(8)

Figure 3.

(a) Oscillation scheme used in experiment (black line) and its fitting function that is input into the analytical solution (red line) and (b) comparisons of the theoretical predictions on output voltage with experimental measurements [25].

In Figure 3b, the voltage output obtained by Eq. (5) is compared with the experimental results from Ref. [16]. Excellent agreements could be observed between the experimental and analytical results due to different loading frequencies, especially the positive part of the voltage output. For the negative part of the results, the analytical results are slightly larger than the experimental ones, particularly for the low contact frequency case. The results presented above clearly indicate the accuracy of the model and method developed in this work.

#### 2.1.2 Sliding-mode TENG

For lateral sliding-mode TENG, the basic structure is shown in Figure 4.

For the lateral sliding-mode TENG, the contact area is S, the sliding distance is x tð Þ, the length of the tribo-pair is l, and the width is w. Since l is always much larger than d<sup>1</sup> and d<sup>2</sup> as the tribo-pair, and x is always smaller than 0:9l, it is difficult to keep tribo-pair in perfect alignment. In this condition, the total capacitance C can be estimated as C ¼ ε0w lð Þ � x =d0. Utilizing the charge distribution shown above and Gauss theorem, the Voc can be estimated as Voc ¼ σxd0=ε0w lð Þ � x . So the V-Qx relationship can be shown as

$$V = -\frac{Q}{C} + V\_{\alpha^\*} = -\frac{Qd\_0}{\varepsilon\_0 w(l-\infty)} + \frac{\sigma \mathbf{x} d\_0}{\varepsilon\_0 w(l-\infty)}\tag{7}$$

For conductor-to-dielectric TENGs when the edge effect can be neglected, the relationship is similar by turning d<sup>2</sup> into zero. As a result, we can get the output voltage of sliding-mode TENG as

Figure 4.

Theoretical model of sliding-mode TENG. (a) Dielectric-to-dielectric sliding-mode TENG and (b) conductorto-dielectric sliding-mode TENG [26].

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

$$\begin{split} V &= \frac{\sigma d\_0}{\varepsilon\_0} \left[ \frac{l}{l - \varkappa(t)} \exp \left( -\frac{d\_0}{\varepsilon\_0 \text{RS}} \int\_0^t \frac{l}{l - \varkappa(t)} \mathbf{d}t' \right) \\ &+ \frac{d\_0}{\varepsilon\_0 \text{RS}} \frac{l}{l - \varkappa(t)} \int\_0^t \exp \left( \frac{d\_0}{\varepsilon\_0 \text{RS}} \int\_t^{t'} \frac{l}{l - \varkappa(\delta)} d\delta \right) \mathbf{d}t' - \mathbf{1} \right] \end{split} \tag{8}$$

Based on these output voltage expressions, we can get the average power output for TENG as

$$P\_{\rm eff} = \frac{1}{T} \int\_{t}^{t+T} \frac{V^2}{R} \,\mathrm{d}t \tag{9}$$

Here,T is the time span of one mechanical loading cycle.

This V-Q-x relationship provides a method for evaluation of the electric output of sliding-mode TENG, which may be easily extended as a methodology for sliding freestanding tribo-layer mode TENG. To make sure the accuracy and precision of the proposed V-Q-x relationship of sliding-mode TENG, the experiments from Ref. [17] are carried out for validation. The materials and scale parameters are shown in Table 2.

In the experiments, the tribo-pair is fixed on a horizontal tensile loading platform for reciprocating lateral sliding process. The sliding process of the tribo-pair was imposed by the dynamic testing machine with a symmetric accelerationdeceleration mode (Figure 5c). The analytical results of the voltage output obtained by Eq. (8) agree very well with the experimental result.

Based on the V-Q-x relationships presented from Eqs. (5)–(8), the output performance of TENG is predictable when device structure, material parameters, and motion process are clear. With these equations, the influence of each parameter is clear, while all others are certain. For example, we can change the one parameter such as load resistance while all the others kept unchanged. As a result, we can get the output characteristic of the target TENG device and find out the optimized load resistance.

With this method, parametric analyses are carried out to characterize the output performance of TENGs with different working conditions. Niu et al. studied the output characteristics of contact-mode [18], sliding-mode [19], single-electrode


#### Table 2. Parameters of sliding-mode TENG [17].

In Figure 3b, the voltage output obtained by Eq. (5) is compared with the experimental results from Ref. [16]. Excellent agreements could be observed between the experimental and analytical results due to different loading frequencies, especially the positive part of the voltage output. For the negative part of the results, the analytical results are slightly larger than the experimental ones, particularly for the low contact frequency case. The results presented above clearly indicate the accuracy of the model and method developed in this work.

(a) Oscillation scheme used in experiment (black line) and its fitting function that is input into the analytical solution (red line) and (b) comparisons of the theoretical predictions on output voltage with experimental

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

For lateral sliding-mode TENG, the basic structure is shown in Figure 4. For the lateral sliding-mode TENG, the contact area is S, the sliding distance is x tð Þ, the length of the tribo-pair is l, and the width is w. Since l is always much larger than d<sup>1</sup> and d<sup>2</sup> as the tribo-pair, and x is always smaller than 0:9l, it is difficult to keep tribo-pair in perfect alignment. In this condition, the total capacitance C can be estimated as C ¼ ε0w lð Þ � x =d0. Utilizing the charge distribution shown above and Gauss theorem, the Voc can be estimated as Voc ¼ σxd0=ε0w lð Þ � x . So the V-Q-

<sup>C</sup> <sup>þ</sup> Voc ¼ � Qd<sup>0</sup>

ε0w lð Þ � x

For conductor-to-dielectric TENGs when the edge effect can be neglected, the relationship is similar by turning d<sup>2</sup> into zero. As a result, we can get the output

Theoretical model of sliding-mode TENG. (a) Dielectric-to-dielectric sliding-mode TENG and (b) conductor-

<sup>þ</sup> <sup>σ</sup>xd<sup>0</sup> ε0w lð Þ � x

(7)

2.1.2 Sliding-mode TENG

Figure 3.

measurements [25].

x relationship can be shown as

voltage of sliding-mode TENG as

to-dielectric sliding-mode TENG [26].

Figure 4.

74

<sup>V</sup> ¼ � <sup>Q</sup>

2.2.1 Figure of merit for quantifying the output performance of TENG

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator

[22]. The expression of FOMP is

DOI: http://dx.doi.org/10.5772/intechopen.86992

closed loop in the V-Q curve as

Figure 6.

77

E ¼ PT ¼

ðT 0

periodic cycles is named as "cycles for energy output" (CEO).

"cycles for maximized energy output" (CMEO) by Ref. [22].

VIdt ¼

ð<sup>t</sup>¼<sup>T</sup> t¼0

The V-Q curve is usually obtained from the working process of TENG by the V-Q-x relation. In general, the working process of any type of TENG contains four steps as a cycle (Figure 6a). As Figure 6b shows, the V-Q curve becomes steady after a few initial cycles. The output energy per cycle derived from these steady

For each CEO, the difference between the maximum and the minimum trans-

ferred charges in its steady state is defined as the total cycling charge QC. As Figure 6b shows, Q<sup>C</sup> is always smaller than the maximum transferred charge QSC:max. If the Q<sup>C</sup> could be maximized to QSC:max, we may achieve the maximum power output. Having noticed that the Q<sup>C</sup> ¼ QSC:max only occurs in short-circuit condition, we add a switch in parallel with the external load and design the following repeated steps to achieve instantaneous short-circuit conditions during operations: step 1, the tribo-pair separates from x ¼ 0 to x ¼ xmaxat switch off; step 2, turn switch on to enable Q<sup>C</sup> ¼ QSC:max; step 3, the tribo-pair displaces from x ¼ xmax to x ¼ 0 at switch off; and step 4, turn the switch on to enable Q<sup>C</sup> ¼ 0. With this process, we may obtain the V-Q plot in Figure 6c. These cycles are named

(a) Working process of TENG, (b) voltage-charge (V-Q) plot for CEO, and (c) V-Q plot for CEMO [22].

VdQ ¼ ∮VdQ (11)

Based on the V-Q-x relationship, a series of optimization strategies for independent parameters have been proposed. But their optimization target is the maximum output voltage or power, and the physical properties of the tribo-pair are not taken into account. To take into consideration the influence of other parameters of the device, the performance figure of merit (FOMP) is developed as a new standard. It stands for the maximum power density of TENG, which represents a quantitative standard to reflect the output capability of TENGs with different configurations

> Em Axmax

(10)

FOMP ¼ 2ε<sup>0</sup>

Here, Em is the largest possible output energy per cycle, xmax is the maximum displacement of tribo-pair, and A is the contact area. The FOMP is considered as a universal standard to evaluate varieties of TENGs, since it is directly proportional to the greatest possible average output power regardless of the mode and size of TENG [22]. To obtain the FOMP, the power output per cycle of TENG is required first. The output energy per cycle E can be calculated through the encircled area of the

Figure 5.

