Introduction
The eco-friendly solution for supplying devices in wireless sensor networks is represented by energy harvesters that are able to locally convert into electricity otherwise wasted forms of energy available in the surrounding environment [1], [2]. Even if many sources of energy can be exploited, like sun, wind, vibrations, electromagnetic fields, and so on, the most ubiquitous one is water. For this reason, many research efforts have been recently devoted to scavenging energy from raindrops [3]. The traditional method consists in collecting raindrops into reservoirs and in generating electricity through microturbines and electromagnetic generators. However, turbines are inefficient with low water supply and generators are bulky and heavy. Energy harvesters exploiting the kinetic energy of free-falling droplets impacting on piezoelectric cantilevers were investigated in [4] and [5], but very low energy levels, around 1 nJ per droplet [4], can be extracted mainly due to the resonant nature of piezoelectric cantilevers.
To overcome the above drawbacks, several water energy-harvesting techniques that rely on the generation and transfer of interfacial charges have been developed in the last years [6], [7]. Reverse electrowetting energy harvesting, exploiting the mechanical variation of the electrical double layer (EDL) capacitance of a droplet squeezed between two electrodes by an external mechanical source, has emerged as an efficient technology able to generate power at low modulation frequencies [8], [9]. The electrostatic harvesters developed in [10] and [11] are also based on the capacitance variation induced by a conductive droplet freely sliding on an electret film sputtered onto interdigital electrodes.
As an alternative to exploiting variable capacitances, water droplets can be used for generating triboelectricity through the contact electrification between the water and a triboelectric insulating polymer film. The conjunction of contact electrification and electrostatic induction in a single electrode harvesting device led to a promising droplet-based triboelectric nanogenerator (TENG) [12], featuring a simple structure, low-cost material fabrication, and good low-frequency characteristics. The desire for a superhydrophobic insulating surface, which ensures timely refresh of contact sites from droplets, and the drawback of a superhydrophobic surface, which reduces the effective contact area of impinging droplets, led to the development of a porous surface for TENG devices [13], exhibiting higher power together with optical transparency, useful for hybrid generation from raindrop and sunlight [14].
A significant increase in the power generated by a TENG was reached by the so-called droplet electrical generator (DG) [15], through the addition of a second electrode over the insulating film, to obtain a closed-loop electrical system. Further significant improvements in the generated power have been made possible by optimizing the effects of electrode geometry and droplet parameters (volume, conductivity, and dropping frequency) [16] and by tailoring parameters, such as dielectric layer thickness, droplet ion concentration, and external load [17]. These improvements have resulted in a harvested energy per droplet of 650 nJ, with an average power density per pulse of 357 W/
Even if the above DG devices represent a real turning point in the development of raindrop-based energy harvesters, an efficient electrical model that can be employed for designing the optimal electronic interface for these devices is still missing. Indeed, the electrical power that can be extracted by energy harvesters is a function not only of the energy source and the harvesting device, but it strongly depends also on the harvester’s electrical load [18], [19], [20]. First attempts to develop electrical models of DGs based on their physical behavior were proposed in [15], [16], and [17], but they are not simple enough for design purposes. An interesting numerical analysis was developed in [21], but it is only focused on the spreading process of a droplet impacting the insulating surface.
To overcome the above gaps, in this article, a compact electrical model of a DG device is developed by showing that it can be reduced to a triboelectric-based electret nanogenerator [22], made up of a dielectric layers with injected charges, which separates two conductive electrodes moving with respect to each other. The resulting model makes it easier to understand, visualize, and simulate the DG device behavior in different operating conditions and allows checking the performance of different electronic circuits without the need of making a physical DG [23]. Thus, the model can be usefully exploited for the analysis and the design of energy harvesters based on DGs.
