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J. Low Power Electron. Appl. 2013, 3(2), 194-214; doi:10.3390/jlpea3020194
Published: 18 June 2013
Abstract
: 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.1. Introduction
Figure 1 shows a schematic of a Piezoelectric (PZ) Energy Harvesting Device (EHD) that is the subject of this research. This structure is referred to as a cantilever structure, and is used to amplify the amplitude of the source vibration [1]. Previous work has shown that maximum output power is achieved when the cantilever has a high Q resonance at the frequency of the source vibration. However, the frequency of the source vibration is not usually matched to the resonant frequency of the EHD. The source vibration may vary with time. This paper addresses the problem of electronically tuning the PZ EHD to achieve maximum power in situations where the source vibration is not a stable frequency matched to the mechanical resonant frequency of the EHD.
In the interest of simplicity, we will analyze the structure in Figure 2. The results achieved through analysis of this structure can be generalized to the cantilever structure through the addition of geometrical constants.
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 [2]. In the succeeding sections of this paper, these concepts will be presented in more detail. Section 6 shows that BF can be used to effectively optimize the power output from a PZ EHD. In this paper, we have changed some of the notation that we used in [2], in order to conform to generally accepted usage.
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 et al. [3] have described concepts very similar to #1 and #2 above in the context of electro-magnetic EHDs.
2. Frequency Tuning by Voltage
The material equations for PZ material can be written as follows [1].
The parameters are defined below.
δ = mechanical strain (displacement/length)
σ = mechanical stress (force/area)
Y = Young’s Modulus (force/area)
d = piezo-electric (PZ) coefficient (m/volt)
PZ coupling constant
E = electric field (volt/m)
D = electrical displacement (coulomb/m^{2})
ε = dielectric constant (coul/volt-m)
k_{m} = mechanical (short-circuit) spring constant
mechanical (short-circuit) resonant frequency
electrical (σ = 0) capacitance
motion constrained (δ = 0) capacitance.
L = inductance
motion constrained resonant frequency
η = mechanical damping factor (force/velocity)
mechanical Q-factor
load conductance
normalized load conductance
internal conductance
normalized internal conductance
Refer to the device of Figure 2. When the output is shorted, E = 0, and the mechanical stiffness is given by Young’s modulus, . The short-circuit, resonance frequency is given by . However, when the output is in the open circuit condition, D = 0; the open-circuit stiffness is given by , where ρ is the dimensionless PZ coupling constant, defined in Table 1. From this, it can be shown that the open-circuit and short-circuit resonant frequencies ω_{oc} and ω_{m} are related by the equation . This relationship is well-known, and has been used to experimentally determine the coupling constant ρ [1].
Similarly, we define the electrical capacitance for the case when there is no stress (σ = 0); and we define the motion constrained capacitance for the case when δ = 0.
The equations for the PZ EHD shown in Figure 2 are given below
These equations can be solved for V(ω) and X(ω) as shown below.
When the source vibration frequency ω equals the mechanical resonant frequency ω_{m}, Equation (5) for output voltage reduces to a familiar form.
This results in the familiar circuit model for the PZ EHD, shown at the left of Figure 3.
If a purely resistive load is connected to the EHD, the device capacitance C_{mc} degrades output power. As a result, an inductor (or an effective inductor) is added to the output circuit for the purpose of cancelling the capacitive admittance and achieving maximum average power to the load.
In succeeding sections, simulations of voltage and output power are shown as a function of frequency. In [2] simulations were shown for ρ = 0.2; Q_{m} = 50; and . Throughout this paper, simulations are shown for ρ = 0.05; Q_{m} = 20; and . These values are more representative of today’s commercial devices.
Figure 4 shows the magnitude of the output voltage as a function of frequency, calculated using Equation (5), for the case of no inductor (ω_{mc} = 0). On the vertical axis, voltage is normalized to the maximum open-circuit voltage at the mechanical resonant frequency . Frequency, on the horizontal axis, is normalized to ω_{m}. For , Figure 4 confirms the predictions of Equation (7). For , the normalized voltage magnitude is 0.707; and for , the normalized voltage magnitude is 0.447. However, for w ≠ 1, Figure 4 shows interesting behavior, especially for . For , the voltage shows a resonance at .
Figure 4 shows that, for large , the voltage peaks at the mechanical or short-circuit resonance frequency ω_{m}. However, for small , the device characteristics change. When , the peak voltage occurs at . This can be understood as follows. For large , the electric field is effectively shorted. As decreases, the electric field in the EHD increases, and alters the effective spring constant of the cantilever beam.
It is tempting to assume from the above that frequency tuning is possible only in the narrow range [4]. However, as we will see below, the addition of a resonant electrical circuit allows the voltage to swing below zero and above V_{oc}, thereby enabling a wider tuning range.
