Issue
Sci. Tech. Energ. Transition
Volume 79, 2024
Decarbonizing Energy Systems: Smart Grid and Renewable Technologies
Article Number 30
Number of page(s) 8
DOI https://doi.org/10.2516/stet/2024026
Published online 07 May 2024

© The Author(s), published by EDP Sciences, 2024

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Harvesting all the available sources of energy in the environment (mechanical, thermal, electromagnetic) has become a fundamental issue with the multiplication of connected objects in the environment, such as sensor networks, which makes the development of self-powered electronic devices for wireless communication [1, 2]. Thermal energy is abundant, but its conversion to electrical energy is limited by the Carnot efficiency [3]. Harvesting mechanical energy from the environment to produce electrical energy is more promising, as the conversion efficiency can in principle reach 100%, which is a determinant advantage to power low-power systems. Moreover, the constraint of developing environmentally friendly devices focused research since 2012 on the development of triboelectric generators to convert mechanical to electrical energy [4, 5]. The principal advantage of triboelectric nanogenerators, commonly named TENGs, is that they can be fabricated from low cost environmentally friendly materials. The fabrication of TENGs requires thin layers of materials, which facilitates the fabrication and reduces the cost.

So far, many applications of TENGs already emerged, such as medical applications [6, 7]. Also, TENGs were developed for harvesting the energy of wind and ocean waves[8, 9]. The hybridization of a TENG with a piezoelectric generator was also explored [10].

Basically, a contact-separation (CS) TENG is represented by Figure 1. The generator consists of a capacitor whose upper electrode is mobile and the lower electrode is fixed and grounded. A dielectric layer of thickness hd and relative permittivity εr is attached to the lower electrode. The CS-TENG is also characterized by the triboelectric charge density σ distributed on an area Ad. Due to the numerous parameters to optimize, a profound understanding of the conversion mechanism of a TENG is required in order to achieve optimal performance. If a TENG is devoted to harvesting energy, it seems relevant to focus the study on the maximization of energy output. In this context, Zi et al. [11] defined a figure-of-merit (FOM) for a TENG, which has the advantage of allowing the comparison of performance of TENGs operating with different modes. Also, the additional advantage of FOM is to characterize the TENG performance independently of the triboelectric charge density, σ. However, the performance of a triboelectric generator may be characterized by different criteria, depending on the intended application of the TENG. Vasandani et al. [12] focused on the maximization of the output voltage of a CS-TENG. Using the design of experiment (DOE) methodology, the authors of [12] studied the effect of three parameters (the device area Ad, the maximum vibration displacement zmax and the dielectric thickness hd), while the TENG was in an open circuit. Adly et al. [13] chose to maximize the voltage per volume Vd of the device (Vd = Ad × zmax) by varying the resistance R and the dielectric thickness hd, using the design of experiment methodology. Zi et al. [11] presented results which demonstrate the existence of an optimum displacement zmax of the electrode which maximizes efficiency.

thumbnail Fig. 1

Diagram of a contact-separation (CS) triboelectric nanogenerators.

Many studies highlighted the importance of the description of the cyclic operation of a TENG by a voltage-charge diagram (U-Q diagram) for the evaluation of the energy transferred to the load resistance [1417]. Meng and Chen [17] pointed out that the conversion efficiency of a TENG could reach 100%, provided that a switch is used during the triboelectric cycle. If the resistive load R is disconnected from the TENG during the upward movement of the electrode, electrostatic energy is stocked within the electric field. Then, this electrostatic energy can be released into the load before the electrode goes back in contact with the dielectric.

The actual efficiency of the TENG depends on many parameters (at least four parameters are required to characterize a CS-TENG). In addition, the vibration characteristics (frequency, amplitude) are parameters to be considered for optimization of efficiency. It seems difficult to take account of all the parameters in an optimization process. This paper attempts to take into account all the parameters in the optimization process. The theoretical study that is presented concerns the optimization of a CS-TENG.

The paper first presents the theory of a CS-TENG, where it is shown that all the parameters of a TENG can be reduced to only two dimensionless parameters named α and β. In Section 2, the determination of optimum parameters α and β is conducted for two different situations: (i) the TENG is permanently connected to a load resistance R, and (ii) a switch is used to connect or disconnect the load R during the triboelectric cycle. Section 3 illustrates how the results of the preceding section can be used to design an optimized TENG, once the size of the device, frequency and amplitude of the vibrations are specified.