Moreover, a simple approach to the identification of the parameters of the equivalent electric circuit of the DG is shown, overcoming the limitations of the typical identification approaches [22]. Differently from typical triboelectric-based electret nanogenerators, the considered DG is characterized by open circuit voltages up to hundreds of volts and currents that can be as small as a fraction of nanoamperes. Besides, typical DG voltage waveforms can exhibit rates of voltage increase as high as hundreds of volts per microsecond. These characteristics require very demanding measuring electronic interfaces or, as an alternative, they impose to take into account the effect of the measuring interface on the device under test. Moreover, usual procedures for parameter identification assume that the trend over time of the variable capacitance is known, while in the considered case such a trend depends on how the contact between the droplet and the electrode evolves in time, and it is unknown. In this article, an identification procedure is presented, which takes into account the ohmic-capacitive impedance of the probe and is able to estimate the time waveform of the variable capacitance.
Finally, by exploiting the presented model, an analytical expression of the energy extracted by the DG is deduced as a function of the characteristics of the droplet generator and of the bridge rectifier. The relationship, which provides valuable insight into the harvesting mechanism, can be usefully exploited by the system designers.
The rest of the article is organized as follows. In Section II, the equivalent circuit of a DG is presented. In Section III, an identification procedure is outlined, whereas in Section IV, an analytical estimation of the energy extracted for each droplet is proposed. Finally, simulations and experiments are presented in Section V.
Equivalent Circuit of a DG
A DG is made up of a solid dielectric layer with trapped charges near the upper surface, a lower conductive layer acting as the lower electrode, and finally, a conductive terminal placed above the dielectric acting as the upper electrode, as shown in Fig. 1(a) [15]. A typical material employed for the dielectric layer is represented by the PTFE (PolyTetraFluoroEthylene, also called Teflon), an electret material with high charge storage capability and stability. The negative charges prestored in the dielectric surface are formed through continuous droplet impingement or precharging on its initial neutral state, in order to obtain a saturated surface charge density, which is essential for high output voltages and efficient energy harvesting. The whole structure has an inclination of about 45° to allow the falling droplets to hit the surface, flow on the surface, make contact with the upper electrode, and flow away, as illustrated in Fig. 1(b).
(a) Schematic of a DG. (b) Operation over time of a droplet generator. i) Droplet falling; ii) droplet flowing on the surface; iii) droplet contacting the upper electrode; and iv) droplet leaving the upper electrode.
The negative charges trapped in the PTFE induce positive charges into the lower electrode, forming a charged capacitance. When the droplet hits and spreads on the PTFE surface, positive charges are also induced in the flowing droplet and an additional capacitance,
(a) Structure. (b) Equivalent circuit of a DG when the droplet contacts the upper terminal.
The three capacitances appearing in the equivalent electrical circuit of the droplet generator can be modeled as capacitors with flat and parallel faces. In particular, with reference to the PTFE/lower-electrode capacitance, it should be highlighted that the negative trapped charges and the positive charges induced in the lower electrode are present in the entire device area. However, if \begin{equation*} C_{p}=\frac {\varepsilon _{\mathrm {PTFE}} A_{p}}{d_{p}} \tag{1}\end{equation*}