3. Coupled Oscillators
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 in the simulations above, the degradation is not large. However, in some cases, the PZ EHD has a large capacitance, which can substantially degrade output power at ω = ω_{m}. In Figure 5, we simulate Equation (5) for the case ω_{mc} = ω_{m}. This is equivalent to adding an inductor that cancels the reactive admittance of the capacitor at ω = ω_{m}. The inductor performs as expected at ω = ω_{m}. The output voltage equals . Moreover, for large , the output voltage continues to show a resonant peak at ω = ω_{m}; and the voltage falls away sharply away from ω = ω_{m}.
However, for ω ≠ ω_{m}, Figure 5 shows a surprising result when is small. Two peaks occur at and . These resonances result from the poles in the denominator of Equation (5) when .
Solution to the pole-splitting equation above, for the case ω_{mc} = ω_{m} and ρ = 0.05 gives the values of w_{+} and w_{−} above. Note that, when ω_{mc} = 0 (case of no inductor), the solutions of Equation (9) are w_{−} = 0 and . Pole splitting, which describes coupled modes of the mechanical and electrical resonators, occurs only for small . When is large, the electric field in the PZ material is screened, and coupling is suppressed.
The roots of the pole-splitting equation are shown in Figure 6 for several values of coupling constant ρ. In Figure 5, we selected ω_{mc} = ω_{m}, in order to optimize output power at ω = ω_{m}. But, we discovered that, if we increased the load resistance, we could also optimize output voltage at w_{+} and w_{−}. (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 ω_{mc}.
We can gain further insight into the pole-splitting by returning to Equations (5) and (6). For small , the following relationship holds between X(ω) and V(ω).
Using Equation (10) the force of the spring can be written as
The above equation shows that, by tuning L, we can vary ω_{mc} and vary the effective spring constant. When ω < ω_{mc}, V(ω) has a phase of 180° relative to X(ω); and the voltage reduces the effective spring constant. When ω > ω_{mc}, V(ω) has a phase of 0° relative to X(ω); and the voltage increases the effective spring constant. When ω = ω_{mc}, the magnitude of X(ω) is zero. If we adjust ω_{mc} such that the corresponding root of Equation (9) equals the source vibration frequency, then Equation (11) reduces to
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 ω_{−} and ω_{+} are 3.5× to 5.5× higher than . The simulations in this paper assume that the source magnitude Z is held constant as the frequency changes. As a result, the input acceleration increases in proportion to ω^{2}, and . The important result is that the peak voltages away from resonance can be somewhat higher than , and the higher voltage enables frequency tuning.
4. Optimizing Output Power
In the previous section, we showed that voltage can be made to peak at frequencies ω_{−} and ω_{+}, which are different from ω_{m}. Figure 7 shows that power is also maximized at these frequencies. The curve for indicates that output power peaks at ω = ω_{m}, at the power given by Equation (8). The curve for shows a peak at ω = ω_{m} at a degraded power level. The curve for shows two peaks at ω_{−} and ω_{+} that have output power comparable to . When is further reduced to , the output power at ω_{−} and ω_{+} decreases.
Figure 7 suggests that output power can be optimized at frequencies different from ω_{m} by adjusting the external inductor (or effective inductor). Equation (5) provides a general expression for V(ω). Varying the parameter in Equation (5) is equivalent to varying the inductor. Maximizing power with respect to ω_{mc} gives the expression
When we use Equation (13) to determine the value of ω_{mc} that optimizes power at each frequency, the resulting voltage and power are shown in Figure 8, Figure 9.
Equation (13) suggests that we need two strategies for optimizing power, depending on the source frequency ω. For frequencies near ω_{m}, (Region 2), Equation (13) reduces to
Note that, when ω = ω_{m}, Equation (14) reduces further to ω_{mc} = ω_{m}, which is equivalent to matching the capacitive admittance (refer to Figure 3). When ω is above or below ω_{m} (Regions 1 and 3), Equation (13) reduces to the pole-splitting Equation (9). Far from ω_{m}, output power is maximized at the pole frequencies [roots of Equation (9)]. However, as ω approaches ω_{m}, interaction between the poles shifts the max-power frequency [given by Equation (13)] slightly away from the pole frequency.
Within the region around the mechanical resonant frequency (Region 2), output power can be optimized by using (14) for reactive admittance and . In Regions 1 and 3, power is optimized by using the pole splitting Equation (9) for reactive admittance and . For the parameters used in this example (Q_{m} = 20), Δw ≈ 0.025.
Cammarano et al. [3] have derived an equation very similar to Equation (13): Equation (8) in [3]. They observe that the power conditioning system at the output of the EHD can be used to synthesize the complex load impedance required by Equation (13), and they comment on the challenge of reducing the power of such systems. Chang et al. [5] have implemented a switch-mode power conditioning system to the output of a PZ EHD, and have demonstrated the ability to harvest energy from two sources simultaneously: ω = ω_{m} and ω = 1.2ω_{m}.
