can be achieved by decreasing the effective inductance, which can be achieved by setting _{mc}

These results are summarized in

Strategy For Power Optimization in the Three Frequency Regions of Operation.

Frequency region |
_{eff} |
_{mc} |
Phase of voltage | Optimum power |
---|---|---|---|---|

Region 1:
_{m} |
>0 | _{mc} |
~ +90^{o} |
_{L}_{in} |

Region 2:
_{m} |
≈0 | _{mc}_{m} |
~0^{o} |
_{L}_{in} |

Region 3:
_{m} |
<0 | _{mc} |
~ −90^{o} |
_{L}_{in} |

Maximizing power in the three regions can be envisioned in term of an effective inductor. Alternatively, it can be envisioned in terms of setting the phase of the voltage _{S}_{L}_{in}^{o} (Region 1) and −90° (Region 3) relative to the phase of _{S}_{L}_{in}

The foregoing analysis suggests that the Bias-Flip technique can be used to synthesize an inductor, by flipping the polarity of the voltage in such a way that

Simulations were performed starting from the differential equations for

In the case of no Bias-Flip,

Using the voltage waveform, we calculated average output power. This is shown normalized to

Normalized average power as a function of frequency. In Regions 1 and 3, the Bias-Flip technique (red and blue dashed lines) improves output power by ~100X compared to the case of no inductor and no BF (green line). Moreover, it gives output power that is comparable to the maximum power achievable with an optimized inductor (red and blue solid lines).

Analysis of the voltage waveforms reveals another aspect of self-tuning. In Region 1, the bias flips from negative to positive at

In the preceding sections, we have explained the principles for electrically tuning of PZ EHDs. These principles are summarized below.

Equation (11) shows that the effective spring constant of the mechanical resonator is a function of voltage. If the load conductance is large, the voltage is kept small, and the resonator responds only at the mechanical resonant frequency _{m}_{L}

In Regions 1 and 3, output power is maximized by maximizing input power (force x velocity), transferred from the source to the EHD. At frequencies below _{m}_{m}^{o} relative to the source vibration.

This optimum phase relationship can, in theory, be achieved using a tunable inductor, whose value can be obtained from Equation (13). A large tunable inductor is not generally practical. However, the Bias-Flip technique can be used to emulate a large, tunable inductor. Previous work has shown that the BF technique can be used to optimize the output power at _{m}

The authors acknowledge informative exchanges with Samuel Chang of MIT.

The authors declare no conflict of interest.