# Tuning Techniques for Piezoelectric and Electromagnetic Vibration Energy Harvesters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Piezoelectric and Electromagnetic Resonant Vibration Energy Harvesters

#### 2.1. Modeling and Maximum Power Extraction

_{P}(t) between the tip of the cantilever and the clamping point leading to the generation of electric energy. As shown in Figure 2b, in an E-RVEH a permanent magnet is connected to a spring and, due to vibrations, it moves out of phase with respect to the generator housing to which a coil is fixed. The relative displacement x

_{E}(t) taking place between the magnet and the coil gives rise to the conversion of mechanical energy into electric energy.

_{P}(which is a combination of the equivalent piezoelectric cantilever mass and of the tip mass [76]) and m

_{E}(oscillating magnet mass) are depicted by means of suitable arrows.

_{P}∙ẋ

_{P}(t) and c

_{E}∙ẋ

_{E}(t) are the viscous damping forces (c

_{P}and c

_{E}are the viscous damping coefficients). k

_{P}∙x

_{P}(t) and k

_{E}∙x

_{E}(t) are the elastic forces (k

_{P}is the equivalent stiffness of the piezoelectric cantilever and k

_{E}is the equivalent stiffness of the spring). θ

_{P}∙v

_{P}(t) is the force due to the piezoelectric inverse effect that opposes to the strain of the piezoelectric material (θ

_{P}is the piezoelectric electromechanical coupling coefficient). θ

_{E}∙i

_{E}(t) is the electromagnetic force that opposes to the movement of the magnet (θ

_{E}is the electromechanical coupling coefficient of the coil) [74,75].

_{P}(t) and x

_{E}(t) [74,75]

_{piezo}and R

_{piezo}, respectively are the capacitance and the resistance at the output of the piezoelectric layers. i

_{piezo}(t) is the current generated by the piezoelectric effect whose expression is given in (3). The parallel of L

_{L_P}, R

_{L_P}, and C

_{L_P}represents a generic linear load that is connected to the P-RVEH and i

_{P}(t) and v

_{P}(t), respectively are the current and voltage across such a load. In Figure 4b, L

_{coil}and R

_{coil}, respectively are the inductance and the resistance of the harvester coil. ε

_{coil}(t) is the electromotive force induced in the coil whose expression is given in (4). The series of L

_{L_E}, R

_{L_E}, and C

_{L_E}represents a generic linear load that is connected to the E-RVEH and i

_{E}(t) and v

_{E}(t), respectively are the current and voltage across this load. It should be highlighted that the load parallel connection in Figure 4a and the load series connection in Figure 4b have been chosen, without any loss of generality, in order to get two completely dual systems.

_{vib}and amplitude A

_{vib}(ÿ(t) = A

_{vib}∙cos(2πf

_{vib}∙t)), the Equations (1), (2), (3) and (4) can be written in the frequency domain and respectively become (5), (6), (7) and (8)

_{P}(f

_{vib}), V

_{P}(f

_{vib}), X

_{E}(f

_{vib}), I

_{E}(f

_{vib}), I

_{piezo}(f

_{vib}), and E

_{coil}(f

_{vib}), respectively are the Fourier transforms of x

_{P}(t), v

_{P}(t), x

_{E}(t), i

_{E}(t), i

_{piezo}(t), and ε

_{coil}(t). j is the imaginary unit.

_{P}(f

_{vib}) and X

_{E}(f

_{vib}) can be obtained

_{nat_P}and f

_{nat_E}are the undamped natural frequencies of the SDOF systems shown in Figure 3.

_{L_P}(G

_{L_P},B

_{L_P},f

_{vib}) and P

_{L_E}(R

_{L_E},X

_{L_E},f

_{vib}) at a given frequency, are [77]:

_{L_Popt}(f

_{vib}) and P

_{L_Eopt}(f

_{vib}) provided to the optimal loads, at a given frequency, can be obtained by substituting expressions (24) and (26) in (22) and (25) and (27) in (23):

_{vib}) is reported. The unique symbol P

_{Lopt}is used in Figure 5 for the maximum load power of both types of RVEHs.

_{max}in Figure 5) is provided to the optimal loads only when the vibration frequency is equal to the frequencies f

_{Popt}and f

_{Eopt}(a unique symbol f

_{opt}is used in Figure 5). By differentiating (28) and (29) with respect to f

_{vib}and by equating to zero the obtained expressions it is possible to find the values of the frequencies f

_{Popt}and f

_{Eopt}.

_{Popt}and f

_{Eopt}are coincident with the undamped natural frequencies of the SDOF RVEHs’ models and are typically also called mechanical resonance frequencies. It is worth noting here that, by observing the equivalent circuits of Figure 4, the above mechanical resonance frequencies can be obtained by means of suitable frequency scans of proper electrical quantities at the output of the RVEHs. In particular, in the case of P-RVEHs, the f

_{Popt}is equal to the resonance frequency of the short circuit current. Instead, in the case of E-RVEHs, f

_{Eopt}is equal to the resonance frequency of the open circuit voltage.