5. Bias-Flip Technique
For a typical, discrete EHD, C_{mc} ≈ 100nF, and the inductor required to match this reactance at 100 Hz is impractically large: L ≈ 25H. However, it has been shown that the Bias-Flip technique can be used to synthesize a reactive impedance for effective impedance matching [6,7]. This technique is suitable to ULP miniaturization. It utilizes a very small inductor together with ULP microelectronics to emulate an inductor that is large and tunable. The BF technique has been shown to be effective in maximizing the output power of PZ EHDs at ω = ω_{m} [7]. In this section, we will describe the Bias-Flip technique in the context of the equivalent circuit of Figure 3 describing a PZ EHD operating at the mechanical resonance frequency.
The BF technique is illustrated in Figure 10. In the BF circuit, the large inductor is replaced by a small inductor, connected by MOS switches. When the switches are closed, a high frequency tank circuit is formed. After ½ period of oscillation of this tank circuit, the switches are opened, and the voltage on the capacitor has “flipped” adiabatically from +V to −V. In this paper, the switches are assumed to be ideal and lossless.
Refer to Figure 3. When an ideal inductor is used together with a matched resistive load R_{L} = R_{in}, the maximum average output power is given by . [See Equation (8)] In Figure 11, we show how effective the ideal Bias-Flip circuit is in achieving maximum power.
In the worst case of very large C_{mc}, the output power is degraded by several orders of magnitude, when no inductor is used. However, the Bias-Flip approach delivers power , which is 8 / π^{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 C_{mc} is large [7].
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 [7].
The rectification circuit analyzed in [7] is shown in Figure 12. For simplicity, we assumed that the EHD is operating at the mechanical resonance frequency, and we use the equivalent circuit of Figure 3. The output AC voltage v(t) is rectified in the diode bridge and stored on the capacitor C_{RECT} that is maintained at voltage V_{RECT} by the Energy Management Circuit. The analysis below assumes ideal diodes.
Operation of the Bias-Flip rectifier is described with reference to Figure 13.
When the capacitance is zero, as shown in Figure 13a, Bias-Flip is not required. until t = t_{on}; at which time, v(t) becomes clamped at V_{RECT}. Between t_{on} and t_{off} power is supplied to the storage unit. At t = t_{off}, the diodes turn off, and v(t) returns to zero following the curve . The presence of non-zero C_{mc} degrades transferred power: Figure 13b. When the current turns positive, there is a negative bias on C_{mc} that must be discharged before the voltage can swing positive. This delays diode turn-on, and forces a reduction in V_{RECT}, both of which degrade transferred power. This degradation can be corrected by adiabatically flipping the bias on C_{mc} when the current changes sign, as illustrated in Figure 13c.
The energy transferred per cycle depends on V_{RECT}. When V_{RECT} is low, the power transfer interval t_{off} − t_{on} is long, but the power is low. When V_{RECT} is high, the power transfer is high, but the transfer interval is short. In fact, for V_{RECT} above a maximum value the diodes do not turn on, and no power is transferred. Figure 14 shows the power transfer as a function of V_{RECT}, for various values of C_{mc}. This simulation is made using the values and .
Figure 15 shows that, for large C_{mc}, output is severely degraded. However, the Bias-Flip circuit is effective in recovering most of the lost output power. This analysis confirms the conclusions of Ramadass and Chandrakasan [7] that for an EHD, operating at resonance, the Bias-Flip circuit is effective in canceling the reactive impedance of the device, and achieving near-optimum output power. In the next section, we will demonstrate that the Bias-Flip technique can be used to form an effective LC tank circuit that, when coupled to the EHD can tune the resonant frequency.
6. Bias Flip for Frequency Tuning
In Section 5, we confirmed the effectiveness of the BF technique for power optimization at the mechanical resonant frequency. When ω = ω_{m}, and the equivalent circuit of Figure 3 applies, we can optimize power to the load by “canceling” or “matching” the reactive admittance of the capacitor with an inductor. We select an inductor value such that . Matching the reactive admittance aligns the phase of the voltage across the load with the phase of the current. This works because the current source is assumed to be ideal. Varying the reactive load does not change the current from the source. We confirmed the finding of [7] that the BF inductor is effective in canceling capacitive admittance at ω_{m}. In this section, we will demonstrate that the BF inductor can also be used to tune the resonant frequency of the EHD and optimize power at frequencies substantially different from ω_{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 z(t) to be zero. , and the velocity of the source is . The source velocity has a phase of +90^{o}. The force acting on the EHD is given by Equation (3). Our goal is to maximize.