(a) The TENG device in testing, (b) time history of output voltage, (c) simulation of dimensionless sliding process and (d) comparison of time history of output voltage between theoretical results and experimental results [26].

mode [20], and freestanding-mode [21] TENG under different load resistances, products of velocities, contact area sizes, effective dielectric thicknesses, and gap distances. These works obtained the effect of a series of independent parameters on the output characteristics including load resistance, maximum gap or slid distance, moving speed, device capacitance, and device structure parameters. These works, by numerical calculation of the real-time output characteristics, presented the suitable value range of these preceding parameters with common TENG device design and provided excellent guidance for structural design and optimization strategies for TENG devices.

#### 2.2 Optimization approach based on figure of merit

For various applications, four basic modes of TENGs have been developed. Each mode has its own structure and various triboelectric material choices for the tribopair. That makes it difficult to characterize and compare the output performance of TENG. A universal standard has to be introduced to quantify the performance of the TENGs, regardless of its operation mode. To solve this problem, the figure of merit of TENG is proposed [22]. It gives a quantitative evaluation of TENG's performance from both structure's and the materials' points of view [22–24]. The application of figure of merit leads to a more efficient design and optimization approach of various TENG structures in practical applications. It may help to establish a series of standards for developing TENGs toward practical applications and industrialization.

The figure of merits of TENG includes performance figure of merit related to the structure and material figure of merit related to surface charge density. The theoretical derivation and simulation of these two figures of merit will be discussed in Sections 2.2.1 and 2.2.2, respectively.

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

#### 2.2.1 Figure of merit for quantifying the output performance of TENG

Based on the V-Q-x relationship, a series of optimization strategies for independent parameters have been proposed. But their optimization target is the maximum output voltage or power, and the physical properties of the tribo-pair are not taken into account. To take into consideration the influence of other parameters of the device, the performance figure of merit (FOMP) is developed as a new standard. It stands for the maximum power density of TENG, which represents a quantitative standard to reflect the output capability of TENGs with different configurations [22]. The expression of FOMP is

$$\text{FOM}\_{\text{P}} = \text{2}\varepsilon\_{0} \, \frac{E\_{m}}{A \varkappa\_{\text{max}}} \tag{10}$$

Here, Em is the largest possible output energy per cycle, xmax is the maximum displacement of tribo-pair, and A is the contact area. The FOMP is considered as a universal standard to evaluate varieties of TENGs, since it is directly proportional to the greatest possible average output power regardless of the mode and size of TENG [22]. To obtain the FOMP, the power output per cycle of TENG is required first.

The output energy per cycle E can be calculated through the encircled area of the closed loop in the V-Q curve as

$$E = \overline{P}T = \int\_0^T V \mathbf{Id} \mathbf{t} = \int\_{t=0}^{t=T} V \mathbf{d}Q = \oint V \mathbf{d}Q \tag{11}$$

The V-Q curve is usually obtained from the working process of TENG by the V-Q-x relation. In general, the working process of any type of TENG contains four steps as a cycle (Figure 6a). As Figure 6b shows, the V-Q curve becomes steady after a few initial cycles. The output energy per cycle derived from these steady periodic cycles is named as "cycles for energy output" (CEO).

For each CEO, the difference between the maximum and the minimum transferred charges in its steady state is defined as the total cycling charge QC. As Figure 6b shows, Q<sup>C</sup> is always smaller than the maximum transferred charge QSC:max. If the Q<sup>C</sup> could be maximized to QSC:max, we may achieve the maximum power output. Having noticed that the Q<sup>C</sup> ¼ QSC:max only occurs in short-circuit condition, we add a switch in parallel with the external load and design the following repeated steps to achieve instantaneous short-circuit conditions during operations: step 1, the tribo-pair separates from x ¼ 0 to x ¼ xmaxat switch off; step 2, turn switch on to enable Q<sup>C</sup> ¼ QSC:max; step 3, the tribo-pair displaces from x ¼ xmax to x ¼ 0 at switch off; and step 4, turn the switch on to enable Q<sup>C</sup> ¼ 0. With this process, we may obtain the V-Q plot in Figure 6c. These cycles are named "cycles for maximized energy output" (CMEO) by Ref. [22].

Figure 6.

(a) Working process of TENG, (b) voltage-charge (V-Q) plot for CEO, and (c) V-Q plot for CEMO [22].

mode [20], and freestanding-mode [21] TENG under different load resistances, products of velocities, contact area sizes, effective dielectric thicknesses, and gap distances. These works obtained the effect of a series of independent parameters on the output characteristics including load resistance, maximum gap or slid distance, moving speed, device capacitance, and device structure parameters. These works, by numerical calculation of the real-time output characteristics, presented the suitable value range of these preceding parameters with common TENG device design and provided excellent guidance for structural design and optimization strategies

(a) The TENG device in testing, (b) time history of output voltage, (c) simulation of dimensionless sliding process and (d) comparison of time history of output voltage between theoretical results and experimental

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

For various applications, four basic modes of TENGs have been developed. Each mode has its own structure and various triboelectric material choices for the tribopair. That makes it difficult to characterize and compare the output performance of TENG. A universal standard has to be introduced to quantify the performance of the TENGs, regardless of its operation mode. To solve this problem, the figure of merit of TENG is proposed [22]. It gives a quantitative evaluation of TENG's performance from both structure's and the materials' points of view [22–24]. The application of figure of merit leads to a more efficient design and optimization approach of various TENG structures in practical applications. It may help to establish a series of standards for developing TENGs toward practical applications

The figure of merits of TENG includes performance figure of merit related to the structure and material figure of merit related to surface charge density. The theoretical derivation and simulation of these two figures of merit will be discussed in

2.2 Optimization approach based on figure of merit

for TENG devices.

Figure 5.

results [26].

and industrialization.

76

Sections 2.2.1 and 2.2.2, respectively.

The CMEO is related to load resistance of the output circuit: the higher the resistance, the higher the output energy per cycle. Therefore, the maximized output energy could be obtained when an infinitely large resistor as

$$E\_m = \frac{1}{2} Q\_{\text{SC.max}} \left( V\_{\text{OC.max}} + V\_{\text{max}}' \right) \tag{12}$$

Through FOMP, we introduced a universal standard to quantify the power output performance of the TENG regardless of its operation mode and materials. With this standard, we are able to evaluate the performance of the TENGs in different structures/modes and achieve the optimization approach for structure design and working parameter setting. Using Eq. (11), we may obtain the FOMP for different TENGs with same tribo-pair and contact area through analytical and simulation method. Taking the maximum FOMP as an optimization index, we will achieve the optimized structure/mode for a TENG. On the other hand, for a known TENG, we can find out its maximum FOMP and thus determine the best suitable working condition like load resistance, xmax, etc. Through these analyses based on FOMP, the following conclusions have been found: (1) xmax influences the output of TENG directly, and when xmax grows, the total transferred charges will increase and so does the output power; (2) the power output performance of TENG triggered by contact-separation action is better than that of sliding action; and (3) reducing the parasitic capacitance can help to increase the power output of TENG, and also larger-area generators with larger air gap capacitance are much more robust to

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator

DOI: http://dx.doi.org/10.5772/intechopen.86992

Using FOMm, we have introduced a quantitative standard to evaluate the maximum surface charge density for a triboelectric material. With FOMm of different materials, we are able to determine their triboelectric charge polarity. And with different FOMm of tribo-pairs, we can find the optimal choice for a TENG. With this standard, we found some useful regulations: (1) the optimized choice of materials in a tribo-pair should be in opposite polarities, and (2) the larger the polarity differential between the materials, the bigger the FOMm of the tribo-pair [22].

For TENGs, their electric output performance is simultaneously and coherently influenced by a group of factors including the dimensions of the electrodes and insulators, electrical properties of the materials and the loading processing, etc. These parameters are interlinked with each other; therefore changing one parameter with the others fixed may break the optimized condition of the device and require new adjustment in other parameters and further optimization. It is necessary to simulate the output performance of a TENG via theoretical models based on multiparameter analysis rather than specific cases with only one variable considered. In addition, TENGs could be used as sensor or energy harvester in either macro- or microscale, within which the physical properties of materials and device are quite different. Hence, the dimensionless analysis method is more feasible for

Here, to realize the optimization of the device, we developed a series of normal-

For the optimization of contact-separation mode TENG, a set of expressions for normalized electric voltage and power in dimensionless forms are proposed in this section. In these dimensionless expressions, the effects of all the parameters

ized expressions for output voltage and output power in dimensionless forms, which provide a group of scaling laws between the normalized electric output and two independent compound variables. These scaling laws can facilitate the analysis of the effects in different aspects of the device simultaneously and provide accordance for optimal design of TENG by considering the effects of all factors simultaneously. The optimal electric output could be obtained through the proposed formulations with all parameters of the TENG considered as variables [25, 26].

2.3 Optimization strategy for TENG based on multiparameter analysis

parasitic leakage [22–24].

parameter analysis of TENG.

