Performance investigation of a dissipative dielectric elastomer generator by the damping model

Abstract

This paper proposes an analytical model to investigate the performance of a dissipative dielectric elastomer generator (DEG) by considering the damping effect and the leakage current. Based on the analytical model, we study the effect of the cycle period, the maximum stretch ratio, and the pre-stretch ratio on the performance parameters of DEGs: the energy density, the average power density, and the conversion efficiency. The simulation results show that increasing the cycle period contributes to improving the energy density and the conversion efficiency. Moreover, the energy density and the average power density increase with the increase in the maximum stretch ratio. In addition, as the pre-stretch ratio increases, the energy density and the average power density can decrease. These methods and conclusions can better guide the optimal design and assessment of DEGs.

1 Introduction

The dielectric elastomer generator (DEG) consists of a soft elastomeric membrane sandwiched between two compliant electrodes. A DEG is, in essence, a soft variable capacitor that converts stored mechanical energy into electrical energy by transferring charges from a low potential to a high potential. Dielectric elastomers (DEs) are attractive smart materials due to their intrinsic advantages, such as high-energy density, lightweight, flexibility, large deformation and low cost [1,2,3,4,5,6]. It has been reported that DEGs can be used to harvest electrical energy from diverse mechanical energy sources, including flowing water [7], ocean waves [7, 8], wind [9], and human motion, such as heel striking [6] and knee bending [10, 11]. In the past decade, many DEGs with different materials and harvesting circuits have been reported in the literature [4,5,6,7,8,9,10,11,12]. Pelrine et al. first proposed a DEG embedded in a shoe and then developed a DEG installed on a buoy [6, 7]. Jean-Mistral et al. presented a DEG located behind a person’s knees [10]. Kaltseis et al. developed a DEG with balloon geometry [8]. A DEG that uses a self-priming circuit as the inverse charge pump has also been reported [12].

The capacitance of a DEG increases as it is stretched in area and compressed in thickness. The achievable energy density depends on how much the capacitance can be changed during an energy harvesting cycle. This capacitance change depends strongly on the loading form, such as uniaxial, pure-shear, and equal-biaxial. The more that a DEG deforms, the greater the capacitance change is, and the higher energy density that it can achieve. Once the loading form is set, the maximum energy density converted by a DEG is limited by the failure modes of the materials, i.e., electrical breakdown (EB), electromechanical instability (EMI), loss of tension (LT), and rupture of the elastomer by excessive stretching [13,14,15,16,17]. To date, the highest energy density achieved experimentally is 0.78 J/g [18], which is obviously lower than the theoretical maximum energy density of 1.7 J/g [14]. The conversion efficiency of a DEG is mainly limited by viscous loss and leakage current [2, 19]. However, most of the nonlinear models used for DEGs ignore the material viscosity and the leakage current and consider only the hyperelasticity [13,14,15,16,17]. This leads to a significant difference between the experimental results and the theoretical prediction, which needs to be further investigated. The leakage current can cause energy dissipation, which can decrease the energy density of the DEG. When a DEG is subjected to excessive leakage current, it cannot even generate electric energy [19]. Lochmatter et al. proposed a visco-hyperelastic film model to predict the electromechanical performance of a DE actuator [20]. Foo et al. investigated the dissipative performance of DEGs by utilizing Hong’s [21] viscoelastic model [22]. Li et al. studied the energy conversion of a viscoelastic DEG under inhomogeneous deformation [23]. Zhou et al. investigated the performance of a viscoelastic DEG by considering the fatigue life [24]. Chen et al. studied the temperature effect on the performance of a dissipative DEG [25] and then analyzed the dynamic and quasi-static performances of a dissipative DE at different temperatures [26]. Deng et al. investigated the effect of pre-stretch and temperature on the actuation performance of a dissipative DE [27]. In addition, the dynamic characteristics of a DE actuator were analyzed by considering the damping effect [28, 29].The damping effect was also taken into account while studying the tunable output force induced by a DE actuator [30].

In this paper, we develop an analytical model to investigate the performance of a dissipative DEG by considering the damping effect and the leakage current. Utilizing the analytical model, we analyze the energy conversion of DEGs under the constant voltage scheme and the triangular scheme [18], and the simulation results show good agreement with the existing experimental results. To further improve the performance of DEGs, we study the effect of the cycle period, the maximum stretch ratio, and the pre-stretch ratio on the performance parameters of DEGs: the energy density, the power density, and the conversion efficiency. The methods and the results can better provide guidance for the optimal design and assessment of DEGs.