2 Theory

2.1 Triboelectric cycle

Triboelectric generators are energy converters that can potentially convert 100% of mechanical energy into electrical energy. The triboelectric charges are assumed to be uniformly distributed over the upper surface of the dielectric with density −σ (assume that triboelectric charges are negative). The total triboelectric charge is −Q0 = −σAd, where Ad is the area of the capacitor. The density of triboelectric charge −σ is assumed constant during operation. It will be shown in this section that if this assumption is not fulfilled, the results of the study are not changed.

The harvesting of vibrations is based on the oscillation of the upper electrode which periodically comes in contact with the dielectric. The device has a cyclic operation, which is depicted in Figure 2a. The cycle has four steps described as follows:

1. In the initial state of the generator, the mobile electrode is in contact with the dielectric (z = 0) and the capacitor is in short-circuit. The upper electrode carries a total positive charge Q0 which neutralizes the triboelectric charge −Q0. The lower electrode has no charge so the capacitor voltage U is zero. Due to the two opposite charges, an attractive electrostatic force exists between the upper electrode and the dielectric. From this state, the mobile electrode will have an upward movement during which the charges remain constant. To do so, the capacitor is in an open circuit during the displacement of the electrode.

2. In this state, the mobile electrode has reached the position z = zmax. During the displacement from z = 0 to z = zmax, the electrostatic force is counterbalanced by an external force which does mechanical work. This results in an increase in the electrostatic potential energy of the capacitor. The electrostatic energy increases linearly with the displacement z of the electrode. In this state, the capacitor has stored electrostatic energy:(1)

3. In this state, the position of the mobile electrode is still z = zmax. Before the electrode starts to return, the charge of the mobile electrode is cancelled, by driving the charge Q0 from the upper electrode to the lower electrode, using an external circuit. The voltage goes from to during the charge transfer. Note that the external circuit cannot be a simple resistance but an active circuit.

4. In this state, the position of the mobile electrode is again z = 0. During the backward movement, the charge was maintained to zero, so that no work could be done by the external force during the step from z = zmax to z = 0. In order to recover the initial state, the charge Q0 has to be driven back from the lower electrode to the upper electrode.

thumbnail Fig. 2

Triboelectric cycle: a) steps of ideal triboelectric cycle, b) voltage-charge diagram of the cycle.

Now, a balance sheet of the energy transferred to the external circuit during the triboelectric cycle previously described is drawn. Figure 2b represents the cycle in a voltage-charge diagram (U-Q diagram). The electrical energy transferred to the load R during a cycle is:(2)

The transferred energy, represented by the area of the cycle in Figure 2b is:(3)

This output energy Eref is taken as the reference output energy, as it is the maximum energy that can be transferred per cycle. If the external circuit does not have the ideal behavior, for example, a resistance R permanently connected, it will be shown that the output energy Ecycle is inferior to Eref. So the efficiency of energy transfer to the external harvester circuit is defined by:(4)

Note that the above definition of the efficiency coincide with the FOM defined by Zi et al. [11], which is:(5)

Indeed, . Substitution of the last expression in equation (4) leads to equation (5). Note that the reference energy defined by Meng and Chen [17] is slightly different than in equation (3), as the authors consider a different reference cycle, where negative external work is done during the backward translation of the mobile electrode and they considered that this work has to be removed from the balance sheet.

2.2 Dynamics equation of the CS-mode triboelectric generator

In Figure 1, a switch is present in the circuit. Firstly, considering that the switch remains permanently closed, the resistance R is continuously connected to the TENG. Section 3.2 deals with the case where the switch is closed and opened intermittently depending on certain switching levels.

2.2.1 Differential equation for the charge Q

In Appendix A, it is shown that the charge Q carried by the mobile electrode evolves according to the following differential equation of first order:(6)

The motion of the electrode is assumed to be described by a periodic function z(t). The solution Q(t) of equation (6) is the sum of a transient solution and a steady-state solution. The analysis of the energy exchange between the generator and the load R can be done using the U-Q diagram [11, 17]. The voltage U across R and the current I = −dQ/dt in the receiver convention for the load R gives an instantaneous power dissipation P(t) = −U dQ/dt. The energy dissipated in the load R over a cycle, during a steady state regime is given by equation (2) so that the efficiency η of the energy transfer from mechanical to electrical is equation (4). The problem is to find the set of parameters of the CS-TENG which provides the maximum efficiency η. First of all, it is mandatory to identify the parameters that intrinsically govern the electrical dynamics of the charge Q in equation (6). It can be seen that equation (6) is a priori determined by four parameters: σ, Ad, hd/εr (effective dielectric thickness) and R. Moreover, the motion z(t) of the mobile electrode in equation (6) must be specified. This motion is assumed to be sinusoidal: , where zmax is the peak-to-peak amplitude and f is the vibration frequency.