Moreover, the PTFE/droplet capacitance \begin{equation*} C_{i}=\frac {\varepsilon _{D} A_{i}}{\lambda _{\mathrm {EDL}}} \tag{2}\end{equation*}
Finally, the droplet/upper-electrode capacitance \begin{equation*} C_{d}\left ({t}\right)=\frac {\varepsilon _{D} A_{d}\left ({t}\right)}{\lambda _{\mathrm {EDL}}} \tag{3}\end{equation*}
\begin{equation*} C_{v}\left ({t}\right)=\frac {C_{i} C_{d}\left ({t}\right)}{C_{i}+C_{d}\left ({t}\right)}. \tag{4}\end{equation*}
It is interesting to observe that, due to the time variation of
The negative charge density, \begin{equation*} V_{p0}=\frac {\vert \sigma _{p0}\vert d_{p} }{\varepsilon _{\mathrm {PTFE}}}=\frac {Q_{p0}}{C_{p}} \tag{5}\end{equation*}
\begin{equation*} Q_{p0}=Q_{p}\left ({t}\right)+Q_{v}\left ({t}\right) \tag{6}\end{equation*}
\begin{equation*} v_{L}\left ({t }\right)=v_{p}\left ({t }\right)-v_{v}\left ({t }\right)=\frac {Q_{p}\left ({t }\right)}{C_{p}}-\frac {Q_{v}\left ({t }\right)}{C_{v}\left ({t }\right)} \tag{7}\end{equation*}
\begin{equation*} v_{L}\left ({t }\right)=\frac {Q_{p0}}{C_{p}}-\frac {Q_{v}\left ({t }\right)}{C_{t}\left ({t }\right)} \tag{8}\end{equation*}
\begin{equation*} C_{t}\left ({t }\right)=\frac {C_{p} C_{v}\left ({t}\right)}{C_{p}+C_{v}\left ({t}\right)}. \tag{9}\end{equation*}
According to (8) and (5), the equivalent electric circuit in Fig. 2(b) can be reduced to the compact circuit in Fig. 3(a), made up of a constant voltage source with a value
(a) Compact equivalent circuit of a DG. (b) Qualitative trend of the load voltage for a resistive-capacitive load impedance.
Identification of Model Parameters for a Physical DG
Parameter identification is aimed at estimating, for a physical DG, the value of the initial voltage on the PTFE capacitance
In the considered case of a DG, the impedance of the measurement system cannot be neglected. The usual probe impedance of oscilloscopes is not large enough to be considered infinite and very-high impedance voltage buffers are not usually available for voltages of hundreds of volts and slew rates of several tens of volts per microsecond. For these reasons, a resistive-capacitive load, modeling the measurement system in addition to an external load, is considered at the DG electrical terminals, as shown in Fig. 3(a).
The describing equations of the equivalent circuit in Fig. 3(a) are \begin{align*}&\frac {v_{L}}{R_{L}}+C_{L} \frac {d v_{L}}{d t}=\frac {d}{d t}\left [{C_{t}(t) v_{t}}\right] \tag{10a}\\[-0.5em]{}\smash {\left \{{\vphantom {\begin{matrix}.\\.\\.\\.\\ \end{matrix}}}\right.}& \\[-0.5em]& V_{p 0}=v_{L}+v_{t} \tag{10b}\end{align*}
\begin{equation*} \frac {v_{L}}{R_{L}}+C_{L}\frac {dv_{L}}{dt}=V_{p0}\frac {d}{dt}C_{t}\left ({t }\right)-\frac {d}{dt}\left [{ C_{t}\left ({t }\right) v_{L} }\right] \tag{11}\end{equation*}
\begin{align*} \left [{ C_{L}+C_{t}\left ({t }\right) }\right]\frac {dv_{L}}{dt}&= -\frac {v_{L}}{R_{L}}\left [{ 1+R_{L}\frac {d}{dt}C_{t}\left ({t }\right) }\right] \\ &\qquad +V_{p0}\frac {d}{dt}C_{t}\left ({t}\right)\!. \tag{12}\end{align*}
Equation (12) is a first-order nonlinear differential equation describing the time evolution of the load voltage
When the load impedance is mainly capacitive, the fast initial raising of \begin{align*} \begin{cases} \displaystyle C_{L} V_{L 1}=C_{t 1} V_{t 1} \\ \displaystyle V_{p 0}=V_{L 1}+V_{t 1} \end{cases} \tag{13}\end{align*}
\begin{equation*} Q_{p0}=\left ({C_{p}+C_{L}+\frac {C_{p }C_{L}}{C_{v1}} }\right)V_{L1} \tag{14}\end{equation*}