_{Popt}and f

_{Eopt}of the RVEHs, the optimal loads assume simplified expressions:

#### 2.2. Definition of Tuning Techniques

_{opt}), it has to be used only in those applications that are characterized by a dominant frequency (f

_{vib}) that is coincident with such a mechanical resonance frequency. During the last years, some papers have been proposed in the scientific literature on the optimal design of RVEHs based on the characteristics of the vibrations [78,79,80]. As an example, cantilever beam shape optimization has been studied in great detail in case of P-RVEHs [81,82,83]. Unfortunately, in most practical cases, the frequencies of the exploitable sinusoidal vibrations are time-varying or even such vibrations are not sinusoidal but are characterized by a random behavior with a wide frequency spectrum where the energy is distributed [84,85,86]. Therefore, in practical applications, the increase of the effective operating frequency range of RVEHs is mandatory. In the literature, a number of papers describing control methods or architectures that are specifically designed with the aim of increasing the effective operating frequency range of RVEHs, has been proposed [87,88,89,90,91,92,93,94,95]. In particular, harvesters arrays, nonlinear harvesters, and mechanical tuning techniques (MTTs) are the most widely analyzed optimization methods. There is general agreement on the fact that MTTs are much more promising with respect to the other two options. This is essentially due to the low volumetric power density of harvesters arrays and to the considerable complexity of nonlinear harvesters [95].

_{opt}= f

_{vib}in all the operating conditions). Obviously, as it is evident from the previous Modeling Section, another requirement is mandatory for maximizing the extraction of power from RVEHs. In fact, even if the mechanical resonance frequency of a RVEH is always coincident with the vibration frequency, the extracted power is dependent on the RVEH load impedance. In particular, according to Section 2.1 (see Equations from (24) to (27) and from (34) to (37)) and on the basis of the maximum power transfer theorem [77], the optimal load impedance of a RVEH, that is the load impedance able to maximize power drawn from the RVEH itself, changes with f

_{vib}. Therefore, a second objective to fulfill is the matching of the RVEH with its optimal load impedance, frequency by frequency. This is the aim of the so-called electrical tuning techniques (ETTs) [96,97,98,99,100,101,102]. It is worth noting that the application of the ETTs of course does not change the mechanical resonance frequency of a RVEH. ETTs allow the maximization of the extraction of power at each frequency; but the vibration frequency that allows to obtain the global maximum power (P

_{MAX}) always is the mechanical resonance frequency. The only way to change such a frequency is by means of a MTT. In order to better clarify such a statement, it is useful to analyze the following figures.

_{L}(R

_{Lopt}(f

_{k}),X

_{Lopt}(f

_{k}),f

_{vib}) provided to the fixed load (R

_{Lopt}(f

_{k}),X

_{Lopt}(f

_{k})) (k = 1, 2,… n). Each color curve P

_{L}(R

_{Lopt}(f

_{k}),X

_{Lopt}(f

_{k}),f

_{vib}) has been obtained in correspondence of a different fixed load (R

_{Lopt}(f

_{k}),X

_{Lopt}(f

_{k})) that is just the optimal load in correspondence of the frequency f

_{k}(see Equations from (24) to (27)). Therefore, each color curve is maximized only when f

_{vib}= f

_{k}. The envelope of the maximums P

_{ETT}(f

_{vib}) ≡ P

_{Lopt}(f

_{vib}) ≡ P

_{L}(R

_{Lopt}(f

_{vib}),X

_{Lopt}(f

_{vib}),f

_{vib}) of the color curves (dashed black curve in Figure 6a) represents the maximum power, as a function of the frequency, that can be gotten with the application of the ETT. It is clear that the frequency that allows to obtain the maximum global harvesting of power is unique and coincident with the mechanical resonance frequency f

_{opt}. In conclusion, ETTs allow only the maximization of the extraction of power at a given vibration frequency f

_{vib}from a given RVEH with a fixed mechanical resonance frequency f

_{opt}. ETTs do not change the mechanical resonance frequency of the RVEH. As stated above, the only way to change such a frequency is by means of a MTT. In other words, the black dashed curve P

_{ETT}(f

_{vib}) of Figure 6a indicates the best performance that can be obtained with the application of only ETT without MTT.

_{opt}(gray curve). What changes among the color curves is the mechanical resonance frequency f

_{opt_k}(k = 1, 2,…n). In other words, the color curves represent a sort of horizontal translation, at different resonance frequencies, of the central grey curve of Figure 6b that coincides with the central gray curve of Figure 6a. The envelope of the maximums of such color curves (dashed black curve in Figure 6b) represents the maximum power P

_{MTT}(f

_{vib}) ≡ P

_{L}(R

_{Lopt}(f

_{opt}),X

_{Lopt}(f

_{opt}),f

_{vib}) that can be harvested, as a function of the frequency, by applying MTT without ETT. It is clear that, in this case, the performance of the RVEH is improved, with respect to the case of Figure 6a, because the mechanical characteristics of RVEH are changed and therefore it always resonates in correspondence of the vibration frequency f