The problem to solve is the following: once the motion of the electrode, i.e. zmax and f fixed, find the parameters of the TENG which provides the maximum efficiency η. To solve this problem, it is useful to transform equation (6) into dimensionless form. The advantage of this transformation is to reveal which parameters intrinsically determine efficiency η.

2.2.2 Dimensionless dynamical equation

Firstly, the dimensionless time is obtained by dividing the real time t by the vibration period T = 1/f. Secondly, the dimensionless coordinate is obtained by dividing the real coordinate z by the peak-to-peak amplitude zmax. In the following, the symbols t and z will represent the dimensionless time and coordinate. Then, the dimensionless displacement of the electrode is written:(7)

The electric charge Q and the voltage U have to be replaced by the dimensionless charge q = Q/Q0 and voltage u = U/Umax, where the charge reference Q0 = σAd is the total triboelectric charge, which is assumed constant. The voltage is normalized to the maximum voltage . Using equations (2) and (3), the efficiency can be rewritten in terms of the dimensionless variables u and q:(8)

In the Appendix A, it is shown that the dimensionless voltage is: u = qz − β(1 − q), where β = hd/(εr zmax) is the ratio of effective dielectric thickness to the peak-to-peak amplitude of vibrations. Taking account that the integration of β(1 − q) over a cycle is zero, as it must be for a state function, the expression of η is simplified as follows:(9)

Equation (9) shows that the efficiency of a CS-TENG is only determined by the evolution of the dimensionless charge q and coordinate z. In Appendix A, it is shown that the differential equation for the variable q is:(10)

The first dimensionless parameter of equation (10) is α = R/R0, which is defined as the normalized resistance with the reference resistance R0 = T/C0, where C0 = ε0εrAd/hd is the reference capacitance of the TENG. The capacitance C0 is the capacitance of the TENG when the mobile electrode is in contact with the dielectric, i.e. z = 0. If the resistance R is permanently connected to the TENG (switch permanently closed), equation (10) shows that the evolution of q(t) is only determined by the two dimensionless parameters α and β. Consequently the efficiency η only depends on these two parameters. Equation (10) shows that the evolution of the dimensionless charge q(t) does not depend on the triboelectric charge density σ. So, the search for the maximum of the function η(αβ) has to be performed in a two-dimensional space (2D). The search of optimum parameters αopt and βopt which lead to the maximum of η will be studied in Section 3.1. In the case where the switch is opened and closed in synchronization with the electrode motion, at least one additional parameter SL (switching level) must be used to specify when the switch must open and close. Then, the optimum parameters αopt and βopt are functions of parameter SL. This situation will be discussed in Section 3.2. Once the optimum values αopt and βopt are found, it is easy to go back to the real parameters hd and R of the TENG.

2.2.3 Calculation of efficiency η

The numerical resolution of equation (10) can be performed with the xcos module of Scilab, an alternative to the Simulink module of Matlab. Once the two parameters α and β are chosen, equation (10) is solved to find the evolution of the dimensionless charge q(t) with initial charge q(t = 0) = 1. The solution q(t) displays a transient regime whose duration is of the order of the time constant RC0. The parameter , can be considered as the normalized time constant, so the calculation must be performed until the system reaches the steady state regime of the TENG (t ≥ 2α), which requires several periods of vibration if α < 1 (see Fig. 4 and 5). The efficiency η is calculated using equation (9) over one period in the steady state regime.

3 Optimization

Due to the numerous parameters governing the functioning of a TENG (see Table 1), optimization of TENG efficiency would be a priori a hard task. Fortunately, the normalization of variables and parameters has led to a drastic simplification of the problem to be solved. Now, the problem is reduced to the search of only two optimum parameters αopt and βopt.

Table 1

Definition of basic parameters of the TENG.

In this section, the situation where the TENG is always connected to the load resistance resistance R is first considered (switch permanently closed). Then, the case of a switch that is opened and closed in synchronization with the motion of the electrode is studied.