2 Building the analytical model

With reference to Fig. 1, a membrane of a DEG is of dimensions L₁, L₂ and L₃ in the undeformed state. When forces P₁ and P₂ are applied to the membrane and a voltage Ф is applied to the electrodes, each electrode gets charge Q, and the membrane deforms to a homogeneous state of dimensions l₁, l₂, and l₃. The stretch ratios of the membrane in the x, y and z directions are defined by λ₁ = l₁/L₁, λ₂ = l₂/L₂, and λ₃ = l₃/L₃, respectively. The membrane is considered incompressible, namely λ₁λ₂λ₃ = 1. The nominal stresses are defined by s₁ = P₁/(L₂L₃) and s₂ = P₂/(L₁L₃).

Fig. 1

a A DEG in the undeformed state. b Under forces P₁ and P₂, and a voltage Ф, the DEG deforms to a charged and stretched state, accompanied by a leakage current i_leak through the thickness direction

The induced electric field leads to a leakage current through the thickness direction, and thus, the DE is modeled as a parallel capacitor and a resistor, as shown in Fig. 2. The charge on the electrodes can be written as Q = CФ, where C is the capacitance of the DE. The capacitance can be expressed as $C={\varepsilon _0}{\varepsilon _{\text{r}}}{L_1}{L_2}L_{2}^{{ - 1}}\lambda _{1}^{2}\lambda _{2}^{2}$, where the vacuum permittivity ε₀ = 8.85 × 10⁻¹² F/m and ε_r is the dielectric constant of the elastomer. In addition, the dielectric constant is a function of the temperature T and the stretch ratios λ₁ and λ₂, i.e., ${\varepsilon _{\text{r}}}=\left( {{\varepsilon _\infty }+{A \mathord{\left/ {\vphantom {A T}} \right. \kern-0pt} T}} \right)\left( {1+b\left( {{\lambda _1}+{\lambda _2} - 2} \right)+e{{\left( {{\lambda _1}+{\lambda _2} - 2} \right)}^2}+c{{\left( {{\lambda _1}+{\lambda _2} - 2} \right)}^3}} \right)$, where ε_∞ = 2.1 is the limiting value of the dielectric constant, and the coefficients for polyacrylate very high bond (VHB) elastomer are A = 960K, b = − 0.1658, e = 0.04086, and c = − 0.003027 [31]. Furthermore, the DEG works at room temperature T = 293 K.

Fig. 2

The DE is modeled as a parallel capacitor C and a resistor σ_c(E), and the leakage current i_leak flows through the resistor σ_c(E). The red part denotes the charging circuit, and the turquoise part denotes the discharging circuit. During the charging process, the switch K is closed to position 1, and during the discharging process, the switch K is closed to position 2. According to Kirchhoff’s current law (KCL), the current in the conducting wire i is the sum of the rate of the charge on the DE dQ/dt and the leakage current i_leak

As shown in Fig. 2, during the charging process, the charge flowing out from the battery partly accumulates on the electrodes and partly leaks through the membrane. During the discharging process, the charge flowing out from the electrodes partly is harvested and partly leaks through the membrane. The electrodes and the wires are regarded as perfect conductors. Therefore, the current in the wire connected to the DE is expressed as follows:

$$i=\frac{{{\text{d}}Q}}{{{\text{d}}t}}+{i_{{\text{leak}}}}=\Phi \frac{{{\text{d}}C}}{{{\text{d}}t}}+C\frac{{{\text{d}}\Phi }}{{{\text{d}}t}}+{i_{{\text{leak}}}},$$

(1)

where i_leak is the leakage current through the thickness direction. The leakage current density is defined as j_leak = i_leak/(L₁L₂λ₁λ₂). The relation between the leakage current density and the electric field is j_leak = σ_c(E)E, where σ_c(E) is the conductivity and is only dependent on the electric field E and E is defined as E = Фλ₁λ₂/L₃. The leakage current of a DE is approximately ohmic in a relatively small electric field. However, as the electric field increases, the leakage current becomes non-ohmic and increases exponentially [2, 19]. Thus, the conductivity σ_c(E) is expressed as σ_c(E) = σ_c0exp(E/E_B), where σ_c0 is the conductivity in a low electric field that equals 3.23 × 10⁻¹⁴ S/m and E_B is an empirical constant that equals 40 MV/m [2]. The leakage current can be written as follows:

$${i_{{\text{leak}}}}={\sigma _{{\text{c}}0}}\frac{{\Phi {L_1}{L_2}\lambda _{1}^{2}\lambda _{2}^{2}}}{{{L_3}}}\exp \left( {\frac{E}{{{E_{\text{B}}}}}} \right).$$

(2)

