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Circuital Modeling of a Droplet Electrical Generator | IEEE Journals & Magazine | IEEE Xplore

Circuital Modeling of a Droplet Electrical Generator


Abstract:

A droplet electrical generator (DG) is an energy harvester able to scavenge energy from water droplets sliding on its surface. A compact electrical model of a droplet gen...Show More

Abstract:

A droplet electrical generator (DG) is an energy harvester able to scavenge energy from water droplets sliding on its surface. A compact electrical model of a droplet generator is here presented together with a black-box identification procedure. Even if previous research works have shown the great potential of the droplet generator in terms of extracted power and have investigated the optimization of the device, a simple equivalent electrical circuit is not available, which is fundamental for predicting its behavior and for designing its electronic interface, devoted to maximizing the power extraction under varying source and load conditions. A detailed identification procedure for the model parameters is also presented, in order to overcome the issues due to the high voltage, low current, high bandwidth, and unknown time-waveform of the time-varying capacitances. It is also shown how the proposed model allows the designer to predict analytically and numerically the energy that can be extracted by the generator. Finally, experimental tests are presented to show that the proposed model and the outlined procedure are able to effectively predict with good accuracy the system behavior under different operating conditions.
Published in: IEEE Sensors Journal ( Volume: 23, Issue: 12, 15 June 2023)
Page(s): 13028 - 13036
Date of Publication: 27 April 2023

ISSN Information:

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CCBY - IEEE is not the copyright holder of this material. Please follow the instructions via https://creativecommons.org/licenses/by/4.0/ to obtain full-text articles and stipulations in the API documentation.
SECTION I.

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/\text{m}^{2} in [17].

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.

SECTION II.

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).

Fig. 1. - (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.
Fig. 1.

(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, C_{i} , is formed at the droplet/PTFE interface. Before the droplet contacts the upper electrode, there is no charge flow through the above capacitances. By contrast, when the upper electrode and PTFE are connected by the droplet, another capacitance, C_{d} , is established at the droplet/upper electrode interface, as shown in Fig. 2(a), leading to the equivalent electric circuit shown in Fig. 2(b) [15], [16], [17]. Thus, if an electrical load is connected between the electrical terminals, a closed circuit is formed, which allows a current to flow through the series of the three capacitances and the external load.

Fig. 2. - (a) Structure. (b) Equivalent circuit of a DG when the droplet contacts the upper terminal.
Fig. 2.

(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 A_{p}(t) is the contact area between the droplet and the PTFE, only the charges in A_{p}(t) contribute to the current flow when the circuit is closed. It should also be noted that when the droplet slides over the surface, A_{p}(t) has a very limited variation, so that A_{p}(t)\approx A_{p} . Hence, the PTFE/lower-electrode capacitance is limited to the droplet contact area, C_{p}(t) , can be considered nearly constant and it can be expressed as \begin{equation*} C_{p}=\frac {\varepsilon _{\mathrm {PTFE}} A_{p}}{d_{p}} \tag{1}\end{equation*}

View SourceRight-click on figure for MathML and additional features. where \varepsilon _{\mathrm {PTFE}} is the dielectric constant of the PTFE, d_{p} is the distance between the trapped charges and the lower electrode.

Moreover, the PTFE/droplet capacitance C_{i} can be expressed as \begin{equation*} C_{i}=\frac {\varepsilon _{D} A_{i}}{\lambda _{\mathrm {EDL}}} \tag{2}\end{equation*}

View SourceRight-click on figure for MathML and additional features. where \varepsilon _{D} is the dielectric constant of the droplet, A_{i}=A_{p} is the PTFE/droplet contact area and \lambda _{\mathrm {EDL}} is the Debye width of the EDL formed in the droplet to neutralize the charged surface.

Finally, the droplet/upper-electrode capacitance C_{d} can be expressed as \begin{equation*} C_{d}\left ({t}\right)=\frac {\varepsilon _{D} A_{d}\left ({t}\right)}{\lambda _{\mathrm {EDL}}} \tag{3}\end{equation*}

View SourceRight-click on figure for MathML and additional features. where A_{d}(t) is the contact area between the droplet and the upper electrode. A_{d}(t) is null before the contact, reaches a maximum in the time interval when the droplet flows on the upper electrode, and finally, returns to zero when the droplet goes away. The current can flow through the load only in the time interval when A_{d}(t) is not null. The series of the two droplet capacitances C_{i} and C_{d}(t) can be modeled by a single time-varying capacitance \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*}
View SourceRight-click on figure for MathML and additional features.

It is interesting to observe that, due to the time variation of A_{d}(t) , C_{v}(t)=0 before the contact of the droplet with the upper electrode and after the end of the contact. Moreover, the time waveform of the varying capacitance C_{v}(t) is usually characterized by a very fast increase due to the impact of the droplet on the upper electrode, followed by a slow decay due to the slow separation of the droplet from the upper terminal. Since the thickness of the PTFE is several orders of magnitude larger than the EDL width at the droplet interfaces, the maximum values taken by the capacitances formed at the droplet interfaces, C_{i} and C_{d} , are far higher than the value of the capacitance C_{p} , leading to \max (C_{v}(t))\gg C_{p} .