_{vib}. However, by observing the dashed black curve in Figure 6b, it is evident that there is still an optimal frequency. It is the untuned resonance frequency f

_{opt}. This happens because a fixed load has been considered and such a load is just the optimal one at f

_{opt}. This aspect can be clarified by observing that the optimal load at f

_{opt}(see Equations from (34) to (37)) has a resistive part that is independent from the resonance frequency and hence it does not need a regulation while applying MTT. Instead, the reactive part of the optimal load at f

_{opt}, that compensates the reactive part of the output impedance of the considered RVEH (C

_{piezo}for P-RVEH and L

_{coil}for E-RVEH), depends on the frequency and hence it would need a regulation while applying MTT. Therefore, a fixed not optimized load, leads to the nonhorizontal shape of the maximum extracted power P

_{MTT}(f

_{vib}) (black dashed curve in Figure 6b).

_{MTT-ETT}(f

_{vib}) ≡ P

_{max}≡ P

_{L}(R

_{Lopt}(f

_{vib}),X

_{Lopt}(f

_{vib}),f

_{vib}) that can be extracted, frequency by frequency, with the joint application of both a MTT and an ETT.

_{ETT}(f

_{vib}) represented by the black dashed curve in Figure 6a. The application of a MTT without any ETT leads to the extraction of the power P

_{MTT}(f

_{vib}) represented by the black dashed curve in Figure 6b. The joint application of both an ETT and a MTT leads to the extraction of the power P

_{MTT-ETT}(f

_{vib}) represented by the black dashed curve in Figure 6c.