When the stretch ratios λ₁ and λ₂ vary by δλ₁ and δλ₂, the tensile forces do the work of P₁L₁δλ₁ and P₂L₂δλ₂. When the charge varies by δQ, the applied voltage does the work of ФδQ. When the forces and the voltage are applied to the membrane, the variation of the charge is

$$\begin{aligned} \delta Q & =\frac{{{\varepsilon _0}{\varepsilon _{\text{r}}}{L_1}{L_2}}}{{{L_3}}}\lambda _{1}^{2}\lambda _{2}^{2}\delta \Phi +\Phi \frac{{{\varepsilon _0}{\varepsilon _{\text{r}}}{L_1}{L_2}}}{{{L_3}}}\left( {2{\lambda _1}\lambda _{2}^{2}\delta {\lambda _1}+2{\lambda _2}\lambda _{1}^{2}\delta {\lambda _2}} \right) \\ & \quad +\Phi \frac{{{\varepsilon _0}{L_1}{L_2}}}{{{L_3}}}\lambda _{1}^{2}\lambda _{2}^{2}\left( {\frac{{\partial {\varepsilon _{\text{r}}}}}{{\partial {\lambda _1}}}\delta {\lambda _1}+\frac{{\partial {\varepsilon _{\text{r}}}}}{{\partial {\lambda _2}}}\delta {\lambda _2}} \right). \\ \end{aligned}$$

(3)

The damping forces along the x-direction and y-direction are expressed as dL₁L₂L₃x(dλ₁/dt) and dL₁L₂L₃y(dλ₂/dt), respectively, where d is the damping constant per unit volume of the DE [28, 29]. When the stretch ratios, λ₁ and λ₂ vary by δλ₁ and δλ₂, the damping forces work along the x-direction and y-direction

$$\begin{gathered} 2{\text{d}}{L_1}{L_2}{L_3}\frac{{{\text{d}}{\lambda _1}}}{{{\text{d}}t}}\delta {\lambda _1}\int_{0}^{{{{{L_1}} \mathord{\left/ {\vphantom {{{L_1}} 2}} \right. \kern-0pt} 2}}} {x{\text{d}}x\int_{{ - {{{L_2}} \mathord{\left/ {\vphantom {{{L_2}} 2}} \right. \kern-0pt} 2}}}^{{{{{L_2}} \mathord{\left/ {\vphantom {{{L_2}} 2}} \right. \kern-0pt} 2}}} {\frac{{{\text{d}}y}}{{{L_2}}}=} \frac{{{\text{d}}L_{1}^{3}{L_2}{L_3}}}{4}\frac{{{\text{d}}{\lambda _1}}}{{{\text{d}}t}}\delta {\lambda _1}} \hfill \\ 2{\text{d}}{L_1}{L_2}{L_3}\frac{{{\text{d}}{\lambda _2}}}{{{\text{d}}t}}\delta {\lambda _2}\int_{0}^{{{{{L_2}} \mathord{\left/ {\vphantom {{{L_2}} 2}} \right. \kern-0pt} 2}}} {y{\text{d}}y\int_{{ - {{{L_1}} \mathord{\left/ {\vphantom {{{L_1}} 2}} \right. \kern-0pt} 2}}}^{{{{{L_1}} \mathord{\left/ {\vphantom {{{L_1}} 2}} \right. \kern-0pt} 2}}} {\frac{{{\text{d}}x}}{{{L_1}}}=\frac{{{\text{d}}L_{1}^{3}{L_2}{L_3}}}{4}\frac{{{\text{d}}{\lambda _2}}}{{{\text{d}}t}}\delta {\lambda _2}} } . \hfill \\ \end{gathered}$$

(4)

For an arbitrary variation in the system, the variation in the free energy of the membrane is equal to the work done by the voltage, the tensile forces and the damping forces, which is shown as

$${L_1}{L_2}{L_3}\delta W=\Phi \delta Q+{P_1}{L_1}\delta {\lambda _1}+{P_2}{L_2}\delta {\lambda _2} - \frac{{{\text{d}}L_{1}^{3}{L_2}{L_3}}}{4}\frac{{{\text{d}}{\lambda _1}}}{{{\text{d}}t}}\delta {\lambda _1} - \frac{{{\text{d}}L_{2}^{3}{L_1}{L_3}}}{4}\frac{{{\text{d}}{\lambda _2}}}{{{\text{d}}t}}\delta {\lambda _2}.$$

(5)

Using the Gent model [32], the free energy function of the DE is expressed as

$$W= - \frac{{\mu {J_{\text{m}}}}}{2}\ln \left( {1 - \frac{{\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{1}^{{ - 2}}\lambda _{2}^{{ - 2}} - 3}}{{{J_{\text{m}}}}}} \right)+\frac{{{\varepsilon _0}{\varepsilon _{\text{r}}}}}{2}{\left( {\frac{\Phi }{{{L_3}}}} \right)^2}\lambda _{1}^{2}\lambda _{2}^{2},$$

(6)

where µ is the shear modulus of the DE and J_m is a dimensionless parameter referring to the dependence of the DE on chain-extension limit.