2.3.1 Contact-separation mode TENG

79

Here QSC:max is the short-circuit transferred charge, VOC:max is the maximum open-circuit voltage, and V<sup>0</sup> max is the maximum achievable absolute voltage. The equation for Em of TENG operated in any conditions can also be expressed as

$$E\_m = \int\_0^{t\_1} P\_1(t) \, \mathrm{d}t + \int\_{t\_1}^{t\_2} P\_2(t) \, \mathrm{d}t + \int\_{t\_2}^{t\_3} P\_3(t) \, \mathrm{d}t + \int\_{t\_3}^{t\_4} P\_4(t) \, \mathrm{d}t \tag{13}$$

The average power output P is then

$$\overline{P} = \frac{E\_m}{T} \approx \frac{\overline{\nu} E\_m}{2\varkappa\_{\text{max}}} \tag{14}$$

Here v is the average velocity of the relative motion in tribo-pair. Thus, we can define the FOMP depending on the parameters of Em, xmax, and A and get its expression as Eq. (10).

#### 2.2.2 Figure of merit for quantifying material characteristic

From Section 2.1, we notice that the transfer charge Q and output voltage V are proportional to the surface charge density σ in the V-Q-x relations of TENG. The increase of σ will directly enhance the possible average output power of TENG significantly as the power output is proportional to σ2. Yet for FOMP as Eq. (10) shows, the material characteristic is not taken into consideration. Thus we define the material figure of merit (FOMm) as a standard to quantify the charge density of general surface as

$$\text{FOM}\_{\text{m}} = \sigma^2 \left( \text{C}^2/\text{m}^4 \right) \tag{15}$$

The FOMm is only determined by σ and the component related to the material properties alone. It can be used to evaluate the triboelectric performance of the materials in contact [22].

The accurate value of σ for one kind of material is measured by putting it into a testing TENG device and using electrometer to measure the total transfer charge. To avoid the influence of uncertain contact conditions, the testing TENG device utilizes liquid metals as the other material of the tribo-pair to get the maximum possible surface charge of the material. Through this method, the FOMm of a series of commonly used materials has been acquired, and their position in triboelectric series compared with the liquid metal has been figured out. These results in detail can be found in Ref. [22] and its following works.

#### 2.2.3 Application of figure of merits

The figure of merits provides a series of quantitative standards to evaluate the working performance of TENG. Their application enables more efficient design and optimization of various TENGs in practical applications. The optimization works based on figure of merits are likely to establish the principles for TENG design and develop TENGs toward practical applications and industrialization.

#### Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

Through FOMP, we introduced a universal standard to quantify the power output performance of the TENG regardless of its operation mode and materials. With this standard, we are able to evaluate the performance of the TENGs in different structures/modes and achieve the optimization approach for structure design and working parameter setting. Using Eq. (11), we may obtain the FOMP for different TENGs with same tribo-pair and contact area through analytical and simulation method. Taking the maximum FOMP as an optimization index, we will achieve the optimized structure/mode for a TENG. On the other hand, for a known TENG, we can find out its maximum FOMP and thus determine the best suitable working condition like load resistance, xmax, etc. Through these analyses based on FOMP, the following conclusions have been found: (1) xmax influences the output of TENG directly, and when xmax grows, the total transferred charges will increase and so does the output power; (2) the power output performance of TENG triggered by contact-separation action is better than that of sliding action; and (3) reducing the parasitic capacitance can help to increase the power output of TENG, and also larger-area generators with larger air gap capacitance are much more robust to parasitic leakage [22–24].

Using FOMm, we have introduced a quantitative standard to evaluate the maximum surface charge density for a triboelectric material. With FOMm of different materials, we are able to determine their triboelectric charge polarity. And with different FOMm of tribo-pairs, we can find the optimal choice for a TENG. With this standard, we found some useful regulations: (1) the optimized choice of materials in a tribo-pair should be in opposite polarities, and (2) the larger the polarity differential between the materials, the bigger the FOMm of the tribo-pair [22].

#### 2.3 Optimization strategy for TENG based on multiparameter analysis

For TENGs, their electric output performance is simultaneously and coherently influenced by a group of factors including the dimensions of the electrodes and insulators, electrical properties of the materials and the loading processing, etc. These parameters are interlinked with each other; therefore changing one parameter with the others fixed may break the optimized condition of the device and require new adjustment in other parameters and further optimization. It is necessary to simulate the output performance of a TENG via theoretical models based on multiparameter analysis rather than specific cases with only one variable considered. In addition, TENGs could be used as sensor or energy harvester in either macro- or microscale, within which the physical properties of materials and device are quite different. Hence, the dimensionless analysis method is more feasible for parameter analysis of TENG.

Here, to realize the optimization of the device, we developed a series of normalized expressions for output voltage and output power in dimensionless forms, which provide a group of scaling laws between the normalized electric output and two independent compound variables. These scaling laws can facilitate the analysis of the effects in different aspects of the device simultaneously and provide accordance for optimal design of TENG by considering the effects of all factors simultaneously. The optimal electric output could be obtained through the proposed formulations with all parameters of the TENG considered as variables [25, 26].

#### 2.3.1 Contact-separation mode TENG

For the optimization of contact-separation mode TENG, a set of expressions for normalized electric voltage and power in dimensionless forms are proposed in this section. In these dimensionless expressions, the effects of all the parameters

The CMEO is related to load resistance of the output circuit: the higher the resistance, the higher the output energy per cycle. Therefore, the maximized output

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

QSC:max VOC:max þ V<sup>0</sup>

Here QSC:max is the short-circuit transferred charge, VOC:max is the maximum

P2ð Þt dt þ

<sup>T</sup> <sup>≈</sup> vEm 2xmax

Here v is the average velocity of the relative motion in tribo-pair. Thus, we can

From Section 2.1, we notice that the transfer charge Q and output voltage V are proportional to the surface charge density σ in the V-Q-x relations of TENG. The increase of σ will directly enhance the possible average output power of TENG significantly as the power output is proportional to σ2. Yet for FOMP as Eq. (10) shows, the material characteristic is not taken into consideration. Thus we define the material figure of merit (FOMm) as a standard to quantify the charge density of general surface as

FOMm <sup>¼</sup> <sup>σ</sup><sup>2</sup> <sup>C</sup><sup>2</sup>

The FOMm is only determined by σ and the component related to the material properties alone. It can be used to evaluate the triboelectric performance of the

The accurate value of σ for one kind of material is measured by putting it into a testing TENG device and using electrometer to measure the total transfer charge. To avoid the influence of uncertain contact conditions, the testing TENG device utilizes liquid metals as the other material of the tribo-pair to get the maximum possible surface charge of the material. Through this method, the FOMm of a series of commonly used materials has been acquired, and their position in triboelectric series compared with the liquid metal has been figured out. These results in detail

The figure of merits provides a series of quantitative standards to evaluate the working performance of TENG. Their application enables more efficient design and optimization of various TENGs in practical applications. The optimization works based on figure of merits are likely to establish the principles for TENG design and

develop TENGs toward practical applications and industrialization.

ðt3 t2

equation for Em of TENG operated in any conditions can also be expressed as

<sup>P</sup> <sup>¼</sup> Em

define the FOMP depending on the parameters of Em, xmax, and A and get its

ðt2 t1

max

max is the maximum achievable absolute voltage. The

P3ð Þt dt þ

� � (12)

ðt4 t3

=m<sup>4</sup> � � (15)

P4ð Þt dt (13)

(14)

energy could be obtained when an infinitely large resistor as

Em <sup>¼</sup> <sup>1</sup> 2

P1ð Þt dt þ

2.2.2 Figure of merit for quantifying material characteristic

can be found in Ref. [22] and its following works.

2.2.3 Application of figure of merits

78

open-circuit voltage, and V<sup>0</sup>

expression as Eq. (10).

materials in contact [22].

Em ¼

ðt1 0

The average power output P is then

#### Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

involved have been investigated comprehensively and simultaneously. The normalized expressions for output voltage and power may facilitate the optimization based on the scaling laws by tuning different physical properties simultaneously rather than those only focusing on one physical property either in dimensional or dimensionless forms [25].

Vð Þτ ε<sup>0</sup> σd<sup>0</sup>

81

¼ Vd

þ d0T RSε<sup>0</sup>

A d0 ; RSε<sup>0</sup> d0T � �

> 1 þ A d0 xð Þτ � � ð<sup>τ</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86992

PeffTε<sup>0</sup> <sup>σ</sup>2d0<sup>S</sup> <sup>¼</sup> Pd

generator area, or oscillation period.

well with experiments (markers) in Figure 6.