## 3. Indicators for the Classification and Comparison of Mechanical Tuning Techniques

- Indicator Name: RVEH.
- Description: It indicates the type of RVEH to which the considered MTT is applied. If RVEH is “P” (“E”) it means that the considered MTT is applied to a P-RVEH (an E-RVEH). RVEH can be also equal to “Hybrid P/E” if the considered MTT is applied to a hybrid piezoelectric–electromagnetic RVEH.
- Indicator Name: Direction.
- Description: It indicates the oriented direction that, starting from the untuned resonance frequency, a MTT is able to exploit. In particular, if direction is “BOTH” the MTT is able to move the mechanical resonance frequency both in the increasing and in the decreasing direction. If instead direction is “RIGHT” (“LEFT”) the MTT is able to move the mechanical resonance frequency only in the increasing (decreasing) direction.
- Indicators Name: Δf
_{R}and Δf_{L}. - Description: These percentage indicators provide information concerning the range of frequency where the MTT can be applied with respect to the untuned mechanical resonance frequency. They are defined as follows$${\mathsf{\Delta}\mathrm{f}}_{\mathrm{R}}=\frac{{\mathrm{f}}_{\mathrm{R}}-{\mathrm{f}}_{\mathrm{opt}}}{{\mathrm{f}}_{\mathrm{opt}}}\xb7100\%$$$${\mathsf{\Delta}\mathrm{f}}_{\mathrm{L}}=\frac{{\mathrm{f}}_{\mathrm{opt}}-{\mathrm{f}}_{\mathrm{L}}}{{\mathrm{f}}_{\mathrm{opt}}}\xb7100\%$$
_{R}(f_{L}) is the maximum (minimum) achievable frequency in the right (left) direction starting from the untuned mechanical resonance frequency f_{opt}. The definition of both indicators is obviously possible only for MTTs that can exploit both directions of tuning. - Indicators Name: ΔP
_{R}and ΔP_{L}. - Description: These percentage indicators provide information concerning the reduction of the extracted power that is obtained at f
_{R}and f_{L}with respect to f_{opt}. They are defined as follows:$${\mathsf{\Delta}\mathrm{P}}_{\mathrm{R}}=\frac{{\mathrm{P}}_{\mathrm{R}}-{\mathrm{P}}_{\mathrm{MAX}}}{{\mathrm{P}}_{\mathrm{MAX}}}\xb7100\%$$$${\mathsf{\Delta}\mathrm{P}}_{\mathrm{L}}=\frac{{\mathrm{P}}_{\mathrm{L}}-{\text{}\mathrm{P}}_{\mathrm{MAX}}}{{\mathrm{P}}_{\mathrm{MAX}}}\xb7100\%$$_{R}(P_{L}) is the maximum extractable power in correspondence of f_{R}(f_{L}). Moreover, in this case, the definition of both indicators is obviously possible only for MTTs that can exploit both directions of tuning. Of course, in order to be able to compare MTTs, it is fair to provide also the amplitude A_{vib}of the input acceleration. - Indicator Name: Implementation.
- Description: It describes the way the considered MTT is implemented. In fact, the mechanical resonance frequency of an RVEH can be varied by acting on many different variables. Some of these are mechanical quantities, other are electrical quantities. Therefore, implementation can be “MECHANICAL” for mechanically implemented MTTs or “ELECTRICAL” for electrically implemented MTTs. In particular, in a mechanically implemented MTT the working principle of the tuning relies on the regulation of a mechanical variable (as an example a distance between magnets or the position of a clamp) even if such a variable is tuned by means of an electrical system (as an example an electrical actuator). What is important in this definition is the fundamental variable driving the working principle and not its particular implementation. Instead, in an electrically implemented MTT the operating principle of the tuning is based on the regulation of an electrical quantity (as an example the voltage of a piezoelectric actuator).
- Indicator Name: Actuation.
- Description: It indicates the way the considered MTT is actuated. In particular, if actuation is “MANUAL” the MTT needs the intervention of an operator that manually acts on the tuning mechanism. Instead, if it is “AUTOMATIC” the MTT adopts an actuator (as an example a motor) or an electrical signal in order to prevent the human intervention.
- Indicator Name: Control.
- Description: If control is equal to “OPEN LOOP” then the MTT needs a precharacterization of the RVEH (e.g., for the definition of a look-up table) and it is affected by errors due to any possible parameter change in the system. If control is equal to “CLOSED LOOP” then the MTT is more robust and does not need a precharacterization, but it needs sensors in order to implement the feedback circuitry and hence more energy is required for its operation. Obviously, the only type of actuation that can be controlled in a closed loop is the AUTOMATIC one.
- Indicator Name: Supply.
- Description: It provides a picture of the type of power demand characterizing the considered MTT. In particular, if supply is “PASSIVE”, the considered MTT needs a significant amount of power only to move the RVEH’s resonance frequency but it is able to indefinitely maintain the new resonance frequency value without any additional power consumption. In such a case, the control system energy consumption mainly depends on the energy required for the tuning step and, hence, on the rate of resonance frequency adjustments that the operating conditions require. If instead supply is “ACTIVE”, besides the initial power required in order to move the resonance frequency, the MTT continues to require power in order to maintain the new resonance frequency. If supply is “SEMI-ACTIVE”, the MTT needs an initial significant amount of power for moving the resonance frequency and it is able to maintain, only for a limited period of time, such a new value of resonance frequency without any other additional power demand. However, due to unavoidable continuous drifts of the resonance frequency, a periodic refresh is needed with additional power demands. It is worth noting that, the supply indicator can be defined only when the actuation indicator is equal to AUTOMATIC (without human intervention).
- Indicator Name: Tuning period.
- Description: This is a crucial indicator when considering the power consumption. The main requirement for a MTT is of course as low as possible power consumption. A MTT that consumes an average power greater than the harvested one is obviously useless in practice. In fact, the main purpose of a RVEH is to power both the MTT controller and the load in any vibrations’ conditions. The tuning period indicator provides the minimum period of time that is needed by the REVEH in order to store the amount of energy needed by the MTT controller for a given resonance frequency adjustment in the considered acceleration conditions. Obviously, in the presence of vibration frequency shifts, for a proper operation of a MTT the tuning period must be much shorter than the period of the vibration’s frequency shifts. The shorter the tuning period is, the faster the MTT will be able to react to vibrations’ frequencies changes. There are papers in which this aspect is not discussed at all. The tuning period indicator can be defined only when the actuation indicator is equal to AUTOMATIC.
- Indicator Name: Vibrations.
- Description: This indicator is focused on the type of vibrations that the considered MTT and the corresponding RVEH are able to exploit. In particular, the basic classification that is considered in this paper is between “SINUSOIDAL” and “NOT-SINUSOIDAL” vibrations. This is a crucial aspect from the practical point of view. Various vibration sources characterized by different harmonic contents exist in practical applications. Examples of vibration sources are walking people, moving trains or cars, and domestic or industrial working machines [84,85,86]. Sinusoidal vibrations are rarely encountered in real world applications. Even if vibrations in practical applications are usually periodic, random, or single event motions (e.g., impacts) [84,85,86], for simplicity reasons, most research papers focusing on RVEHs and in particular on MTTs deal with purely sinusoidal vibration sources.

## 4. Overview of Mechanical Tuning Techniques

#### 4.1. Magnetic Forces Based MTTs

#### 4.2. Piezoelectric Actuators Based MTTs

#### 4.3. Axial Loads Based MTTs

#### 4.4. Clamp Position Change Based MTTs

#### 4.5. Variable Reluctance Based MTTs

#### 4.6. Variable Center of Gravity Based MTTs

## 5. Discussions and Open Issues

_{max}and the y-axis is the corresponding value of P

_{MAX}. The various MTTs are identified with different colors. In particular, black asterisk symbols are used for “Magnetic Forces Based MTTs”, red asterisk symbols for “Piezoelectric Actuators Based MTTs”, green asterisk symbols for “Axial Loads Based MTTs”, cyan asterisk symbols for “Clamp Position Change Based MTTs”, blue asterisk symbols for “Variable Reluctance Based MTTs”, and yellow asterisk symbols for “Variable Center of Gravity MTTs”.

_{R}, Δf

_{R}, ΔP

_{L}, and Δf

_{L}are summarized. The x-axis represents the values of Δf

_{R}(in the case of asterisk markers) or Δf

_{L}(in the case of circular markers) and the y-axis represents the values of ΔP

_{R}(in the case of asterisk markers) or ΔP

_{L}(in the case of circular markers). The various MTTs are identified with the same colors as in Figure 26.