Thermodynamics dictates that a stable equilibrium state should minimize the free energy of the system [33]. Based on Eqs. (3), (5) and (6), the constitutive relation of the DEG system is as follows:

$$\begin{aligned} {s_1} & =\mu \frac{{{\lambda _1} - \lambda _{1}^{{ - 3}}\lambda _{2}^{{ - 2}}}}{{1 - {{\left( {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{1}^{{ - 2}}\lambda _{2}^{{ - 2}} - 3} \right)} \mathord{\left/ {\vphantom {{\left( {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{1}^{{ - 2}}\lambda _{2}^{{ - 2}} - 3} \right)} {{J_{\text{m}}}}}} \right. \kern-0pt} {{J_{\text{m}}}}}}} - {\varepsilon _0}{\varepsilon _{\text{r}}}{\left( {\frac{\Phi }{{{L_3}}}} \right)^2}{\lambda _1}\lambda _{2}^{2} \\ & \quad - \frac{{{\varepsilon _0}}}{2}{\left( {\frac{\Phi }{{{L_3}}}} \right)^2}\lambda _{1}^{2}\lambda _{2}^{2}\frac{{\partial {\varepsilon _{\text{r}}}}}{{\partial {\lambda _1}}}+\frac{{{\text{d}}L_{1}^{2}}}{4}\frac{{{\text{d}}{\lambda _1}}}{{{\text{d}}t}}, \\ \end{aligned}$$

(7-a)

$$\begin{aligned} {s_2} & =\mu \frac{{{\lambda _2} - \lambda _{2}^{{ - 3}}\lambda _{1}^{{ - 2}}}}{{1 - {{\left( {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{1}^{{ - 2}}\lambda _{2}^{{ - 2}} - 3} \right)} \mathord{\left/ {\vphantom {{\left( {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{1}^{{ - 2}}\lambda _{2}^{{ - 2}} - 3} \right)} {{J_{\text{m}}}}}} \right. \kern-0pt} {{J_{\text{m}}}}}}} - {\varepsilon _0}{\varepsilon _{\text{r}}}{\left( {\frac{\Phi }{{{L_3}}}} \right)^2}{\lambda _2}\lambda _{1}^{2} \\ & \quad - \frac{{{\varepsilon _0}}}{2}{\left( {\frac{\Phi }{{{L_3}}}} \right)^2}\lambda _{1}^{2}\lambda _{2}^{2}\frac{{\partial {\varepsilon _{\text{r}}}}}{{\partial {\lambda _2}}}+\frac{{{\text{d}}L_{2}^{2}}}{4}\frac{{{\text{d}}{\lambda _2}}}{{{\text{d}}t}}. \\ \end{aligned}$$

(7-b)

In the special case when equal biaxial stresses are applied to the membrane, namely, s₁ = s₂ = s, the stretch ratios are equal, namely λ₁ = λ₂ = λ. In addition, we assume that the lengths of the membrane in the x-direction and y-direction are equal, namely L₁ = L₂ = L. Therefore, Eqs. (7-a, 7-b) becomes

$$\begin{aligned} s & =\mu \frac{{\lambda - {\lambda ^{ - 5}}}}{{1 - {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} \mathord{\left/ {\vphantom {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} {{J_{\text{m}}}}}} \right. \kern-0pt} {{J_{\text{m}}}}}}} - {\varepsilon _0}{\varepsilon _{\text{r}}}{\left( {\frac{\Phi }{{{L_3}}}} \right)^2}{\lambda ^3} \\ & \quad - \frac{{{\varepsilon _0}}}{2}{\left( {\frac{\Phi }{{{L_3}}}} \right)^2}{\lambda ^4}\frac{{\partial {\varepsilon _{\text{r}}}}}{{\partial \lambda }}+\frac{{{\text{d}}{L^2}}}{4}\frac{{{\text{d}}\lambda }}{{{\text{d}}t}}, \\ \end{aligned}$$

(8)

where the dielectric constant ${\varepsilon _{\text{r}}}=\left( {{\varepsilon _\infty }+{A \mathord{\left/ {\vphantom {A T}} \right. \kern-0pt} T}} \right)\left( {1+2b\left( {\lambda - 1} \right)+4e{{\left( {\lambda - 1} \right)}^2}+8c{{\left( {\lambda - 1} \right)}^3}} \right)$.