\begin{equation*} Q_{p0}\cong \left ({C_{p}+C_{L} }\right) V_{L1}. \tag{15}\end{equation*}
If the peak load voltage, \begin{align*} \begin{cases} \displaystyle Q_{p0}=\left ({C_{p}+C_{L\_{}a} }\right) V_{L1\_{}a}\\ \displaystyle Q_{p0}=\left ({C_{p}+C_{L\_{}b} }\right) V_{L1\_{}b}. \end{cases} \tag{16}\end{align*}
By solving system (16) with respect to the unknowns \begin{align*} \begin{cases} \displaystyle C_{p}=\frac {C_{L\_{}a}V_{L1\_{}a}-C_{L\_{}b}V_{L1\_{}b}}{V_{L1\_{}b}-V_{L1\_{}a}}\\ \displaystyle Q_{p0}=\left ({C_{p}+C_{L\_{}b} }\right) V_{L1\_{}b}. \end{cases} \tag{17}\end{align*}
Once the values of
Let us now determine the time evolution of \begin{equation*} C_{L}\gg \frac {C_{p} C_{v}\left ({t }\right)}{C_{p}+C_{v}\left ({t }\right)}=C_{t}\left ({t }\right) \tag{18}\end{equation*}
\begin{equation*} v_{L}\ll v_{t}\approx V_{p0}. \tag{19}\end{equation*}
Thus, (10a) can be simplified as \begin{equation*} \frac {v_{L}}{R_{L}}+C_{L}\frac {dv_{L}}{dt}=V_{p0}\frac {d}{dt}C_{t}\left ({t}\right)\!. \tag{20}\end{equation*}
Equation (20) is a linear differential equation in the unknown \begin{equation*} q\left ({t }\right)=V_{p0} C_{t}\left ({t}\right)\!. \tag{21}\end{equation*}
Equation (20) shows that a DG connected to a load with very small impedance behaves like a current source of value
Starting from the measured time waveform \begin{equation*} Q\left ({s }\right)=\frac {1+s R_{L}C_{L}}{s R_{L}} V_{L}\left ({s}\right) \tag{22}\end{equation*}
\begin{equation*} T\left ({s }\right)=\frac {1+s R_{L}C_{L}}{s R_{L}} \tag{23}\end{equation*}
\begin{equation*} C_{t}\left ({t }\right)=\frac {q\left ({t }\right)}{V_{p0}}. \tag{24}\end{equation*}
For simulation purposes, the function \begin{equation*} C_{t}\left ({t }\right)\approx k_{c} e^{-\left ({t / \tau _{d} }\right)^{\alpha _{d}}} \left [{ 1-e^{-\left ({t/ \tau _{r} }\right)^{\alpha _{r}}} }\right] \tag{25}\end{equation*}
Once determined
Energy Prediction
The developed model can be efficiently employed for predicting the energy extracted by the DG. The simplest electronic interface for the DG is the diode bridge in Fig. 4(a), employed for rectifying the generated ac current [16].
(a) Schematic of a DG loaded by a diode bridge rectifier. (b) Time evolution of voltage/charge of the variable capacitance
Before any droplet reaches the PTFE surface,
When a droplet touches the upper electrode, the capacitance
Then, the capacitance starts to decrease, leading to a voltage increase that reverse biases the diodes. No current flows at the DG terminals and the charge
Therefore, the diode bridge rectifier implements a rectangular charge-voltage cycle for the variable capacitance \begin{equation*} W_{t}=2 C_{tM}\left ({V_{p0}-V_{s}-2V_{D} }\right)\left ({V_{s}+2V_{D}}\right)\!. \tag{26}\end{equation*}
Since the energy lost for the leakage currents due to diodes, the storage capacitor, the capacitor load, and so on, is \begin{equation*} W_{L}=\frac {V_{s} I_{L}}{f} \tag{27}\end{equation*}
\begin{align*} W_{S}&=4 W_{M} \cdot \\ &\quad \times \,\left [{\left ({1-\frac {V_{s}+2V_{D}}{V_{p0}} }\right)\frac {V_{s}+2V_{D}}{V_{p0}}-\frac {1}{2}\frac {V_{s}}{V_{p0}}\frac {I_{L}/f}{C_{tM}V_{p0}} }\right] \tag{28}\end{align*}
Equation (28) shows that the maximum converted energy depends on the product of the initial voltage on the PTFE, \begin{equation*} V_{S\_{}{\mathrm {opt}}}=\frac {V_{p0}}{2} \left ({1-\frac {I_{L}/f}{2C_{tM}V_{p0}} }\right)-2V_{D}. \tag{29}\end{equation*}
Equations (28) and (29), directly deduced by the DG model, allow the designers to get a precious initial estimate for sizing the droplet generator and its electronic interface.