- First of all, it is possible to observe that, in all the papers focusing on MTTs, no attention at all is paid to ETTs. As it has been evidenced in Section 2, for the optimal exploitation of a RVEH both tuning techniques should be applied. Therefore, the need exists for a study of the joint application of both MTTs and ETTs. In addition, since in most papers reported in Figure 26 and Figure 27 a pure resistive load is considered, it is possible to state they are associated to underestimated results as their power performance is concerened.
- An important observation is worth noting. All the MTTs are based on the fact that RVEHs can be schematically represented (as discussed in Section 2.1) by means of spring-mass-damper systems. The mechanical resonance frequency of such systems (see (30) and (31)) depends on the values of the mass and of the stiffness and therefore it can be varied by acting on such two parameters values. In particular, for obvious reasons, it is easier to adopt a MTT that, during the RVEH operation, changes the stiffness rather than the value of the oscillating mass. In fact, nearly all the analyzed MTTs are based on the change of the stiffness. The application of a MTT for the increase of the resonance frequency by varying only the stiffness of the spring-mass-damper system is preferable also from another point of view. In fact, at least in principle, on the basis of (30) and (31), the increase of f
_{res}by means of the variation of the stiffness does not affect the maximum extractable power P_{MAX}. Instead, the increase of f_{res}by acting on the movable mass, requires the reduction of the value of such a mass with a consequent reduction of the maximum extractable power P_{MAX}as shown in (32) and (33). It is worth noting instead that, in the case of application of a MTT for reducing f_{res}, the variation of the moving mass (although unpractical) should be preferred with respect to that of the stiffness. In fact, at least in principle, on the basis of (32) and (33), the value of such a mass should be increased with a consequent increase of the maximum extractable power P_{MAX}. - From the analysis reported in Section 4 also another important aspect must be underlined. The tuning periods of the MTTs usually have too high values that can lead to questionable practical applicability of these techniques. In fact, a large tuning period means that a change in the frequency can be carried out only with a low speed, leading to quite slow MTTs that are able to track only vibrations characterised by relative slow dynamics. Therefore, an important objective for future research activities must be the reduction of the values of the tuning periods.
- It is clear that MTTs that are based on a closed-loop control are surely more robust and do not need a precharacterization of the system. However, they need sensors in order to implement the feedback circuitry and hence require much energy for their operation. Instead the MTTs that are based on an open-loop control require less energy but need a precharacterization of the system (e.g., for the definition of a LUT) and are less robust since they are affected by possible errors due to system parameters change. Trade-off solutions between closed-loop and open-loop adopting LUTs with “learning capability” seem to be very promising [25,26,38]. The MTTs that are more suitable for closed-loop controls are the piezoelectric actuators based MTTs since they can be simply controlled by acting on a voltage. In principle, it could be also possible to implement a MTT that is controlled in a closed-loop by acting on a current flowing in a proper coil in a magnetic forces based MTT. No system of this type has been proposed in the literature yet.
- A further aspect to underline is that, at the moment, all the MTTs are designed and tested in the case of purely sinusoidal vibrations. However, as stated in Section 3, purely sinusoidal vibrations are nearly impossible to be found in practical applications. Actual vibrations are usually periodic (with a fundamental component plus harmonics), random (with an energy content that is distributed over a wide frequency spectrum), or single event motions (as in the case of impacts) [84,85,86]. This is a crucial aspect and is an open issue for nearly all the RVEHs applications. In particular, as it can be observed from Section 4, many MTTs are based on the measurement of the vibration frequency. In the presence of purely sinusoidal vibrations such a task is quite simple. However, when the input vibrations are non-sinusoidal the task becomes much more complex and the detection of the vibration zero-crossings could mislead the MTT. A solution to such a problem could be a perturbative approach such as the one that is implemented in maximum power point tracking applications for RVEHs [98,100]. The perturbative approach could get rid of the measurement of the vibration frequency, since it could be based on the measurement of the extracted power and could adapt the RVEH’s resonance frequency in order to maximize such a power. Moreover, in this case the gap in the literature on such an important issue needs to be filled.
- Another important observation concerns the compliance of MTTs with miniaturization. In fact, miniaturization is of crucial importance in order to make RVEHs equipped with MTTs and suitable for wireless sensors networks or biosensors applications. Obviously, all the MTTs that are implemented by using cumbersome motors or mechanical actuators are not compatible with miniaturized systems and therefore further research on such a topic is necessary.
- Among all the analyzed MTTs, a very interesting property is the “passive self-tuning” capability that characterizes the variable center of gravity based MTT proposed in [45,46]. In particular, it has an auto-tune resonance frequency property due to the continuous change in the center of gravity following the vibration frequency change. This is a very important mechanical property that leads to a system that does not need any external control circuitry and its associated energy consumption. Such an interesting property could be the starting point for the future identification of other types of structures with auto-tuning capabilities.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Publication profile regarding vibration energy harvesters (Source Google Scholar, Keyword Search: “Vibration Energy Harvesting”).