When the tensile stress disappears, namely, s = 0, the condition is known as LT. Combining the condition s = 0 with Eq. (8), the parametric equation of LT in the voltage-charge work-conjugate plane is obtained as

$$\left\{ \begin{aligned} \frac{\Phi }{{{L_3}}} & =\sqrt {\frac{{\mu \frac{{\lambda - {\lambda ^{ - 5}}}}{{1 - {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} \mathord{\left/ {\vphantom {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} {{J_{\text{m}}}}}} \right. \kern-0pt} {{J_{\text{m}}}}}}}+\frac{{{\text{d}}{L^2}}}{4}\frac{{{\text{d}}\lambda }}{{{\text{d}}t}}}}{{{\varepsilon _0}{\varepsilon _{\text{r}}}{\lambda ^3}+\frac{{{\varepsilon _0}}}{2}{\lambda ^4}\frac{{\partial {\varepsilon _{\text{r}}}}}{{\partial \lambda }}}}} \\ \frac{Q}{{{L^2}}} & ={\varepsilon _0}{\varepsilon _{\text{r}}}\frac{\Phi }{{{L_3}}}{\lambda ^4} \\ \end{aligned} \right..$$

(9)

According to Eq. (8), Ф can be expressed as a function of λ and s [34]. Then, by making the derivative of Ф(λ, s) with respect to λ be zero, the expression for EMI can be obtained as follows:

$$\left\{ \begin{aligned} \frac{\Phi }{{{L_3}}} & =\sqrt {\mu \frac{{\left( {1 - 5{\lambda ^{ - 6}}} \right)\left( {1 - {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} \mathord{\left/ {\vphantom {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} {{J_{\text{m}}}}}} \right. \kern-0pt} {{J_{\text{m}}}}}} \right)+4{{{{\left( {\lambda - {\lambda ^{ - 5}}} \right)}^2}} \mathord{\left/ {\vphantom {{{{\left( {\lambda - {\lambda ^{ - 5}}} \right)}^2}} {{J_{\text{m}}}}}} \right. \kern-0pt} {{J_{\text{m}}}}}}}{{\left( {3{\varepsilon _0}{\varepsilon _{\text{r}}}{\lambda ^2}+3{\varepsilon _0}{\lambda ^3}\frac{{\partial {\varepsilon _{\text{r}}}}}{{\partial \lambda }}+\frac{{{\varepsilon _0}}}{2}{\lambda ^4}\frac{{{\partial ^2}{\varepsilon _{\text{r}}}}}{{\partial {\lambda ^2}}}} \right){{\left( {1 - {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} \mathord{\left/ {\vphantom {{\left( {2{\lambda ^2}+{\lambda ^{ - 4}} - 3} \right)} {{J_{\text{m}}}}}} \right. \kern-0pt} {{J_{\text{m}}}}}} \right)}^2}}}} \\ \frac{Q}{{{L^2}}} & ={\varepsilon _0}{\varepsilon _{\text{r}}}\frac{\Phi }{{{L_3}}}{\lambda ^4} \\ \end{aligned} \right..$$

(10)

In addition, the voltage of EB can be expressed as Ф = E_EBL₃λ⁻², where E_EB is the breakdown field strength and E_EB = 3 × 10⁸ V/m [1]. To avoid damage to the DEG, the applied voltage across the electrodes in the following simulations is lower than the critical voltage of EB and EMI, and thus, the curves of EB and EMI are not depicted in the subsequent figures.

3 Results and discussion

Utilizing Eq. (7), the damping model is fitted to the uniaxial and equal biaxial tensile tests applied to VHB at different stretching rates [35, 36] as shown in Fig. 3, which exhibits good agreement between the model and the experiment. The red line is the fitting result of the uniaxial tensile test and the blue line is the fitting result of the equal biaxial tensile test. The common parameters used to fit the damping model are µ = 68 kPa and d = 8.46 × 10⁷ kg/(sm³). The dimensionless parameter J_m = 100 [17].

Fig. 3

The damping model is fitted to the experimental data [35, 36]

In our analysis, the dielectric relaxation is ignored because the time scale of operation of the DE is much larger than the dielectric relaxation time [2]. Thus, in a dissipative DEG, the energy is dissipated by the damping effect and the leakage current. Under an equal biaxial loading form, according to Eq. (4), the dissipated energy due to the damping effect W_damp in a cycle is as follows:

$${W_{{\text{damp}}}}=\frac{{{\text{d}}{L^2}}}{2}\int {\dot {\lambda }{\text{d}}\lambda } ,$$

(11)

where the deformation rate $\dot {\lambda }={{{\text{d}}\lambda } \mathord{\left/ {\vphantom {{{\text{d}}\lambda } {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}}$.

The dissipated energy due to the leakage current is given as follows:

$${W_{{\text{ele}}}}=\frac{1}{{{L^2}{L_3}}}\int {\Phi {i_{{\text{leak}}}}} {\text{d}}t.$$

(12)

The electrical energy generated in a cycle is expressed as follows:

$${E_{{\text{density}}}}= - \frac{1}{{{L^2}{L_3}}}\int {\Phi i} {\text{d}}t.$$

(13)

The input mechanical energy in a cycle is:

$${W_{{\text{mech}}}}=2\int s {\text{d}}\lambda ,$$

(14)

and the electromechanical conversion efficiency of a DEG is defined as η = E_density/W_mech.