¼ �1 þ 1 þ

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator

0 exp

A d0 ; RSε<sup>0</sup> d0T � � or

A d0 xð Þτ � � exp � <sup>d</sup>0<sup>T</sup>

> d0T RSε<sup>0</sup>

The relations in Eqs. (16)–(20) also reveal that the effective output voltage (or power) is proportional to the square of surface charge density. These dimensionless parameters and corresponding scaling laws also show a straightforward optimization method for the magnitude of oscillation at the same electrical load resistance,

To verify the dimensionless expressions, we compare the peak dimensionless

As can be seen from Figure 7, the theoretical predictions agree very well with the experimental measurements despite that the latter are achieved with different device structures, mechanical loadings, and circuit conditions. This demonstrates that the established scaling law reveals the underlying general correlation between the physical properties of the device and its output performance, which may provide robust guidelines for optimization strategies, no matter what way we use to make it, theoretical or experimental. Take the varying parameters T and R in the two experimental groups as an instance: from the experimental point of view, they are totally different parameters, of which the variations affect the output performance, respectively. However, we find in the scaling law the unified expression for the two parameters in RSε0=AT (or RSε0=d0T). The individual variation of either T or R has since become the variation of the compound variables. That is to say, the same compound variable may have different combinations of T and R. According to the scaling laws shown in Eqs. (16)–(20), the output voltages should have the same value in the cases with the same compound variables RSε0=AT (or RSε0=d0T) and A=d<sup>0</sup> even if the latter are comprised of parameters valued with different combinations. It indicates the necessity for coupling design of these parameters included in the compound variables to achieve the best output performance of the device. However, the correlations among the parameters and their simultaneous effect on the output performance are hard to be found in traditional optimization techniques based on single-parameter investigations, in neither experimental nor theoretical means. As thus, we can conclude that the scaling law proposed in this paper can be not only used to predict the output performance of a TENG comprehensively and systematically with all parameters being considered simultaneously but also treated as a general and rational optimization criterion for the device toward its best performance. Based on the scaling laws from Eqs. 16 to 20, the output performance of the generator can be optimized by tuning combined parameters or individual physical quantities. The scaling law for the output voltage and power in a form of either in Eqs. (16)–(18) or in Eqs. (19)–(20) provide a universal optimizing strategy

output voltage based on Eqs. (16) and (19) with the experimental results in corresponding dimensionless forms. Two groups of experiments with various thicknesses dglass of glass plate [16] are utilized for comparisons. In Group I, the oscillation frequency 1=T varies from 1 Hz to 6 Hz with the load resistance fixed at R ¼ 100 MΩ, while in Group II, the load resistance R varies from 1 MΩ to 1000 MΩ with the oscillation frequency fixed at 1=T ¼ 5 Hz. The theoretical predictions for the dimensionless voltage Vpeakε0=σA and Vpeakε0=σd<sup>0</sup> (solid lines) both agree very

PeffRε<sup>2</sup> 0 σ2d<sup>2</sup> 0

RSε<sup>0</sup>

d0 ðζ τ

> A d0 ; RSε<sup>0</sup> d0T

� � � � <sup>d</sup><sup>ζ</sup>

ð Þþ <sup>ζ</sup> � <sup>τ</sup> <sup>A</sup>

¼ PdR

τ þ A d0 ðτ 0 xð Þτ dτ

xð Þζ dζ

� � � �

� � (20)

(19)

For the output voltage in a general circuit with the load resistance of R, the various structure, material, mechanical loading, and circuit properties of TENG are investigated in a coherent manner and expressed with two combined parameters. The dimensionless expression for normalized output voltage that depends only on the two combined parameters is derived based on Eq. (5) as

$$\frac{V(\tau)e\_0}{\sigma A} = \overline{V}\_A \left(\tau, \frac{A}{d\_0}, \frac{R\mathcal{S}e\_0}{AT}\right) = -\frac{d\_0}{A} + \left(\frac{d\_0}{A} + \overline{\mathfrak{x}}(\tau)\right) \exp\left[-\frac{AT}{R\mathcal{S}e\_0} \left(\frac{d\_0}{A}\tau + \int\_0^\tau \overline{\mathfrak{x}}(\tau)d\tau\right)\right],\tag{16}$$

$$+ \frac{AT}{R\mathcal{S}e\_0}\frac{d\_0}{A}\left(\frac{d\_0}{A} + \overline{\mathfrak{x}}(\tau)\right)\int\_0^\tau \exp\left[\frac{AT}{R\mathcal{S}e\_0}\left((\zeta-\tau)\frac{d\_0}{A} + \int\_\tau^\zeta \overline{\mathfrak{x}}(\zeta)d\zeta\right)\right]d\zeta$$

where τ ¼ t=T is the dimensionless time and A and T are, respectively, the oscillation amplitude and the period of the separation-contact cycle.

While for the application of energy harvest, the output power should be a key variable for characterizing the performance of the generator. According to the

definition of output power <sup>P</sup>eff <sup>¼</sup> <sup>1</sup> R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ð T <sup>0</sup> <sup>V</sup>2ð Þ<sup>t</sup> <sup>d</sup><sup>t</sup> T r , the dimensionless output power may be given by

$$\frac{P\_{\rm eff} T \varepsilon\_0}{\sigma^2 A S} = \overline{P}\_A \left( \frac{A}{d\_0}, \frac{R S \varepsilon\_0}{A T} \right) \tag{17}$$

Here, we can see that the dimensionless output voltage and power depend only on two combined parameters, i.e., A=d<sup>0</sup> and RSε0=AT, in which A=d<sup>0</sup> characterizes the relative oscillation amplitude, while RSε0=AT reflects the hybrid effects of generator area, electrical resistance, oscillation amplitude, and period of the tribopair. Besides this universal optimization of TENG by tuning multiple parameters at the same time, we may still realize the optimization for individual physical quantity with the assumption of other parameters fixed. However, we cannot figure out how to optimize the oscillation amplitude A to achieve the peak output from Eqs. (16) and (17) since this physical property get involved in both the combined dimensionless parameter and the normalized electric output. For the same reason, we may not work out the effect of the generator area S and the mechanical loading period T on the output power with Eq. (17).

If we want to examine the effect of S on the output power, we may simply multiply Eq. (17) with the factor RSε0=AT and have

$$\frac{P\_{\rm eff} Re\_0^2}{\sigma^2 A^2} = \overline{P}\_{AR} \left(\frac{A}{d\_0}, \frac{RSe\_0}{AT}\right) \tag{18}$$

To investigate the effect of oscillation amplitude and period, another set of dimensionless expressions is derived by multiplying Eqs. (16) and (17) with the factor A=d<sup>0</sup> and the combined parameter RSε0=AT and multiplying Eq. (18) with A<sup>2</sup> =d<sup>2</sup> <sup>0</sup> and RSε0=AT. This set of dimensionless expressions for the scaling laws will be

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

$$\frac{V(\tau)e\_0}{\sigma d\_0} = \overline{V}\_d \left(\frac{A}{d\_0}, \frac{R\mathcal{E}e\_0}{d\_0 T}\right) = -1 + \left(1 + \frac{A}{d\_0}\overline{\mathbf{x}}(\tau)\right) \exp\left[-\frac{d\_0 T}{R\mathcal{E}e\_0} \left(\tau + \frac{A}{d\_0}\int\_0^{\tau} \overline{\mathbf{x}}(\tau)d\tau\right)\right],\tag{19}$$

$$+ \frac{d\_0 T}{R\mathcal{E}e\_0} \left(1 + \frac{A}{d\_0}\overline{\mathbf{x}}(\tau)\right) \int\_0^{\tau} \exp\left[\frac{d\_0 T}{R\mathcal{E}e\_0} \left((\zeta - \tau) + \frac{A}{d\_0}\int\_{\tau}^{\zeta} \overline{\mathbf{x}}(\zeta)d\zeta\right)\right] d\zeta$$

$$\frac{P\_{\text{eff}} \, T \varepsilon\_0}{\sigma^2 d\_0 S} = \overline{P}\_d \left( \frac{A}{d\_0}, \frac{R \text{S} e\_0}{d\_0 T} \right) \text{ or } \frac{P\_{\text{eff}} \, R \varepsilon\_0^2}{\sigma^2 d\_0^2} = \overline{P}\_{d \mathbb{R}} \left( \frac{A}{d\_0}, \frac{R \text{S} e\_0}{d\_0 T} \right) \tag{20}$$

The relations in Eqs. (16)–(20) also reveal that the effective output voltage (or power) is proportional to the square of surface charge density. These dimensionless parameters and corresponding scaling laws also show a straightforward optimization method for the magnitude of oscillation at the same electrical load resistance, generator area, or oscillation period.

To verify the dimensionless expressions, we compare the peak dimensionless output voltage based on Eqs. (16) and (19) with the experimental results in corresponding dimensionless forms. Two groups of experiments with various thicknesses dglass of glass plate [16] are utilized for comparisons. In Group I, the oscillation frequency 1=T varies from 1 Hz to 6 Hz with the load resistance fixed at R ¼ 100 MΩ, while in Group II, the load resistance R varies from 1 MΩ to 1000 MΩ with the oscillation frequency fixed at 1=T ¼ 5 Hz. The theoretical predictions for the dimensionless voltage Vpeakε0=σA and Vpeakε0=σd<sup>0</sup> (solid lines) both agree very well with experiments (markers) in Figure 6.