**Figure 2.**(

**a**) Schematic model of a piezoelectric resonant vibration energy harvester (P-RVEH). (

**b**) Schematic model of an electromagnetic (E-RVEH).

**Figure 5.**Typical shape of the power extracted by a RVEH (P-RVEH or E-RVEH), when loaded with the optimal impedance as a function of the vibration frequency.

**Figure 6.**Typical trends vs. the normalized vibration frequency of the powers provided to the load by a RVEH (P-RVEH or E-RVEH) with the application of: (

**a**) ETT, (

**b**) MTT, (

**c**) both ETT and MTT.

**Figure 7.**Schematic representation of the magnetic forces based MTT proposed in [7].

**Figure 9.**Schematic representation of the magnetic forces based MTT proposed in [10].

**Figure 10.**Schematic representation of the magnetic forces based MTT proposed in [11].

**Figure 14.**Schematic representation of the axial loads based MTT proposed in [30].

**Figure 15.**Schematic representation of the axial loads based MTT proposed in [31].

**Figure 17.**Schematic representation of the axial loads based MTT proposed in [34].

**Figure 18.**Schematic representation of the commercial resonance frequency tuning kit [35] exploiting a clamp position change based MTT.

**Figure 19.**Schematic representation of the clamp position change based MTT proposed in [37].

**Figure 20.**Schematic representation of the clamp position change based MTT proposed in [38].

**Figure 21.**Schematic representations of the variable reluctance based MTT proposed in [39]. (

**a**) The flux guide is placed between the two magnets; (

**b**) the flux guide is placed between the poles of the fixed magnet.

**Figure 22.**Schematic representation of the variable reluctance based MTT proposed in [41].

**Figure 23.**Schematic representation of the variable center of gravity based MTT proposed in [43].

**Figure 24.**Schematic representation of the variable center of gravity based MTT proposed in [44].

**Figure 26.**Summary of the power performance of analyzed MTTs. P

_{MAX}as a function of the acceleration amplitude A

_{MAX}. The various MTTs are identified with different colors. Black for “Magnetic Forces Based MTTs”, red for “Piezoelectric Actuators Based MTTs”, green for “Axial Loads Based MTTs”, cyan for “Clamp Position Change Based MTTs”, blue for “Variable Reluctance Based MTTs”, and yellow for “Variable Center of Gravity MTTs”.

**Figure 27.**Summary of the tuning performance of the analyzed MTTs. All the asterisks represent ΔP

_{R}(Δf

_{R}), all the circles represent ΔP

_{L}(Δf

_{L}). The various MTTs are identified with different colors. Black for “Magnetic Forces Based MTTs”, red for “Piezoelectric Actuators Based MTTs”, green for “Axial Loads Based MTTs”, blue for “Variable Reluctance Based MTTs”, and yellow for “Variable Center of Gravity MTTs”.

MTT Name | Papers | Operating Principle Description |
---|---|---|

Magnetic Forces Based | [7,8,9,10,11,12,13,14,15,16,17,18,19,20] | They use the interaction between magnets with the aim of altering the stiffness of a RVEH, thus changing its mechanical resonance frequency. |

Piezoelectric Actuators Based | [21,22,23,24,25,26,27,28,29] | The adjustment of the RVEH resonance frequency is implemented by changing the mechanical stiffness by using piezoelectric actuators. |

Axial Loads Based | [30,31,32,33,34] | They exploit the fact that it is possible to vary the resonance frequency of an oscillating beam by means of axial loads. |

Clamp Position Change Based | [35,36,37,38] | They tune the stiffness of a cantilever beam by changing the position of a clamp supporter placed along such a beam. |

Variable Reluctance Based | [39,40,41,42] | They vary the force between two tuning magnets, and hence the stiffness of the structure, by means of the variation of the position of a magnetically permeable moveable flux guide placed between them. |

Variable Center of Gravity Based | [43,44,45,46,47,48,49] | They exploit the fact that in a cantilever with a tip mass it is possible to change the resonance frequency of the structure by varying the position of the center of gravity. |

Reference | [7] | [8,9] | [10] | [11] | [12,13] |

RVEH | P | E | P | E | E |

Direction | BOTH | LEFT | BOTH | RIGHT | RIGHT |

f_{opt} | 26.2 Hz | 223.1 Hz ^{(2)} | 61 Hz | 4.7 Hz | 67.6 Hz |

Δf_{R} | 22.14% | 42.6% | 91.5% | 45% | |

Δf_{L} | 16.03% | 15.5% | 16.4% | ||

P_{MAX} | 280 µW | 8.45 µW | NOT PROVIDED ^{(3)} | 800 µW | 156.6 µW |

A_{vib} | 80 mg | 125 mg | 10 mg | 60 mg | |

ΔP_{R} | −14.29% | NOT DEFINED ^{(4)} | −60.7% | ||

ΔP_{L} | −3.57% | −23.91% | |||

Implementation | MECHANICAL | MECHANICAL | MECHANICAL | MECHANICAL | MECHANICAL |

Actuation | MANUAL | MANUAL | MANUAL | AUTOMATIC | AUTOMATIC |

Control | OPEN-LOOP | CLOSED-LOOP | |||

Supply | PASSIVE | PASSIVE | |||

Tuning Period | 320 s ^{(1)} | NOT PROVIDED | NOT PROVIDED | 217 s ^{(5)} | 230 s ^{(6)} |

Vibrations | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL |

^{(1)}It is estimated by considering the energy required by the actuator to move the considered prototype device (with tip mass) equal to 85 mJ. Moreover, it is assumed that the device equipped with the MTT provides about 250 μW over the entire tuning range.