Based on the damping model, we analyze the energy conversion of DEGs under the triangular scheme [18]. In this simulation, the geometrical parameters selected are the length L = 35.4 mm and the thickness L₃ = 0.5 mm. As shown in Fig. 4a, state F represents that the membrane is stretched from pre-stretch ratio λ_pre = 2 to maximum stretch ratio λ_max = 5.5 at a deformation rate of dλ/dt = 2.9 s⁻¹, and at state F, switches S₁ and S₂ are open. Then, switch S₁ is closed and the electrodes start to be charged at a constant current. When the voltage between the electrodes increases to 3000 V at state G, switch S₁ is opened to end the charging process. Then, the membrane starts to be relaxed at a deformation rate of dλ/dt = − 7 s⁻¹. As the stretch ratio decreases, the charges on the electrodes can flow to the transfer capacitor. For an ideal DEG, the membrane can be relaxed to the pre-stretch ratio 2 (state E) along the curve of C_p. However, due to the damping effect, LT occurs when the membrane is relaxed to state H along the curve GH. Then, switch S₂ is closed, and the charges on the electrodes and the transfer capacitor start to flow to the harvesting circuit. The membrane can continue to be relaxed from state H to F along the LT curve. The stretch ratio of state H is approximately 4.1, which is slightly greater than the value of 4 reported in the literature [18]. The blue solid lines FDABC are the experimental data [18]. To make the experimental data a closed path, a line is placed between states C and D. The simulation results show that the DEG can achieve an energy density of 0.838 J/g, which is higher than the value of 0.78 J/g reported in the literature [18]. The main reason for this is that there is residual charge on the electrodes at state C in the experimental results [18]. The energy dissipated due to the leakage current is 0.029 J/g, and the energy dissipated by the damping is 1.837 J/g. Therefore, the conversion efficiency of the DEG is 31%, which is higher than the value of 30% reported by the literature [18]. The main reason is that the energy density obtained by the damping model is overestimated. In addition, the energy dissipation caused by friction in this particular experimental setup is not considered in our simulation.

Fig. 4

The triangular scheme of a DEG. a Comparing the simulation results with the experimental data of Shian et al. [18] for the triangular scheme. b The circuit diagram used to control the triangular scheme. The curve FDABC is the experimental data [18], and the closed curve FGHF represents the operation paths of a dissipative DEG. The operation paths of an ideal DEG are the closed curve FGEF

In addition, the damping model is also employed to investigate the energy conversion of DEGs for the constant voltage scheme [36]. Figure 5 shows the operation paths of harvesting energy for the constant voltage scheme in a steady state. The closed curve EFCGE is the operation paths of an ideal DEG. At state C, the membrane starts to be relaxed at a deformation rate of dλ/dt = − 4.2 s⁻¹. As the stretch ratio of the membrane decreases, the voltage between the electrodes can increase until it reaches 5000 V at state G. In interval CG, the charge on the electrodes remains constant. Then, the DEG can be discharged until the stretch ratio decreases to the pre-stretch ratio λ_pre = 1.2 at state E. In interval GE, the voltage remains constant. After state E is reached, the membrane starts to be stretched. As the stretch ratio increases, the voltage can decrease until it reaches 2000 V at state F. In interval EF, the charge also remains constant. Then, the DEG can be charged until the stretch ratio increases to the maximum stretch ratio λ_max = 5.4 at state C. In interval FC, the voltage remains constant. However, the damping and the leakage current can affect the operation paths, and the paths of the dissipative DEG are represented by the closed curve ABCDA. In interval CD, the membrane can be relaxed, and the voltage can increase. Due to the leakage current, the charge on the electrodes can very slowly decrease, and thus, the charge at state D is slightly lower than that at state C. After state D is reached, the membrane can be further relaxed. It is found that due to the damping effect, when the membrane is relaxed to state A, LT occurs, and the relaxing process ends. At state A, the stretch ratio is equal to 2.07, which is slightly greater than the value of 2 reported in the literature [36]. In interval AB, the membrane can be stretched, and the voltage can decrease. After state B is reached, the DEG can be charged until it reaches state C. The simulation results show that the dissipative DEG can achieve an energy density of 0.586 J/g, which is higher than the average energy density of 0.56 J/g for the first eight cycles [36]. The main reason should be that the electrical energy dissipated in the energy harvesting circuit is considered in our simulation. The dissipated energy due to the damping effect is 1.487 J/g. The dissipated energy due to the leakage current is 0.019 J/kg. Therefore, the conversion efficiency of the DEG is 28%, which is close to the average conversion efficiency of 27% [36].