As can be seen from Figure 7, the theoretical predictions agree very well with the experimental measurements despite that the latter are achieved with different device structures, mechanical loadings, and circuit conditions. This demonstrates that the established scaling law reveals the underlying general correlation between the physical properties of the device and its output performance, which may provide robust guidelines for optimization strategies, no matter what way we use to make it, theoretical or experimental. Take the varying parameters T and R in the two experimental groups as an instance: from the experimental point of view, they are totally different parameters, of which the variations affect the output performance, respectively. However, we find in the scaling law the unified expression for the two parameters in RSε0=AT (or RSε0=d0T). The individual variation of either T or R has since become the variation of the compound variables. That is to say, the same compound variable may have different combinations of T and R. According to the scaling laws shown in Eqs. (16)–(20), the output voltages should have the same value in the cases with the same compound variables RSε0=AT (or RSε0=d0T) and A=d<sup>0</sup> even if the latter are comprised of parameters valued with different combinations. It indicates the necessity for coupling design of these parameters included in the compound variables to achieve the best output performance of the device.

However, the correlations among the parameters and their simultaneous effect on the output performance are hard to be found in traditional optimization techniques based on single-parameter investigations, in neither experimental nor theoretical means. As thus, we can conclude that the scaling law proposed in this paper can be not only used to predict the output performance of a TENG comprehensively and systematically with all parameters being considered simultaneously but also treated as a general and rational optimization criterion for the device toward its best performance. Based on the scaling laws from Eqs. 16 to 20, the output performance of the generator can be optimized by tuning combined parameters or individual physical quantities. The scaling law for the output voltage and power in a form of either in Eqs. (16)–(18) or in Eqs. (19)–(20) provide a universal optimizing strategy

involved have been investigated comprehensively and simultaneously. The normalized expressions for output voltage and power may facilitate the optimization based on the scaling laws by tuning different physical properties simultaneously rather than those only focusing on one physical property either in dimensional or dimen-

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

For the output voltage in a general circuit with the load resistance of R, the various structure, material, mechanical loading, and circuit properties of TENG are investigated in a coherent manner and expressed with two combined parameters. The dimensionless expression for normalized output voltage that depends only on

> d0 <sup>A</sup> <sup>þ</sup> <sup>x</sup>ð Þ<sup>τ</sup> � �

where τ ¼ t=T is the dimensionless time and A and T are, respectively, the

While for the application of energy harvest, the output power should be a key variable for characterizing the performance of the generator. According to the

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ð T <sup>0</sup> <sup>V</sup>2ð Þ<sup>t</sup> <sup>d</sup><sup>t</sup> T

> > A d0 ; RSε<sup>0</sup> AT � �

Here, we can see that the dimensionless output voltage and power depend only on two combined parameters, i.e., A=d<sup>0</sup> and RSε0=AT, in which A=d<sup>0</sup> characterizes the relative oscillation amplitude, while RSε0=AT reflects the hybrid effects of generator area, electrical resistance, oscillation amplitude, and period of the tribopair. Besides this universal optimization of TENG by tuning multiple parameters at the same time, we may still realize the optimization for individual physical quantity with the assumption of other parameters fixed. However, we cannot figure out how to optimize the oscillation amplitude A to achieve the peak output from Eqs. (16) and (17) since this physical property get involved in both the combined dimensionless parameter and the normalized electric output. For the same reason, we may not work out the effect of the generator area S and the mechanical loading period T on

If we want to examine the effect of S on the output power, we may simply

To investigate the effect of oscillation amplitude and period, another set of dimensionless expressions is derived by multiplying Eqs. (16) and (17) with the factor

A=d<sup>0</sup> and the combined parameter RSε0=AT and multiplying Eq. (18) with A<sup>2</sup>

RSε0=AT. This set of dimensionless expressions for the scaling laws will be

A d0 ; RSε<sup>0</sup> AT � �

AT RSε<sup>0</sup> exp � AT RSε<sup>0</sup>

A þ

� � � �

ðζ τ

ð Þ <sup>ζ</sup> � <sup>τ</sup> <sup>d</sup><sup>0</sup>

d0 <sup>A</sup> <sup>τ</sup> <sup>þ</sup>

xð Þζ dζ

, the dimensionless output power may

� � � �

ðτ 0 xð Þτ dτ

dζ

(16)

(17)

(18)

=d<sup>2</sup> <sup>0</sup> and

the two combined parameters is derived based on Eq. (5) as

¼ � <sup>d</sup><sup>0</sup> A þ

> 0 exp

oscillation amplitude and the period of the separation-contact cycle.

R

PeffTε<sup>0</sup> <sup>σ</sup><sup>2</sup>AS <sup>¼</sup> PA

r

sionless forms [25].

<sup>σ</sup><sup>A</sup> <sup>¼</sup> VA <sup>τ</sup>;

þ AT RSε<sup>0</sup>

A d0 ; RSε<sup>0</sup> AT � �

> d0 A

definition of output power <sup>P</sup>eff <sup>¼</sup> <sup>1</sup>

the output power with Eq. (17).

multiply Eq. (17) with the factor RSε0=AT and have

PeffRε<sup>2</sup> 0 <sup>σ</sup><sup>2</sup>A<sup>2</sup> <sup>¼</sup> PAR

d0 <sup>A</sup> <sup>þ</sup> <sup>x</sup>ð Þ<sup>τ</sup> � � ð<sup>τ</sup>

Vð Þτ ε<sup>0</sup>

be given by

80

Figure 7.

Validation of the scaling laws for dimensionless peak output voltage through comparisons with experimental measurements with different setups [25].

As mentioned before, the scaling laws in Eqs. (16)–(19) cannot reflect a clear dependence of the output performance on the oscillation amplitude A since it is included in both two dimensionless parameters. Fortunately, with the scaling laws in Eqs. (20) and (21), the effect of A is extracted and can be distinguished quantitatively from the numerical results. Figure 8e–f shows that increasing the oscillation amplitude may enhance the output power or power density monotonically and only a smaller generator area, lower oscillation frequency, or load resistance may

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator

DOI: http://dx.doi.org/10.5772/intechopen.86992

For the optimization of sliding-mode TENG, Zhang [26] established a series of dimensionless expressions of output performance of sliding-mode TENG according

exp � <sup>d</sup>0<sup>T</sup>

d0T <sup>ε</sup>0RS <sup>ð</sup><sup>τ</sup><sup>0</sup> τ

0

B@

0

B@

<sup>ε</sup>0RS <sup>ð</sup><sup>τ</sup> 0

1

τ0

1

CA dτ<sup>0</sup> � 1

dδ

1

CA

(21)

<sup>1</sup> � <sup>A</sup> l x τ<sup>0</sup> ð Þ

1 <sup>1</sup> � <sup>A</sup> l xð Þδ

achieve an optimized output power [25].

Scaling laws for peak output voltage and output power [25].

<sup>¼</sup> <sup>ε</sup>0Vð Þ<sup>τ</sup> σd<sup>0</sup>

> þ d0T ε0RS

<sup>¼</sup> <sup>1</sup> <sup>1</sup> � <sup>A</sup> l xð Þτ

1 <sup>1</sup> � <sup>A</sup> l xð Þτ ðτ 0 exp

2.3.2 Sliding-mode TENG

to Eq. (7) as

VR τ; A l ; ε0RS d0T � �

83

Figure 8.

to enhance the output voltage for sensing application or maximum output power for energy harvesting application [25].

In Figure 8, the maximum peak voltage is obviously increasing monotonically with the relative oscillation amplitude A=d<sup>0</sup> (Figure 8a). While for each given value of A=d0, there exists an optimized RSε0=AT that delivers a maximum dimensionless output power (Figure 8b). It suggests that we may achieve the best output performance of TENG through tuning multiple physical parameters simultaneously. From Figure 8, we can also find that the pinpoint for a peak output power or power density (=0.25) is located around A=d<sup>0</sup> ¼ 2:0 and RSε0=AT ¼ 0:12.

Besides the universal optimization with multiple parameters, we may also use individual parameter analysis by tuning its quantity to enhance the output performance. For instance, from Figure 8b it is found that given all other parameters, there exists an optimal load resistance to achieve a maximum output power. While from Figure 8c with given load resistance and oscillation amplitude, it can be found that larger generator area and higher oscillation frequency may enhance the output power until saturated.

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

#### Figure 8.

to enhance the output voltage for sensing application or maximum output power

Validation of the scaling laws for dimensionless peak output voltage through comparisons with experimental

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

density (=0.25) is located around A=d<sup>0</sup> ¼ 2:0 and RSε0=AT ¼ 0:12.

In Figure 8, the maximum peak voltage is obviously increasing monotonically with the relative oscillation amplitude A=d<sup>0</sup> (Figure 8a). While for each given value of A=d0, there exists an optimized RSε0=AT that delivers a maximum dimensionless output power (Figure 8b). It suggests that we may achieve the best output performance of TENG through tuning multiple physical parameters simultaneously. From Figure 8, we can also find that the pinpoint for a peak output power or power

Besides the universal optimization with multiple parameters, we may also use individual parameter analysis by tuning its quantity to enhance the output performance. For instance, from Figure 8b it is found that given all other parameters, there exists an optimal load resistance to achieve a maximum output power. While from Figure 8c with given load resistance and oscillation amplitude, it can be found that larger generator area and higher oscillation frequency may enhance the output

for energy harvesting application [25].

measurements with different setups [25].

power until saturated.

82

Figure 7.

Scaling laws for peak output voltage and output power [25].