3.1 Resistance permanently connected to the generator

The resistance R is always connected during the cycle. The problem is to search the parameters αopt and βopt which give the maximum efficiency η defined by equation (9). In this case, efficiency η is calculated on a grid of values of α and β in order to plot the contour map of Figure 3. The plot clearly shows that there is only one maximum of efficiency at the point αopt = 0.22 and βopt = 0.83, which gives the maximum efficiency ηmax = 25.4%. The contour line at 20% shows that efficiency η is not very sensitive to variations of parameters α and β. Indeed, if 0.1 < α < 0.5 and 0.25 < β < 2 then η > 20%. For example, if the TENG is designed optimally for a given vibration frequency, then if the frequency has been doubled or halved (α doubled or halved), the efficiency does not fall abruptly, so the TENG remains close to the maximum efficiency. Also, if the amplitude of vibrations has been doubled or halved (β halved or doubled), the efficiency of the TENG remains close to its maximum.

thumbnail Fig. 3

Resistance permanently connected to the TENG. Contour plot of efficiency η versus parameters α and β.

At the optimum point, the evolution of the reduced voltage u(t) and charge q(t) as a function of the time is shown in Figure 4a. The corresponding U-Q cycle is shown in Figure 4b. One can observe that during the forward motion, the charge Q is at its highest level, whereas during the backward motion, the charge Q is at its lowest level. This remark gives a hint on what must be done to improve efficiency. The charge Q must be as large as possible during the upward movement of the electrode, whereas Q must be as close to zero as possible during the downward movement.

thumbnail Fig. 4

Optimal solution of equation (10) without switching. The optimum parameters are: αopt = 0.22 and βopt = 0.83. a) Left vertical axis: dimensionless motion of the electrode z(t) (blue solid line) and charge q(t) (green dot-dash line). Right vertical axis: voltage u(t) (thick red dashed line). b) U-Q cycle in dimensionless variables, showing efficiency of 25.4%, calculated in the steady state regime.

Unfortunately, with the switch permanently closed, efficiency is capped at about 25%. In order to improve TENG efficiency, it is mandatory to use a switch which closes and opens in synchronization with the motion of the electrode.

3.2 Synchronized switching

The switch is opened and closed at particular positions z(t) of the electrode (in dimensionless variable 0 ≤ z ≤ 1). A lower switching level SL is defined. The switch is closed if z(t) ≤ SL or z(t) ≥ SH, else the switch is open. When the switch is open the electric current is zero. So, equation (10) becomes , i.e. q is constant.

For the sake of simplification, a single parameter is chosen for the switching. The lower switching level, SL determines the higher switching level, which is SH = 1 – SL. As the switching level SL gets closer and closer to 0, SH gets closer and closer to 1.

To illustrate the effect of switching on efficiency, a calculation was performed with switching levels SL = 0.05 and SH = 0.95, and parameters α = 0.05 and β = 0.5, which are not the optimum parameters. The calculated efficiency is 70.7%, which is a considerable gain compared to no switching (see Fig. 5). It is reasonable to think that a switching level SL close to zero would increase efficiency, but the closing and opening of the switch at very close times, require a very short switching time and a fast transfer of the electric charge.

thumbnail Fig. 5

Solution of equation (10) at switching levels SL = 0.05 and SH = 0.95, for dimensionless parameters α = 0.05 and β = 0.5. This solution is not optimal for the switching level SL = 0.05. a) Left vertical axis: dimensionless motion of the electrode z(t) (blue solid line) and charge q(t) (green dotdash line). Right vertical axis: voltage u(t) (red dashed line). b) U-Q cycle in dimensionless variables, showing efficiency of 70.7%.

A more systematic study was conducted by varying the switching level SL in the range of [0.2%; 50%]. Note the switching level of 50% means that the switch remains closed all the time, so the load is permanently connected to the TENG. The search of optimum parameters αopt and βopt was conducted by a systematic exploration of a domain of variation of α and β in order to plot a contour map of the function η(α, β), for a fixed value of the switching level SL (see the example of Fig. 6 for SL = 10%). Figure 7a shows the evolution of optimum parameters αopt and βopt as a function of SL and Figure 7b shows the evolution of efficiency ηopt as a function of the switching level SL. Figure 7a shows the linear fit of the data with SL in a log-log plot. So the evolution of the optimum parameters can be approximated by the following functions:(11)and(12)

thumbnail Fig. 6

Resistance permanently connected to the TENG. Contour plot of efficiency η versus parameters α and β.

thumbnail Fig. 7

a) Evolution of the optimum parameters αopt and βopt as a function of the lower switching level SL. The relationships between optimum parameters and SL are obtained from the linear fit of the data in the log-log plot. b) Evolution of the optimum efficiency ηopt as a function of the lower switching level SL.