Fig. 5

The operation paths of harvesting energy for the constant voltage scheme. The closed curve ABCDA is the operation paths of a dissipative DEG, and the closed curve EFCGE is the operation paths of an ideal DEG

To further illustrate the energy conversion of a dissipative DEG, based on the constant voltage scheme, we study the effect of the cycle period T, the maximum stretch ratio λ_max, and the pre-stretch ratio λ_pre on the performance parameters of the DEGs: the energy density E_density, the average power density P_density, the conversion efficiency η, the dissipated energy due to damping effect W_damp, and the dissipated energy due to leakage current W_ele. In addition, due to the damping effect, LT may occur before the stretch ratio of the membrane decreases to the pre-stretch ratio, and thus, the membrane may be not relaxed to the initial position after the relaxing process ends. The stretch ratio at the end of the relaxing process is defined as the minimum stretch ratio λ_min, which is affected by the cycle period, the pre-stretch ratio, and the maximum stretch ratio. The parameters Ф_L and Ф_H are the input and output voltage, respectively, and the voltage difference ΔU = Ф_H − Ф_L. The membrane is stretched at a deformation rate of dλ/dt = 2(λ_max − λ_pre)/T and relaxed at a deformation rate of dλ/dt = 2(λ_pre − λ_max)/T.

We first analyze the effect of the cycle period T on the performance parameters of the DEG, as shown in Fig. 6. In this set of numerical calculations, we set λ_max = 5.4, λ_pre = 1.2, and ΔU = 1500 V. As shown in Fig. 6e, as the cycle period decreases, the deformation rate can increase. The smaller the cycle period is, the greater the variation in the deformation rate. Hence, as the cycle period increases, the minimum stretch ratio λ_min decreases rapidly and then gradually becomes close to the pre-stretch ratio. The larger the minimum stretch ratio is, the greater the residual charge on the electrodes at the end of the relaxing process, and lower the energy density (Fig. 6c). When the minimum stretch ratio is close to the pre-stretch ratio, the energy density tends to be stable if the leakage current is ignored. For the longer cycle period, the leakage current can dissipate more energy, which even results in a decrease in the energy density. In a cycle, the stretch ratio of the membrane can range from λ_min to λ_max at a constant deformation rate $\dot {\lambda }$, and Eq. (11) can become ${W_{{\text{damp}}}}={\text{d}}{L^2}\dot {\lambda }\left( {{\lambda _{{\text{max}}}} - {\lambda _{\hbox{min} }}} \right)$. Therefore, the dissipated energy due to the damping effect is related to the deformation rate $\dot {\lambda }$ and the change in the stretch radio (λ_max − λ_min). With the increase in the cycle period, the deformation rate decreases, but the change in the stretch ratio (λ_max − λ_min) can increase. Therefore, as the cycle period increases, the dissipated energy due to the damping effect can first increase and then decrease, as shown in Fig. 6b. The dissipated energy due to the leakage current first decreases and then increases as the cycle period increases (Fig. 6a). For the shorter cycle period, there is the more charge remaining on the electrodes at the end of the relaxing process, which causes a higher average leakage current in a cycle. Therefore, more energy can be dissipated by the leakage current during the shorter cycle period. The maximum value of the average power density appears at approximately T = 1.25 s (see Fig. 6d). The conversion efficiency increases with the increase in the cycle period, as shown in Fig. 6f. In addition, as the output voltage increases, the energy density, the power density, and the conversion efficiency all increase. The reason for this is mainly that the higher the input voltage is (the constant voltage difference ΔU), the greater the charge on the electrodes at the end of the charging process (Fig. 6c–f). However, for the shorter cycle period, a higher output voltage might result in a lower energy density due to the greater residual charge at the end of the discharging process.

Fig. 6

The effect of the cycle period T on the performance of a DEG. a The dissipated energy due to the leakage current W_ele, b the dissipated energy due to the damping effect W_damp, c the energy density E_density, d the average power density P_density, e the minimum stretch ratio λ_min, and f the conversion efficiency η. The pre-stretch ratio λ_pre, maximum stretch ratio λ_max, and voltage difference ΔU are fixed

Then, the effect of the maximum stretch ratio λ_max on the performance parameters of DEGs is investigated (see Fig. 7). In this group of analyses, we set T = 2 s, λ_pre = 1.2, and Ф_L = 2000 V. For the smaller maximum stretch ratio (λ_max ≤ 3), λ_min = λ_pre, and the membrane can be relaxed to the pre-stretch ratio. In addition, as the maximum stretch ratio increases (λ_max > 3), the minimum stretch ratio also increases, as shown in Fig. 7e. The dissipated energy due to the damping effect increases significantly with the increasing maximum stretch ratio (Fig. 7b). The reason for this is that both the deformation rate and the change in the stretch ratio can increase with the increase in the maximum stretch ratio. For a larger maximum stretch ratio, the dissipated energy due to the leakage current can increase rapidly as the maximum stretch ratio increases and is enhanced by the increasing input voltage (see Fig. 7a). As the maximum stretch ratio and the input voltage increase, the energy density and the average power density of DEGs increase due to the greater charge on the electrodes at the end of the charging process (see Fig. 7c, d). The maximum value of conversion efficiency is at approximately λ_max = 5, and increasing the input voltage contributes to improving the conversion efficiency (Fig. 7f).