^{(2)}The untuned resonance frequency is the one in correspondence of the minimum vertical distance between the tuning magnet and the frame of the silicon spring.

^{(3)}In the paper, nothing is said about the power extracted from this P-RVEH but the tuning is tested with the harvester working only in open circuit conditions.

^{(4)}In the paper, there is no information for defining such an indicator.

^{(5)}It is estimated by considering the energy required by the actuator for one tuning cycle (whose duration is 10 s) to be 52 mJ.

^{(6)}It is estimated by taking into account the energy consumed during the MTT by the actuator equal to 5 mJ. Moreover, it is assumed that the generator provides an average power equal to 120 μW over the entire tuning range.

Reference | [21,22] | [23,24] | [25,26] |

RVEH | E ^{(1)} | E | P |

Direction | BOTH | BOTH | LEFT |

f_{opt} | 299 Hz | 78 Hz | 190 Hz |

Δf_{R} | 8% | 14.1% | |

Δf_{L} | 10.7% | 15.4% | 21% |

P_{MAX} | 60 µW | 1.4 mW | 50 μW |

A_{vib} | 1 g | NOT PROVIDED ^{(3)} | 0.6 g |

ΔP_{R} | −16.7% | NOT DEFINED ^{(4)} | |

ΔP_{L} | 16.7% | NOT DEFINED ^{(4)} | −60% |

Implementation | ELECTRICAL | ELECTRICAL | ELECTRICAL |

Actuation | MANUAL | AUTOMATIC | AUTOMATIC |

Control | CLOSED-LOOP | OPEN-LOOP ^{(6)} | |

Supply | SEMI-ACTIVE | SEMI-ACTIVE | SEMI-ACTIVE |

Tuning Period | 20 s ^{(2)} | NOT PROVIDED ^{(5)} | 22.8 s ^{(7)} |

Vibrations | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL |

^{(1)}When the vibrations’ frequency is equal to the untuned resonance frequency, energy is extracted both from the coil and from the piezoelectric layers.

^{(2)}It is estimated by considering that 200 μJ is the energy that must be transferred to the 18.9 nF piezoelectric actuator’s output capacitance in order to implement a frequency shift across 20 Hz in the tuning range. The harvester power output is equal to 50 μW and the efficiency of the control circuit is equal to 20%.

^{(3)}In the paper, nothing is said about the input acceleration amplitude A

_{vib}.

^{(4)}In the paper, there is no information for defining such indicators.

^{(5)}No tuning period is provided, it can be estimated by considering that the power needed for the discrete control circuit is around 150 mW while the harvested energy is equal to 1.4 mW.

^{(6)}It is OPEN-LOOP but with learning capability that is implemented by means of a feedback control every 90 s.

^{(7)}It has been estimated on the basis of the following considerations. The microcontroller, during the interval of time between two readjustments, requires only 2 μW. The control unit analyzes the vibration frequency and readjusts the actuator voltage, every 22.8 s, if needed. During such a phase, which lasts a couple of ms, the average power consumption of the device can reach 2.6 μW. The exact value depends on the vibration frequency. Then, a voltage to the piezoelectric actuator must be applied. In order to overcome leakage charge effects an additional average power consumption of 8.7 μW is needed.

Reference | [30] | [31] | [32,33] | [34] |

RVEH | P | P | P | P |

Direction | LEFT | BOTH | BOTH | BOTH |

f_{opt} | 250 Hz | 212 Hz | 380 Hz | 29.1 Hz |

Δf_{R} | 10.8% | 4.5% | 112.6% ^{(5)} | |

Δf_{L} | 20% | 62.3% | 23.2% | 79.4% ^{(5)} |

P_{MAX} | 400 μW | 40 μW ^{(1)} | NOT PROVIDED ^{(4)} | 368.9 μW ^{(5)} |

A_{vib} | 1 g | 0.35 g | 1 g | |

ΔP_{R} | −25% ^{(2)} | −80.8% ^{(5)} | ||

ΔP_{L} | −25% | −12.5% ^{(3)} | −67.3% ^{(5)} | |

Implementation | MECHANICAL | MECHANICAL | MECHANICAL | MECHANICAL |

Actuation | MANUAL | MANUAL | MANUAL | MANUAL |

Control | ||||

Supply | ||||

Tuning Period | NOT PROVIDED | NOT PROVIDED | NOT PROVIDED | NOT PROVIDED |

Vibrations | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL |

^{(1)}The authors provided the extracted power only in correspondence of three frequency points: 86, 132, and 154 Hz. 40 μW is the power at 132 Hz that is the central frequency. Moreover, a fixed resistive load (490 kΩ) is used in all the experimental results.