The negative charge density, \sigma _{p0} , present in the PTFE surface makes the capacitance C_{p} electrically charged before any droplet reaches the PTFE surface, determining an initial voltage on the PTFE equal to \begin{equation*} V_{p0}=\frac {\vert \sigma _{p0}\vert d_{p} }{\varepsilon _{\mathrm {PTFE}}}=\frac {Q_{p0}}{C_{p}} \tag{5}\end{equation*}

View SourceRight-click on figure for MathML and additional features. where Q_{p0} is the absolute value of the negative charge present in an area equal to the droplet contact area A_{p} . V_{p0} represents the PTFE surface voltage corresponding to the saturated value \sigma _{p0} of the negative charge density after a continuous droplet impingement, as deeply investigated in [15]. When the droplet contacts the upper electrode, C_{v}(t) increases from zero and the charges Q_{p0} are reorganized between C_{p} and C_{v}(t) by moving through the external load Z_{L} , thus inducing a current. Since the system is electrically neutral at any time, the sum of the charges on the two capacitances is constant, i.e., \begin{equation*} Q_{p0}=Q_{p}\left ({t}\right)+Q_{v}\left ({t}\right) \tag{6}\end{equation*}
View SourceRight-click on figure for MathML and additional features.
where Q_{p}(t) is the charge on C_{p} and Q_{v}(t) is the charge on C_{v}(t) . As shown in Fig. 2(b), it holds \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*}
View SourceRight-click on figure for MathML and additional features.
where v_{L}(t) , v_{p}(t) and v_{v}(t) are the voltages on the load and on the capacitances C_{p} and C_{v}(t) , respectively. Hence, from (6) and (7), it results \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*}
View SourceRight-click on figure for MathML and additional features.
where \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*}
View SourceRight-click on figure for MathML and additional features.

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 V_{p0} and a time-varying capacitance with a value C_{t}(t) . Note that the equivalent circuit in Fig. 3(a) modeling a DG is the same as that usually employed for modeling triboelectric-based electret nanogenerators, which are a particular type of electrostatic kinetic energy harvesters [24]. This model is more compact than the one shown in Fig. 2(b), [15], [16], [17], and, what is more, its parameters can be identified by measurements at the electrical terminals. In other words, as it will be shown in Section III, it avoids the direct estimation of the capacitances by using expressions (1)–​(3), which would inevitably lead to very inaccurate results, also due to the unknown time waveform of the contact area A_{d}(t) .

Fig. 3. - (a) Compact equivalent circuit of a DG. (b) Qualitative trend of the load voltage for a resistive-capacitive load impedance.
Fig. 3.

(a) Compact equivalent circuit of a DG. (b) Qualitative trend of the load voltage for a resistive-capacitive load impedance.

SECTION III.

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 C_{p} , i.e., V_{p0} , and the time evolution of C_{t}(t) , so that the equivalent electrical model in Fig. 3(a) can be used to predict the system behavior for different electrical loads. In this section, it will be shown how to identify the circuit parameters starting from measurements at the electrical terminals of the physical DG.

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*}

View SourceRight-click on figure for MathML and additional features. which can be rewritten as \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*}
View SourceRight-click on figure for MathML and additional features.
and, hence, in the form \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*}
View SourceRight-click on figure for MathML and additional features.

Equation (12) is a first-order nonlinear differential equation describing the time evolution of the load voltage v_{L}(t) as a function of time.

When the load impedance is mainly capacitive, the fast initial raising of C_{v}(t) and, hence, of C_{t}(t) , due to the frontal impact of the droplet on the upper electrode, as described in the previous section, leads to a very fast increase in the load voltage, as qualitatively shown in Fig. 3(b). Let us focus the attention on the time interval between the start of droplet-electrode contact and the voltage peak, that is (t_{0} , t_{1} ). If the value of the load resistance, R_{L} , is high enough so that its current is negligible with respect to the current into the load capacitance, C_{L} , by taking into account that v_{L}(t_{0})=0 and C_{t}(t_{0})=0 , the time integration of system (10) in the time interval (t_{0} , t_{1} ) leads to \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*}

View SourceRight-click on figure for MathML and additional features. where V_{L1}=v_{L}(t_{1}) , V_{t1}=v_{t}(t_{1}) and C_{t1}=C_{t}(t_{1}) . From (13), (5), and (9), it results \begin{equation*} Q_{p0}=\left ({C_{p}+C_{L}+\frac {C_{p }C_{L}}{C_{v1}} }\right)V_{L1} \tag{14}\end{equation*}
View SourceRight-click on figure for MathML and additional features.
where C_{v1}=C_{v}(t_{1}) . By considering the differences in the capacitance thicknesses mentioned in Section II, it results C_{v}(t_{1})=\max (C_{v}(t))\gg C_{p} . Therefore, (14) can be approximated as \begin{equation*} Q_{p0}\cong \left ({C_{p}+C_{L} }\right) V_{L1}. \tag{15}\end{equation*}
View SourceRight-click on figure for MathML and additional features.