As mentioned before, the scaling laws in Eqs. (16)–(19) cannot reflect a clear dependence of the output performance on the oscillation amplitude A since it is included in both two dimensionless parameters. Fortunately, with the scaling laws in Eqs. (20) and (21), the effect of A is extracted and can be distinguished quantitatively from the numerical results. Figure 8e–f shows that increasing the oscillation amplitude may enhance the output power or power density monotonically and only a smaller generator area, lower oscillation frequency, or load resistance may achieve an optimized output power [25].

#### 2.3.2 Sliding-mode TENG

For the optimization of sliding-mode TENG, Zhang [26] established a series of dimensionless expressions of output performance of sliding-mode TENG according to Eq. (7) as

$$\begin{split} \overline{\nabla}\_{R} \left( \tau, \frac{A}{l}, \frac{\varepsilon\_{0} RS}{d\_{0}T} \right) &= \frac{\varepsilon\_{0} V(\tau)}{\sigma d\_{0}} = \frac{1}{1 - \frac{A}{l} \overline{\pi}(\tau)} \exp \left( -\frac{d\_{0}T}{\varepsilon\_{0} RS} \right) \Big|\_{0}^{\tau} \frac{1}{1 - \frac{A}{l} \overline{\pi}(\tau')} \tau' \Big|\_{0}^{\tau} \\ &+ \frac{d\_{0}T}{\varepsilon\_{0} RS} \frac{1}{1 - \frac{A}{l} \overline{\pi}(\tau)} \int\_{0}^{\tau} \exp \left( \frac{d\_{0}T}{\varepsilon\_{0} RS} \int\_{\tau}^{\tau'} \frac{1}{1 - \frac{A}{l} \overline{\pi}(\delta)} d\delta \right) d\tau' - 1 \end{split} \tag{21}$$

Here, A is the maximum sliding distance. When used as energy harvesters, the output power will become the key target to evaluate the output characteristics of TENG. According to the definition of effective output power, the dimensionless output power can be expressed as

$$
\overline{P}\_T \left( \frac{A}{l}, \frac{\varepsilon\_0 RS}{d\_0 T} \right) = \frac{\varepsilon\_0^2 RP\_{\text{eff}}}{\sigma^2 d\_0^2} = \int\_0^1 \overline{V}^2 d\tau \tag{22}
$$

Pd <sup>C</sup>; <sup>T</sup> � � <sup>¼</sup> <sup>T</sup><sup>2</sup>

dimensionless capacitance and the dimensionless time constant.

Scaling laws for dimensionless output voltage and power in case I and case II [26].

listed in Table 2.

Figure 9.

tal data points.

Figure 10.

85

PR <sup>C</sup>; <sup>T</sup> � � <sup>¼</sup> <sup>ε</sup>0TPeff

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator

DOI: http://dx.doi.org/10.5772/intechopen.86992

Peff σ<sup>2</sup>S<sup>2</sup>

Comparisons of dimensionless voltage and power between theoretical results and experimental results [26].

<sup>R</sup> <sup>¼</sup> <sup>1</sup> T2 ð1 0 V2

> T ð1 0 V2

<sup>σ</sup><sup>2</sup>d0<sup>S</sup> <sup>¼</sup> <sup>1</sup>

These equations provide the optimal strategy for TENG design with the two compound parameters related to the TENG device. We may use these equations to study the physical interpretation of dimensionless expressions through the

To verify the dimensionless expressions, we compare the theoretical results with the experimental measurements using the peak output voltage in dimensionless forms [17]. The corresponding parameters used in experiment and simulation are

As Figure 9 shows, compared with the experimental results, the theoretical curves based on scaling laws exhibit good consistent tendency with the experimen-

<sup>R</sup>ð Þτ dτ (26)

<sup>R</sup>ð Þτ dτ (27)

In Eqs. (21) and (22), there are two dimensionless compound parameters A=l and ε0RS=d0T affecting the dimensionless output characteristics of device, which could be described with a group of parameters related to various aspects of TENG device. From these equations, we may understand the effects of the load resistance R, planer area S, loading period T, and maximum sliding distance A on output voltage at the same time. Similarly, based on Eq. (23), we can figure out the optimized parameters for power improvement. Meanwhile, the dimensionless expressions also reflect the relationship between the compound parameters (A=l and ε0RS=d0T) and the output performance.

To make the normalized output performance better understood from the physical point of view, they can also be described with the dimensionless capacitance C and the dimensionless time constant T, in which the expression for C is C ¼ CA=C<sup>0</sup> ¼ 1 � A=l, with C<sup>0</sup> ¼ ε0S=d<sup>0</sup> the capacitance when x ¼ 0 and CA ¼ C xð Þ¼ ¼ A ε0Sð Þ 1 � A=l =d<sup>0</sup> the equivalent capacitance of the device when x ¼ A. Here C, from the physical point of view, represents the ratio of the capacitance with x ¼ A to the capacitance with x ¼ 0. The dimensionless time constant is defined as T ¼ RC0=T ¼ ε0RS=d0T, which reflects the time constant for the firstorder circuits with C ¼ C<sup>0</sup> to the period of the alternating current. The expression for normalized output voltage versus C and T is

$$\overline{W}\_{R}(\tau,\overline{\mathbf{C}},\overline{T}) = \frac{\varepsilon\_{0}V(\tau)}{\sigma d\_{0}} = \frac{1}{1 - (1 - \overline{\mathbf{C}})\overline{\mathbf{x}}(\tau)} \exp\left(-\frac{1}{\overline{T}} \int\_{0}^{\tau} \frac{1}{1 - (1 - \overline{\mathbf{C}})\overline{\mathbf{x}}(\tau')} d\tau'\right) $$

$$+ \frac{1}{\overline{T}} \frac{1}{1 - (1 - \overline{\mathbf{C}})\overline{\mathbf{x}}(\tau)} \int\_{0}^{\tau} \exp\left(\frac{1}{\overline{T}} \int\_{\tau}^{\tau'} \frac{1}{1 - (1 - \overline{\mathbf{C}})\overline{\mathbf{x}}(\delta)} d\delta\right) d\tau' - 1 \tag{23}$$

$$\overline{P}\_T(\overline{\mathcal{C}}, \overline{T}) = \frac{\varepsilon\_0^2 R P\_{\text{eff}}}{\sigma^2 d\_0^2} = \int\_0^1 \overline{V}\_R^2(\tau) d\tau \tag{24}$$

Similar to Eqs. (16)–(18) of contact-mode TENG, in this circumstance, the influence of d<sup>0</sup> on normalized output voltage is not reflected in Eq. (23), neither the effects of d<sup>0</sup> and R on normalized output power from Eq. (24). To find out the influence of these parameters, we propose the following set of expressions for normalized output voltage and power equations as

$$\frac{1}{T}\overline{V}\_d(\tau,\overline{C},\overline{T}) = \frac{T\mathcal{V}(\tau)}{\sigma RS} = \frac{1}{\overline{T}} \frac{1}{1 - (1 - \overline{C})\overline{\mathbf{x}}(\tau)} \exp\left(-\frac{1}{\overline{T}} \int\_0^\tau \frac{1}{1 - (1 - \overline{C})\overline{\mathbf{x}}(\tau')} d\tau'\right) \tag{25}$$

$$+ \frac{1}{\overline{T}^2} \frac{1}{1 - (1 - \overline{C})\overline{\mathbf{x}}(\tau)} \int\_0^\tau \exp\left(\frac{1}{\overline{T}} \int\_\tau^\ell \frac{1}{1 - (1 - \overline{C})\overline{\mathbf{x}}(\delta)} d\delta\right) d\tau' - \frac{1}{\overline{T}} \tag{26}$$

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

#### Figure 9.

Here, A is the maximum sliding distance. When used as energy harvesters, the output power will become the key target to evaluate the output characteristics of TENG. According to the definition of effective output power, the dimensionless

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

¼ ε2 <sup>0</sup>RPeff σ2d<sup>2</sup> 0 ¼ ð1 0 V2

In Eqs. (21) and (22), there are two dimensionless compound parameters A=l and ε0RS=d0T affecting the dimensionless output characteristics of device, which could be described with a group of parameters related to various aspects of TENG device. From these equations, we may understand the effects of the load resistance R, planer area S, loading period T, and maximum sliding distance A on output voltage at the same time. Similarly, based on Eq. (23), we can figure out the optimized parameters for power improvement. Meanwhile, the dimensionless expressions also reflect the relationship between the compound parameters (A=l

To make the normalized output performance better understood from the physical point of view, they can also be described with the dimensionless capacitance C

CA ¼ C xð Þ¼ ¼ A ε0Sð Þ 1 � A=l =d<sup>0</sup> the equivalent capacitance of the device when x ¼ A. Here C, from the physical point of view, represents the ratio of the capacitance with x ¼ A to the capacitance with x ¼ 0. The dimensionless time constant is defined as T ¼ RC0=T ¼ ε0RS=d0T, which reflects the time constant for the firstorder circuits with C ¼ C<sup>0</sup> to the period of the alternating current. The expression

<sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �xð Þ<sup>τ</sup> exp � <sup>1</sup>

<sup>0</sup>RPeff σ<sup>2</sup>d<sup>2</sup> 0 ¼ ð1 0 V2

Similar to Eqs. (16)–(18) of contact-mode TENG, in this circumstance, the influence of d<sup>0</sup> on normalized output voltage is not reflected in Eq. (23), neither the effects of d<sup>0</sup> and R on normalized output power from Eq. (24). To find out the influence of these parameters, we propose the following set of expressions for

<sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �xð Þ<sup>τ</sup> exp � <sup>1</sup>

ðτ 0 exp

1

1 <sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �xð Þ<sup>τ</sup>

ðτ 0 exp T ðτ 0

> T ðτ 0

1 T ðτ0 τ

1 T ðτ0 τ

1 <sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �<sup>x</sup> <sup>τ</sup><sup>0</sup> ð Þ

dδ

<sup>R</sup>ð Þτ dτ (24)

1 <sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �<sup>x</sup> <sup>τ</sup><sup>0</sup> ð Þ

dδ

!