A similar approximate expression can be written for the optimum efficiency:(13)

Equation (13) is valid only if SL ≤ 0.05. For SL between 0.05 and 0.5, a polynomial fit is suitable to model ηopt versus SL.

For each value of switching level SL, the contour map displays a single maximum. For example, the contour map of Figure 6 shows the maximum of efficiency for the switching level SL = 10% at the point of coordinates: αopt = 0.055 and βopt = 0.5, with a maximum ηopt = 62.2%. It is remarkable that, about the optimum point, efficiency is not sensitive to any change in parameters α and β. The contour line at 50% efficiency shows the domain where η ≥ 50%. Thus, a TENG designed at the optimum point remains efficient even if the frequency and amplitude have been doubled or halved.

The practical uses of equations (11) and (12) will be presented in the Section 4.

4 Case studies

In this section, we present the application of the results presented in Section 3. Any dimensionless set of parameters α, β, and SL corresponds to a class of TENGs in the real world, i.e. many generators (in theory an infinity) with very different parameters, i.e. with different parameters: Ad, hd, and different vibration frequencies f and amplitudes zmax, may correspond to the same set of values α and β. The design of three optimized TENG will illustrate the use of dimensionless parameters α and β, each TENG belonging to a different class.

The three generators, each harvesting vibrations of different amplitudes zmax and frequencies f, have different sizes Ad. For the three generators, the same insulating material was chosen, with the dielectric constant εr = 3.4, and the identical triboelectric charge density σ = 10 μCm−2. As highlighted previously, triboelectric charge density σ has no influence on efficiency, but it determines the electrical power delivered by the generator, which is proportional to σ2. Note that σ also determines the electric field in the air gap, which is . This value of Ea is below the breakdown electric field of 3 MV m−1 [18]. Once the frequency f and the amplitude zmax are defined, the optimum dielectric thickness and the optimum load resistance are readily calculated as follows:(14)and(15)

Note that the optimum dielectric thickness is not explicitly frequency-dependent.

In these examples, the vibration amplitudes zmax are chosen by imposing the same acceleration amax = 50g ≈ 500. So, the peak-to-peak amplitude is: .

A minimum time delay between two successive switching Δtmin = 100 is chosen, which is not very fast but sufficient to take into account of the switching time and the time necessary to transfer the electrical charge from one electrode to the other. For each generator, the minimum possible switching level SL is calculated. Then, once the switching level SL, the optimum parameters: αopt, βopt are readily determined using equations (11) and (12). Table 2 summarizes the characteristics of each designed TENG. The average power Pav delivered by the generators is mentioned.

Table 2

Characteristics of the three designed triboelectric generators.

Generator #1:

The frequency of vibrations is 200 Hz. The choice of SL = 0.2% is possible (SH = 99.8%), which provides efficiency η = 96.5%.

Generator #2:

The frequency of vibrations is 2000 Hz. The choice of SL = 10% is possible (SH = 90%), which provides efficiency η = 62.2%.

Generator #3:

The frequency of vibrations is 5000 Hz. No switching is possible at this frequency with Δtmin = 100. The resistance is connected permanently, so efficiency η = 25.4%.

The optimum values of the load resistance R are all of the order of kΩ. It is worth pointing out that if the vibration frequency changes, there is the possibility of conserving αopt if one changes the load resistance R so that αopt remains constant, and then efficiency can remain unchanged. From the results of Section 3, efficiency is not very sensitive to any change in vibration amplitude with a factor between 0.5 and 2.0.

5 Conclusion

A study of CS-TENGs devoted to vibration energy harvesting was presented. The study aims to provide a tool useful to easily design a CS-TENG with the highest efficiency. The design of a CS-TENG necessitates the specification of at least 4 parameters: the device area Ad, the dielectric thickness hd, the load resistance R and the triboelectric charge density σ. Additional parameters are necessary to specify the vibration frequency and amplitude. As far as the designer is only concerned with the efficiency, it was demonstrated that the intrinsic dynamics of a CS-TENG are only determined by two dimensionless parameters: i) α = RC0f, which is the normalized load resistance and ii) , which is the normalized dielectric thickness. In consequence, the optimization of the efficiency of a TENG is considerably simplified by the reduction of the number of parameters to optimize. The case of a sinusoidal motion of the electrode was studied, with frequency f and peak-to-peak amplitude zmax. The first important result is that there is only one optimum solution αopt and βopt which maximizes efficiency. The second important result is that to obtain the maximum efficiency (in principle 100% efficiency achievable), it is mandatory to use switching, which isolates the TENG from the load during the motion of the electrode, except in the vicinity of turning points. Without switching, efficiency would be limited to a value of about 25%. Another important result of the study is the following: in the vicinity of the optimum point, efficiency varies very slowly. An optimized TENG designed for a given frequency and amplitude of vibration is still efficient even if the frequency or the amplitude is doubled or halved. In other words, the efficiency of a TENG can be made practically independent of the conditions of vibration.