^{(2)}It is evaluated by considering P

_{MAX}= P(132 Hz) = 40 μW and P

_{R}= P(154 Hz) = 30 μW.

^{(3)}It is evaluated by considering P

_{MAX}= P(132 Hz) = 40 μW and P

_{L}= P(86 Hz) = 35 μW.

^{(4)}In the paper no attention is paid to the power output. The MTT is tested with the harvester working only in open circuit conditions.

^{(5)}Only numerical results.

Reference | [37] | [38] |

RVEH | P | P |

Direction | BOTH | BOTH |

f_{opt} | 580 Hz ^{(1)} | 121 Hz ^{(1)} |

Δf_{R} | 69% ^{(2)} | 48.8% |

Δf_{L} | 60.3% ^{(2)} | 29.8% |

P_{MAX} | 22 μW ^{(2)} | NOT PROVIDED ^{(4)} |

A_{vib} | 0.1 g | |

ΔP_{R} | NOT DEFINED ^{(3)} | |

ΔP_{L} | NOT DEFINED ^{(3)} | |

Implementation | MECHANICAL | MECHANICAL |

Actuation | MANUAL | AUTOMATIC |

Control | OPEN LOOP ^{(5)} | |

Supply | PASSIVE | |

Tuning Period | NOT PROVIDED | NOT PROVIDED |

Vibrations | SINUSOIDAL | SINUSOIDAL |

^{(1)}The untuned resonance frequency is considered the one in correspondence of the middle position of the support.

^{(2)}Only numerical results.

^{(3)}In the paper there is no information for defining such indicators.

^{(4)}The paper provides only tests with the harvester in open circuit conditions.

^{(5)}It is an open loop control with learning ability.

Reference | [39] | [41] | [42] |

RVEH | E | Hybrid P-E | P |

Direction | LEFT | BOTH | RIGHT |

f_{opt} | 63.6 Hz | 33.5 Hz | 102.5 Hz |

Δf_{R} | 85.1% | 12.8% | |

Δf_{L} | 21.7% | 23.9% | |

P_{MAX} | 166.2 µW | 2.78 mW | NOT PROVIDED ^{(1)} |

A_{vib} | 0.85 g | 0.3 g | |

ΔP_{R} | −42.9% | ||

ΔP_{L} | −30.8% | −28.6% | |

Implementation | MECHANICAL | MECHANICAL | MECHANICAL |

Actuation | MANUAL | MANUAL | MANUAL |

Control | |||

Supply | |||

Tuning Period | NOT PROVIDED | NOT PROVIDED | NOT PROVIDED |

Vibrations | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL |

^{(1)}The paper provides only tests with the harvester in open circuit conditions.

Reference | [43] | [44] | [45,46] |

RVEH | P | P | P |

Direction | BOTH | RIGHT | RIGHT |

f_{opt} | 160 Hz ^{(1)} | 42 Hz ^{(3)} | 21 Hz ^{(4)} |

Δf_{R} | 12.5% | 31% | 66.7% |

Δf_{L} | 18.75% | ||

P_{MAX} | NOT PROVIDED ^{(2)} | 80 μW | 13.18 μW |

A_{vib} | 0.03 g | 1.4 g | |

ΔP_{R} | −43.7% | −69.7% | |

ΔP_{L} | |||

Implementation | MECHANICAL | MECHANICAL | MECHANICAL |

Actuation | MANUAL | MANUAL | AUTOMATIC |

Control | OPEN LOOP | ||

Supply | PASSIVE | ||

Tuning Period | NOT PROVIDED | NOT PROVIDED | ^{(5)} |

Vibrations | SINUSOIDAL | SINUSOIDAL | SINUSOIDAL |

^{(1)}The untuned resonance frequency is considered to be the one in correspondence of a zero position of the movable element.

^{(2)}The paper provides only tests with the harvester in open circuit conditions.

^{(3)}The untuned resonance frequency is considered to be the one in correspondence of the end position of the auxiliary mass.

^{(4)}The untuned resonance frequency is considered to be the one when the effective center of gravity is in the free cantilever end.

^{(5)}The system has a passive auto-tuning capability.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Costanzo, L.; Vitelli, M.
Tuning Techniques for Piezoelectric and Electromagnetic Vibration Energy Harvesters. *Energies* **2020**, *13*, 527.
https://doi.org/10.3390/en13030527

**AMA Style**

Costanzo L, Vitelli M.
Tuning Techniques for Piezoelectric and Electromagnetic Vibration Energy Harvesters. *Energies*. 2020; 13(3):527.
https://doi.org/10.3390/en13030527

**Chicago/Turabian Style**

Costanzo, Luigi, and Massimo Vitelli.
2020. "Tuning Techniques for Piezoelectric and Electromagnetic Vibration Energy Harvesters" *Energies* 13, no. 3: 527.
https://doi.org/10.3390/en13030527