1 <sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �xð Þ<sup>δ</sup>

!

!

1 <sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �xð Þ<sup>δ</sup>

!

dτ<sup>0</sup>

dτ<sup>0</sup> � 1

dτ<sup>0</sup>

<sup>d</sup>τ<sup>0</sup> � <sup>1</sup> T (25)

(23)

and the dimensionless time constant T, in which the expression for C is C ¼ CA=C<sup>0</sup> ¼ 1 � A=l, with C<sup>0</sup> ¼ ε0S=d<sup>0</sup> the capacitance when x ¼ 0 and

dτ (22)

output power can be expressed as

PT A l ; ε0RS d0T � �

and ε0RS=d0T) and the output performance.

for normalized output voltage versus C and T is

normalized output voltage and power equations as

<sup>σ</sup>RS <sup>¼</sup> <sup>1</sup> T

<sup>¼</sup> <sup>1</sup>

1 <sup>1</sup> � <sup>1</sup> � <sup>C</sup> � �xð Þ<sup>τ</sup>

PT <sup>C</sup>; <sup>T</sup> � � <sup>¼</sup> <sup>ε</sup><sup>2</sup>

σd<sup>0</sup>

þ 1 T

VR <sup>τ</sup>;C; <sup>T</sup> � � <sup>¼</sup> <sup>ε</sup>0Vð Þ<sup>τ</sup>

Vd <sup>τ</sup>;C; <sup>T</sup> � � <sup>¼</sup> TVð Þ<sup>τ</sup>

84

þ 1 T2 Comparisons of dimensionless voltage and power between theoretical results and experimental results [26].

$$\overline{P}\_d(\overline{\mathbf{C}}, \overline{T}) = \frac{T^2 P\_{\rm eff}}{\sigma^2 \mathbf{S}^2 R} = \frac{1}{\overline{T}^2} \int\_0^1 \overline{V}\_R^2(\mathbf{r}) d\mathbf{r} \tag{26}$$

$$\overline{P}\_{\rm R}(\overline{\mathbf{C}}, \overline{T}) = \frac{\varepsilon\_0 T P\_{\rm eff}}{\sigma^2 d\_0 \mathbf{S}} = \frac{1}{\overline{T}} \int\_0^1 \overline{V}\_{\rm R}^2(\tau) d\tau \tag{27}$$

These equations provide the optimal strategy for TENG design with the two compound parameters related to the TENG device. We may use these equations to study the physical interpretation of dimensionless expressions through the dimensionless capacitance and the dimensionless time constant.

To verify the dimensionless expressions, we compare the theoretical results with the experimental measurements using the peak output voltage in dimensionless forms [17]. The corresponding parameters used in experiment and simulation are listed in Table 2.

As Figure 9 shows, compared with the experimental results, the theoretical curves based on scaling laws exhibit good consistent tendency with the experimental data points.

Figure 10. Scaling laws for dimensionless output voltage and power in case I and case II [26].

According to the referring experiments [17], we define the two different forms of sliding process to simulate different mechanical loading conditions, including parabolic (Case I) and triangular (Case II) loading pattern. Based on the scaling laws for the correlation between the normalized output performance and the dimensionless compound parameters, some general optimization strategies will be acquired to improve the electric output with Eqs. (23)–(27).

under grant Nos. 11472244, 11621062, and 11772295 and the Fundamental Research

College of Civil Engineering and Architecture, Zhejiang University, Hangzhou,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: zjuzhanghe@zju.edu.cn

provided the original work is properly cited.

Funds for the Central Universities under grant No. 2019QNA4040.

DOI: http://dx.doi.org/10.5772/intechopen.86992

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator

Author details

China

87

He Zhang\* and Liwei Quan

From Figure 10, the optimal combination of A=l and ε0RS=d0T for the best output performance for TENG is found, thus providing a general guideline for device optimization. In addition, we can improve the dimensionless output performance by tuning compound parameters as well as single parameter. We can also use the scaling law from another physical point of view by reflecting the output performance with the dimensionless capacitance C and time constant T. The value of C and T with regard to the best output performance provides the optimization strategy for dimensionless capacitance and time constant.

From Figure 10, it is also obvious that increasing A=l may enhance both the peak output voltage and the maximum output power of TENG. It indicates that the larger the ratio of the maximum sliding distance to the length of the device, the higher the output performance is of the TENG. Additionally, it is found that the optimization strategies for TENG with these two different loading patterns are similar because the scaling laws between output performance and compound parameters in Cases I and II show the same trend.

In Figure 10(c–d), for each A=l, the value of ε0Vpeak=σd<sup>0</sup> increases monotonically with ε0RS=d0T and approaches to a constant when ε0RS=d0T ¼ 1:0, which implies that the optimal combination ε0RS=d0T is over 1. As mentioned before, the scaling laws shown in Figure 10(c–d) may not reflect the explicit relationships between output voltage and equivalent thickness d0, because d<sup>0</sup> is included in both ε0Vpeak=σd<sup>0</sup> and ε0RS=d0T. To achieve the influence of d0, another scaling laws are exhibited in Figure 10(e–f) with Eqs. (23) and (25). Instead of d<sup>0</sup> in Figure 10(c–d), load resistance R, area S, and period T in Figure 10(e–f) are involved in both TVpeak=σRS and ε0RS=d0T, which means the effect of R, S, and T on output voltage may not be reflected in Figure 10(d–f). For each A=l, a peak value of TVpeak=σRS is found, which indicates the existence of the optimal ε0RS=d0T or T.

In Figure 10(g–h) for the relationship between the dimensionless output power and compound parameters, the value of ε<sup>2</sup> 0RPeff=σ<sup>2</sup>d<sup>2</sup> <sup>0</sup> increases with ε0RS=d0T firstly and then reaches a constant. At the same time, the value of T<sup>2</sup> Peff=σ<sup>2</sup>S<sup>2</sup> R decreases with ε0RS=d0T after being a constant when ε0RS=d0T is smaller than 0.1. In Figure 10 (k–l), a peak value is observed for ε0TPeff=σ<sup>2</sup>d0S, which implies the optimal parameter combination for ε0RS=d0T or T. Similar to output voltage, output power may also be enhanced by tuning single parameter. For example, the remaining area S, period T, and equivalent thickness d<sup>0</sup> fixed, the best combination of ε0RS=d0Tand A=l can be obtained for optimizing the output power for TENG by tuning load resistance R [26].

According to Figures 8 and 10, we can conclude that the scaling laws can provide a more comprehensive and rational optimization strategy for both contact-separation mode and sliding-mode TENG based on multiparameter analysis. The results can help enhance the output performance of the device as either a smart sensor or an energy harvester and may render a guideline for designing TENG devices.

#### Acknowledgements

This work was supported by the National Key R&D Program of China under grant No. 2018YFB1600200 and National Natural Science Foundation of China

Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

under grant Nos. 11472244, 11621062, and 11772295 and the Fundamental Research Funds for the Central Universities under grant No. 2019QNA4040.