It would be interesting in future work to apply the method described in this paper to other modes of TENGs, such as the lateral sliding (LS) mode in order to satisfy the need for a new efficient mechanical energy harvester.

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Appendix A

Calculation of the conversion efficiency

The differential equation which governs the evolution of the charge Q carried by the mobile electrode is derived. Figure 1 shows that the voltage at the capacitor terminals is:(A1)

The electric field in the air gap is:(A2)

In the dielectric, the electric induction is:(A3)where Q0 = σAd. So, the electric field within the dielectric is:(A4)

Finally, the voltage at TENG terminals is:(A5)

As U = −R dQ/dt, the differential equation for the charge Q is:(A6)

The efficiency of the triboelectric generator expressed in terms of the voltage U and charge Q is (see Sect. 2.1):(A7)

We define the dimensionless voltage u = U/Umax and charge q = Q/Q0.

Then the efficiency expressed in terms of the dimensionless variables is:(A8)

From equation (A5), we calculate the reduced voltage:(A9)

As ∮β(1 − q)dq = 0 (over a cycle ∮qdq = 0 and ∮dq = 0), the efficiency can be expressed by:(A10)

So, the knowledge of q(t) over a period is sufficient to calculate the efficiency of a CS-TENG.

Another expression for efficiency is derived. Efficiency can be written in the form: η = −∮zdq2. The function g = zq2 is a state function. So dg = q2dz+zdq2 and ∮dg = 0. Then −∮zdq2 = ∮q2dz and finally, one can write efficiency as follows:(A11)

The last expression can be related to the work done by the electrostatic force during a cycle. This force is proportional to q2.The following equality:(A12)expresses the law of conservation of energy over a period. The left expression is proportional to the mechanical work done over a cycle (input) by the external force counterbalancing the electrostatic force. The right expression is proportional to the electrical energy transferred to the load over a cycle (output).

All Tables

Table 1

Definition of basic parameters of the TENG.

Table 2

Characteristics of the three designed triboelectric generators.

All Figures

thumbnail Fig. 1

Diagram of a contact-separation (CS) triboelectric nanogenerators.

In the text
thumbnail Fig. 2

Triboelectric cycle: a) steps of ideal triboelectric cycle, b) voltage-charge diagram of the cycle.

In the text
thumbnail Fig. 3

Resistance permanently connected to the TENG. Contour plot of efficiency η versus parameters α and β.

In the text
thumbnail Fig. 4

Optimal solution of equation (10) without switching. The optimum parameters are: αopt = 0.22 and βopt = 0.83. a) Left vertical axis: dimensionless motion of the electrode z(t) (blue solid line) and charge q(t) (green dot-dash line). Right vertical axis: voltage u(t) (thick red dashed line). b) U-Q cycle in dimensionless variables, showing efficiency of 25.4%, calculated in the steady state regime.

In the text
thumbnail Fig. 5

Solution of equation (10) at switching levels SL = 0.05 and SH = 0.95, for dimensionless parameters α = 0.05 and β = 0.5. This solution is not optimal for the switching level SL = 0.05. a) Left vertical axis: dimensionless motion of the electrode z(t) (blue solid line) and charge q(t) (green dotdash line). Right vertical axis: voltage u(t) (red dashed line). b) U-Q cycle in dimensionless variables, showing efficiency of 70.7%.

In the text
thumbnail Fig. 6

Resistance permanently connected to the TENG. Contour plot of efficiency η versus parameters α and β.

In the text
thumbnail Fig. 7

a) Evolution of the optimum parameters αopt and βopt as a function of the lower switching level SL. The relationships between optimum parameters and SL are obtained from the linear fit of the data in the log-log plot. b) Evolution of the optimum efficiency ηopt as a function of the lower switching level SL.

In the text

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