Experimental Tests
The presented approach to model a DG and to identify its parameters is here applied to a laboratory prototype of a DG made up of a PTFE sheet placed on a sheet of silver paper, acting as a lower electrode. The upper electrode is made by a simple conductive wire placed over the PTFE sheet, as shown in Fig. 5. Much more performing devices can be made by following the construction and optimization criteria shown in [15], [16], and [17], but this is out of the scope of this work. The generation of water droplets is achieved by means of a dripper connected to a tank filled with tap water.
The time waveforms of the load voltage were acquired by a 12 bit oscilloscope, Teledyne Lecroy HDO6054, in the first case when the load was made only by the oscilloscope probe, exhibiting
Measured voltages at the terminals of the DG (a) when it is loaded by the oscilloscope probe only and (b) when it is loaded by an additional capacitor equal to
In order to determine the time evolution of
Time evolution of
The equivalent model with the identified parameters was validated through the comparison of the measured waveforms with the results from SPICE simulations of the proposed model in Fig. 3(a). Since most circuit simulators do not allow to simulate time-varying capacitors, this component was implemented through the equivalent circuit in Fig. 8. Indeed, a two-terminal made up of a capacitor having a constant value \begin{equation*} i_{t}=C_{0} \frac {d}{dt}\left [{ v_{t}-v_{x} }\right]= \frac {d}{dt}\left [{ C_{0}f\left ({t}\right)v_{t} }\right]. \tag{30}\end{equation*}
Emulation of a time-varying capacitor through a series of a constant capacitor and a voltage source.
Thus, the considered two terminals behave like a time-varying capacitor of value
A first comparison of measurements and simulations was performed by connecting different capacitors
Measured and simulated waveforms for different values of the load impedance with
A second set of measurement-simulation comparisons was performed by connecting an additional resistor
Measured and simulated waveforms for different values of the load impedance with
In order to highlight the ability of the model to predict the harvested energy, the comparison between the energy per drop experimentally measured and that predicted by the developed model is shown in Fig. 11 for a load made up of a resistor
Comparison of the energy per droplet measured and predicted by model. Absolute values on the left axis and error on the right axis.
Finally, the identified model of the droplet generator was simulated in the configuration of Fig. 4(a), i.e., connected to a diode bridge rectifier. For the diodes, the low leakage types 1N3595 by Onsemi were considered, whose accurate SPICE models are provided by the manufacturer. The storage capacitor was set to
(a) Time evolution of the voltage and charge of the variable capacitance
Conclusion
A compact equivalent circuit of a DG has been presented and the procedure for determining its parameters from electrical measurements on a physical prototype has been outlined. The obtained model is simple enough to allow the analytical prediction of the extracted energy as a function of the generator characteristics and the load properties. Therefore, the presented results can be exploited by designers for analyzing and building optimized energy harvesters.
In future works, the behavior of the DG will be investigated, when connected to more sophisticated electrical interfaces in order to increase the extracted energy.
NOTE
Open Access funding provided by ‘Università degli Studi della Campania “Luigi Vanvitelli”’ within the CRUI CARE Agreement