### Author details

According to the referring experiments [17], we define the two different forms of sliding process to simulate different mechanical loading conditions, including parabolic (Case I) and triangular (Case II) loading pattern. Based on the scaling laws for the correlation between the normalized output performance and the dimensionless compound parameters, some general optimization strategies will be

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

From Figure 10, the optimal combination of A=l and ε0RS=d0T for the best output performance for TENG is found, thus providing a general guideline for device optimization. In addition, we can improve the dimensionless output performance by tuning compound parameters as well as single parameter. We can also use the scaling law from another physical point of view by reflecting the output performance with the dimensionless capacitance C and time constant T. The value of C and T with regard to the best output performance provides the optimization

From Figure 10, it is also obvious that increasing A=l may enhance both the peak output voltage and the maximum output power of TENG. It indicates that the larger the ratio of the maximum sliding distance to the length of the device, the higher the output performance is of the TENG. Additionally, it is found that the optimization strategies for TENG with these two different loading patterns are similar because the scaling laws between output performance and compound parameters in Cases I

In Figure 10(c–d), for each A=l, the value of ε0Vpeak=σd<sup>0</sup> increases monotonically with ε0RS=d0T and approaches to a constant when ε0RS=d0T ¼ 1:0, which implies that the optimal combination ε0RS=d0T is over 1. As mentioned before, the scaling laws shown in Figure 10(c–d) may not reflect the explicit relationships between output voltage and equivalent thickness d0, because d<sup>0</sup> is included in both ε0Vpeak=σd<sup>0</sup> and ε0RS=d0T. To achieve the influence of d0, another scaling laws are exhibited in Figure 10(e–f) with Eqs. (23) and (25). Instead of d<sup>0</sup> in Figure 10(c–d), load resistance R, area S, and period T in Figure 10(e–f) are involved in both TVpeak=σRS and ε0RS=d0T, which means the effect of R, S, and T on output voltage may not be reflected in Figure 10(d–f). For each A=l, a peak value of TVpeak=σRS is

In Figure 10(g–h) for the relationship between the dimensionless output power

with ε0RS=d0T after being a constant when ε0RS=d0T is smaller than 0.1. In Figure 10 (k–l), a peak value is observed for ε0TPeff=σ<sup>2</sup>d0S, which implies the optimal parameter combination for ε0RS=d0T or T. Similar to output voltage, output power may also be enhanced by tuning single parameter. For example, the remaining area S, period T, and equivalent thickness d<sup>0</sup> fixed, the best combination of ε0RS=d0Tand A=l can be obtained for optimizing the output power for TENG by tuning load resistance R [26]. According to Figures 8 and 10, we can conclude that the scaling laws can provide a more comprehensive and rational optimization strategy for both contact-separation mode and sliding-mode TENG based on multiparameter analysis. The results can help enhance the output performance of the device as either a smart sensor or an energy

This work was supported by the National Key R&D Program of China under grant No. 2018YFB1600200 and National Natural Science Foundation of China

0RPeff=σ<sup>2</sup>d<sup>2</sup>

<sup>0</sup> increases with ε0RS=d0T firstly

R decreases

Peff=σ<sup>2</sup>S<sup>2</sup>

acquired to improve the electric output with Eqs. (23)–(27).

strategy for dimensionless capacitance and time constant.

found, which indicates the existence of the optimal ε0RS=d0T or T.

harvester and may render a guideline for designing TENG devices.

and then reaches a constant. At the same time, the value of T<sup>2</sup>

and compound parameters, the value of ε<sup>2</sup>

Acknowledgements

86

and II show the same trend.

He Zhang\* and Liwei Quan College of Civil Engineering and Architecture, Zhejiang University, Hangzhou, China

\*Address all correspondence to: zjuzhanghe@zju.edu.cn

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Theoretical Prediction and Optimization Approach to Triboelectric Nanogenerator DOI: http://dx.doi.org/10.5772/intechopen.86992

[17] Niu S, Liu Y, Wang S, Lin L, Zhou YS, Hu Y, et al. Theory of sliding-mode triboelectric nanogenerators. Advanced Materials. 2013;25(43):6184-6193

References

[1] Wang ZL. Triboelectric nanogenerators as new energy

[2] Wang ZL. On Maxwell's

1991;353:737-740

2007;221:255-264

1065-1074

88

557-577

technology and self-powered sensors— Principles, problems and perspectives. Faraday Discussions. 2014;176:447-458

Electrical Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

[9] Chen YS, Zhang H, Zhang YY, Li CH, Yang Q, Zheng HY, et al. Mechanical energy harvesting from road pavements under vehicular load using embedded piezoelectric elements. Journal of Applied Mechanics-Transactions of the

[10] Lu CF, Zhang YY, Zhang H, Zhang ZC, Shen MZ, Chen YS. Generalized optimization method for energy conversion and storage efficiency of nanoscale flexible piezoelectric energy harvesters. Energy Conversion and Management. 2018;182:34-40

[11] Zhang H, Ye GR, Zhang ZC. Acoustic radiation of a cylindrical piezoelectric power transformer. ASME Journal of Applied Mechanics. 2013;80

[12] Yang Y, Zhang H, Lin ZH, Zhou YS, Jing Q, Su Y, et al. Human skin based triboelectric nanogenerators for

harvesting biomechanical energy and as self-powered active tactile sensor system. ACS Nano. 2013;7(10):

[13] Ya Y, Hulin Z, Qingshen J, Yusheng Z, Xiaonan W, Zhonglin W. ACS Nano.

[15] Zhang H, Zhang JW, Hu ZW, Quan LW, Shi L, Chen JK, et al. Waistwearable wireless respiration sensor based on triboelectric effect. Nano

[16] Chen JK, Guo HW, Ding P, Pan RZ, Wang WB, Xuan WP, et al. Nano

[14] Changbao H, Chi Z, Xiaohui L, Limin Z, Tao Z, Weiguo H, et al. Selfpowered velocity and trajectory tracking sensor array made of planar triboelectric nanogenerator pixels. Nano

ASME. 2016;83(8)

(6) 061019(1-6)

9213-9222

2013;7(8):7342-7351

Energy. 2014;9:325-333

Energy. 2016;59:75-83

Energy. 2016;30:235-241

displacement current for energy and sensors: The origin of nanogenerators. Materials Today. 2017;20:74-82

[3] Oregan B, Gratzel M. A low-cost, high-efficiency solar cell based on dyesensitized colloidal TiO2 films. Nature.

[4] Brennan L, Renew PO. Biofuels from microalgae—A review of technologies for production, processing, and

extractions of biofuels and co-products. Sustainable Energy Reviews. 2010;14:

[5] Mago PJ, Blunden LS, Bahaj AS. Performance analysis of different working fluids for use in organic Rankine cycles. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy.

[6] Li Z, Zuo L, Luhrs G, Lin L, Qin Y. Electromagnetic energy-harvesting shock absorbers: Design, modeling, and road tests. IEEE Transactions on Vehicular Technology. 2013;62(3):

[7] Zhang H, Shen MZ, Zhang YY, Chen YS, Lu CF. Identification of static loading conditions using piezoelectric sensor arrays. ASME Journal of Applied Mechanics. 2018;85:011008-011005

[8] Chen YS, Zhang H, Zhang ZC, Lu CF. Theoretical assessment on

piezoelectric energy harvesting in smart self-powered asphalt pavements. Journal of Vibration Engineering & Technologies. 2018;6(1):1-10

[18] Niu SM, Wang SH, Lin L, Liu Y, Zhou SY, Hu YF, et al. Theoretical study of contact-mode triboelectric nanogenerators as an effective power source. Energy & Environmental Science. 2013;6:3576

[19] Simiao N, Sihong W, Ying L, Shengyu Z, Long L, Youfan H, et al. A theoretical study of grating structured triboelectric nanogenerators. Energy & Environmental Science. 2014;7:2339

[20] Simiao N, Ying L, Sihong W, Long L, Yusheng Z, Youfan H, et al. Theoretical investigation and structural optimization of single-electrode triboelectric nanogenerators. Advanced Functional Materials. 2014;24: 3332-3340

[21] Simiao N, Yin L, Xiangyu C, Sihong W, Yusheng Z, Long L, et al. Theory of freestanding triboelectric-layer-based nanogenerators. Nano Energy. 2015;12: 760-774

[22] Yunlong Z, Simiao N, Jie W, Zhen W, Wei T, Zhonglin W. Standards and figure-of-merits for quantifying the performance of triboelectric nanogenerators. Nature Communications. 2015;6:8376

[23] Jiajia S, Tao J, Wei T, Xiangyu C, Liang X, Zhonglin W. Structural figureof-merits of triboelectric nanogenerators at powering loads. Nano Energy. 2018;51:688-697

[24] Peng J, Kang SD, Snyder GJ. Optimization principles and the figure of merit for triboelectric generators. Science Advances. 2017;3:eaap 857615

[25] Zhang H, Quan LW, Chen JK, Xu CK, Zhang CH, Dong SR, et al. A general optimization approach for contactseparation triboelectric nanogenerator. Nano Energy. 2019;56:700-707

[26] Zhang H, Zhang CH, Zhang JW, Quan LW, Huang HY, Jiang JQ, et al. A theoretical approach for optimizing sliding-mode triboelectric nanogenerator based on multi-parameter analysis. Nano Energy. 2019. DOI: 10.1016/j. nanoen.2019.04.057

### *Edited by Steven H. Voldman*

As we enter the nanoelectronics era, electrostatic discharge (ESD) phenomena is an important issue for everything from micro-electronics to nanostructures. This book provides insight into the operation and design of micro-gaps and nanogenerators with chapters on low capacitance ESD design in advanced technologies, electrical breakdown in micro-gaps, nanogenerators from ESD, and theoretical prediction and optimization of triboelectric nanogenerators. The information contained herein will prove useful for for engineers and scientists that have an interest in ESD physics and design.

Published in London, UK © 2019 IntechOpen © noLimit46 / iStock

Electrostatic Discharge - From Electrical breakdown in Micro-gaps to Nano-generators

Electrostatic Discharge

From Electrical breakdown in Micro-gaps

to Nano-generators

*Edited by Steven H. Voldman*