Where F_{I} is the imaginary part of F. . From this, we conclude that input average power is maximized when F has phase +90°, matched to source velocity, and X has phase −90°.
Additional insight into maximization of output power is seen in Figure 16. Here, we compare the phase of mechanical displacement X(ω) for three conditions
No inductor. The phase is −90° only at . Below ω_{m}, the phase is ≈0°, and above ω_{m}, the phase is ≈ −180°. Only at ω = ω_{oc} is the force in phase with the source velocity;
Inductor, optimized using Equation (13). Note that the phase approaches −90° above and below ω = ω_{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 ω_{m} results from phase alignment between force and source velocity.
Figure 17 shows output power for 3 conditions.
No inductor: ;
Inductor optimized using Equation (13): ;
Inductor optimized using Equation (13): .
Case #2 illustrates the case where the reactive admittance is chosen to optimize output power, but the load conductance is kept at . Very little improvement is achieved, because the voltage is kept low by the high load conductance, and the voltage is ineffective in modulating the cantilever spring constant. Case #3 shows power improvement of ~50X compared to case #1. When we compensate for the increase in acceleration with frequency, case #3 demonstrates that it is possible to achieve output power at ω ≠ ω_{m} that is comparable to the maximum power at ω_{m}.
Additional insight into the mechanism of frequency tuning can be obtained by transforming the mechanical equations of motion to an equivalent circuit [1,8,9], as shown in Figure 18.
The equations for the mechanical portion of the equivalent circuit are shown below.
Define Z_{m} to be the impedance seen by the voltage source V_{F}(ω) in Figure 18b. Setting Im(Z_{m}) = 0 gives the pole-splitting equation, equivalent to Equation (9), in the limit .
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
These results are summarized in Table 1.
Table 1. Strategy For Power Optimization in the Three Frequency Regions of Operation. |
Frequency region | L_{eff} | ω_{mc} | Phase of voltage | Optimum power |
---|---|---|---|---|
Region 1: ω < ω_{m} | >0 | ω_{mc} > ω | ~ +90^{o} | R_{L} >> R_{in} |
Region 2: ω ≈ ω_{m} | ≈0 | ω_{mc} = ω ≈ ω_{m} | ~0^{o} | R_{L} = R_{in} |
Region 3: ω > ω_{m} | <0 | ω_{mc} < ω | ~ −90^{o} | R_{L} >> R_{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 V(ω). The phase in each of the three regions is given in the table. In Region 2, the voltage across the load V(ω) is in phase with I_{S}(ω), and output power is optimized by setting R_{L} = R_{in}. However, in Regions 1 and 3, max power occurs when V(ω) has a phase of +90^{o} (Region 1) and −90° (Region 3) relative to the phase of I_{S}(ω), and optimum power is achieved by setting R_{L} to be substantially larger than R_{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 X(ω) has a phase of −90° in all three regions of operation. Recall from Equation (10) that, when is small, the phase of X(ω) is related to the phase of V(ω) in a simple way. By adjusting the phase of V(ω) as shown in Table 1, we also adjust the phase of X(ω) to −90°.
Simulations were performed starting from the differential equations for v(t) and x(t) that are comparable to Equations (3) and (4), for the case of no inductor. These equations were solved subject to the boundary conditions.
In the case of no Bias-Flip, . The effect of the BF inductor is to change the phase of v(t) every half-period.
Using the voltage waveform, we calculated average output power. This is shown normalized to in Figure 19. These simulations show that the BF technique is capable of achieving output power, comparable to the optimum power achievable with an optimized inductor. Moreover, the BF technique is self-tuning. If the bias is flipped whenever the source velocity crosses zero (as assumed in this simulation), the desired phase is maintained as the source frequency changes. No calculation is required to solve Equation (13).
Analysis of the voltage waveforms reveals another aspect of self-tuning. In Region 1, the bias flips from negative to positive at t = 0 and from positive to negative at t = T / 2, thereby emulating a +90° phase shift. In Region 3, the reverse happens. The bias flips from positive to negative at t = 0 and from negative to positive at t = T / 2, thus emulating a −90° phase shift.
7. Conclusions
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 ω_{m}. However, for small load conductance G_{L}, the voltage can be used to tune the spring constant, and the resonant frequency of the mechanical oscillator.
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 ω_{m} (Region 1), this occurs when the phase of the voltage is +90° relative to the source vibration, and at frequencies above ω_{m} (Region 3), output power is optimized when the phase of the voltage is -90^{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 ω_{m}, by cancelling the capacitive admittance of the EHD. In this work, we have shown how the BF technique can be used to tune an EHD and harvest energy from frequencies other than the mechanical resonance frequency.
Acknowledgments
The authors acknowledge informative exchanges with Samuel Chang of MIT.
Conflict of Interest
The authors declare no conflict of interest.
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