Fig. 7

The effect of the maximum stretch ratio λ_max on the performance of a DEG. a The dissipated energy due to the leakage current W_ele, b the dissipated energy due to the damping effect W_damp, c the energy density E_density, d the average power density P_density, e the minimum stretch ratio λ_min, and f the conversion efficiency η. The pre-stretch ratio λ_pre, cycle period T, and input voltage Ф_L are fixed

In addition, we study the effect of the pre-stretch ratio on the performance parameters of the DEGs, as shown in Fig. 8. In this set of calculations, T = 2 s, λ_max = 5.4, and Ф_L = 2000 V are fixed. When the pre-stretch ratio λ_pre ≥ 2, λ_min = λ_pre, and the membrane can be relaxed to the initial pre-stretch ratio (Fig. 8e). As the pre-stretch ratio increases, both the deformation rate and the change in the stretch ratio decrease and hence the damping can cause less energy dissipation (see Fig. 8b). As shown in Fig. 8a, with the increase in the pre-stretch ratio and the output voltage, the dissipated energy due to the leakage current can increase because of the enhanced leakage current. As the pre-stretch ratio increases, both the energy density and the average power density decrease. For the smaller pre-stretch ratio, increasing the output voltage can contribute to enhancing the energy density and the average power density. However, as the pre-stretch ratio increases, the enhancement effect can decline, and even increasing the input voltage may lead to lower energy density and average power density (Fig. 8c, d). This is because the dissipated energy due to the leakage current increases rapidly for the larger pre-stretch ratio. The conversion efficiency first increases and then decreases with the increasing pre-stretch ratio, as shown in Fig. 8f. Increasing the output voltage can generally improve the efficiency. However, for the larger pre-stretch ratio, an excessively high output voltage might reduce the conversion efficiency. In addition, as the output voltage rises, the optimal pre-stretch ratio can decrease (see Fig. 8f).

Fig. 8

The effect of the pre-stretch ratio λ_pre on the performance of a DEG. a The dissipated energy due to the leakage current W_ele, b the dissipated energy due to the damping effect W_damp, c the energy density E_density, d the average power density P_density, e the minimum stretch ratio λ_min, and f the conversion efficiency η. The cycle period T, maximum stretch ratio λ_max, and input voltage Ф_L are set

The simulation results indicate that performance parameters such as energy density, average power density, and conversion efficiency cannot reach their maximum values simultaneously; that is, when one performance parameter reaches the maximum value, the others may be at smaller values. By employing the simulation results, we can design and optimize a DEG based on design requirements and environmental constraints. For example, in the literature [36], the operation parameters are selected as λ_max = 5.4, λ_pre = 1.2, and T = 2 s, which makes the DEG generate the highest energy density. However, the power density and the conversion efficiency are lower. It is assumed that the tunable parameter is the cycle period. If the highest power density is desired, then the DEG should operate at T = 1.25 s. In addition, when the cycle period increases to T = 4 s, the highest conversion efficiency is obtained. If the tunable parameter is the maximum stretch ratio, then the DEG operating at λ_max = 5 provides the highest conversion efficiency. Moreover, the highest power density is achieved at λ_max = 5.4. In addition, if the initial pre-stretch is used as the tunable parameter, the DEG should operate at λ_pre = 1.2 to achieve the highest power density. Furthermore, when the DEG operates at λ_pre = 3.5, the highest conversion efficiency can be obtained.

4 Conclusion

In this paper, we develop an analytical model to study the performance of a dissipative DEG by considering the damping effect and the leakage current. Based on the analytical model, the performance of DEGs under different energy harvesting schemes is investigated, and the simulation results show good agreement with the existing experimental results. In addition, based on the constant voltage scheme, we analyze the effect of the cycle period, the maximum stretch ratio, and the pre-stretch ratio on the performance parameters of the DEGs: the energy density, the average power density, and the conversion efficiency.

It can be concluded that the energy density increases as the cycle period or the maximal stretch ratio increases, but an excessively long cycle period might result in a decrease in the energy density. In addition, the energy density can decrease with the increasing pre-stretch ratio. There is a significant positive correlation between the conversion efficiency and the cycle period. Additionally, the conversion efficiency can first increase and then decrease as the pre-stretch ratio or the maximum stretch ratio increases. The average power density first increases and then decreases as the cycle period increases. Moreover, decreasing the pre-stretch ratio or increasing the maximum stretch ratio contributes to enhancing the average power density. In addition, a higher output voltage can generally improve the energy density, the average power density, and the conversion efficiency. However, for a larger pre-stretch ratio or shorter cycle period, increasing the output voltage might reduce the energy density and the conversion efficiency. The conclusions and the methods can better guide the optimal design and assessment of DEGs.