If the peak load voltage, V_{L1} , is measured when the system is connected to two different load capacitors, called C_{L\_{}a} and C_{L\_{}b} , (15) leads to the system \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*}

View SourceRight-click on figure for MathML and additional features.

By solving system (16) with respect to the unknowns Q_{p0} and C_{p} , it results \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*}

View SourceRight-click on figure for MathML and additional features.

Once the values of Q_{p0} and C_{p} are determined, it is possible to estimate the value of the voltage source appearing in the equivalent circuit in Fig. 3(a), i.e., V_{p0}=Q_{p0}{/}C_{p}\mathrm {.}

Let us now determine the time evolution of C_{t}(t) . To this end, let us connect a very large load capacitor C_{L}\gg C_{p} to the physical DG. Under this condition, it results \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*}

View SourceRight-click on figure for MathML and additional features. and, hence, according to (10b) \begin{equation*} v_{L}\ll v_{t}\approx V_{p0}. \tag{19}\end{equation*}
View SourceRight-click on figure for MathML and additional features.

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*}

View SourceRight-click on figure for MathML and additional features.

Equation (20) is a linear differential equation in the unknown v_{L}(t) forced by the time-varying input signal \begin{equation*} q\left ({t }\right)=V_{p0} C_{t}\left ({t}\right)\!. \tag{21}\end{equation*}

View SourceRight-click on figure for MathML and additional features.

Equation (20) shows that a DG connected to a load with very small impedance behaves like a current source of value i_{q}=dq(t)/dt , for which the feedback provided by the load and represented by the last term in (11) can be neglected.

Starting from the measured time waveform v_{L}(t) it is possible to calculate the forcing input q(t) in the Laplace domain, that is \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*}

View SourceRight-click on figure for MathML and additional features. where V_{L}(s) is the Laplace transform of v_{L}(t) and Q(s) is the Laplace transform of q(t) . In other words, the output of a simple linear system with a transfer function \begin{equation*} T\left ({s }\right)=\frac {1+s R_{L}C_{L}}{s R_{L}} \tag{23}\end{equation*}
View SourceRight-click on figure for MathML and additional features.
forced by the measured time waveform v_{L}(t) provides the signal q(t) and the time-varying capacitance waveform \begin{equation*} C_{t}\left ({t }\right)=\frac {q\left ({t }\right)}{V_{p0}}. \tag{24}\end{equation*}
View SourceRight-click on figure for MathML and additional features.

For simulation purposes, the function C_{t}(t) can be efficiently approximated by an exponential function as \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*}

View SourceRight-click on figure for MathML and additional features. where \tau _{r} and \alpha _{r} affect the fast rising, whereas \tau _{d} and \alpha _{d} affect the slow decay. The parameters in (25) can be calculated by best fitting on the obtained waveform for C_{t}(t) in (24).

Once determined V_{p0} and C_{t}(t) , all the parameters appearing in the equivalent circuit in Fig. 3(a) are identified, including the time evolution of the capacitor. By contrast, typical identification procedures assume a sinusoidal evolution of the time-varying capacitor [22], which is very far from the actual behavior of a physical DG.

SECTION IV.

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]. C_{s} is a very large capacitor, C_{s}\gg C_{t} , employed for storing the energy extracted by the DG, whose voltage, V_{s} < { V}_{p0} , does not vary significantly due to a single droplet. The diodes are modeled by a constant voltage, V_{D} , in the forward region of operation.

Fig. 4. - (a) Schematic of a DG loaded by a diode bridge rectifier. (b) Time evolution of voltage/charge of the variable capacitance 
${C}_{t}{(}{t} {)}$
 for a single droplet.
Fig. 4.

(a) Schematic of a DG loaded by a diode bridge rectifier. (b) Time evolution of voltage/charge of the variable capacitance {C}_{t}{(}{t} {)} for a single droplet.

Before any droplet reaches the PTFE surface, C_{t}(t)=0 , the impedance of C_{t} is significantly higher than that of the load at the DG terminals, and, thus, v_{t}=V_{p0} . Moreover, the current i_{t} flowing through the variable capacitance C_{t} is null, and no charge is stored in it, that is q_{t}=0 , as shown by point A in Fig. 4(b).

When a droplet touches the upper electrode, the capacitance C_{t}(t) starts to rise, the diodes are still reverse-biased, and the capacitance-voltage starts to decrease according to the law C_{t}dv_{t}/dt=i_{t}-v_{t}dC_{t}/dt , where i_{t} is only due to the diode leakage currents. However, the charge q_{t} does not increase significantly until the voltage v_{t} reaches a value sufficient to forward bias diodes D_{1},D_{4} , i.e., v_{t}=V_{p0}-V_{s}-2V_{D} , [point B in Fig. 4(b)]. Then, a significant current starts to flow through C_{t} , with a significant increase in its charge q_{t} , while the voltage stays constant. The charge rise stops when the capacitance reaches its maximum value, that is for q_{t}=Q_{tM}=C_{tM}(V_{p0}-V_{s}-2V_{D}) , with C_{tM}=\mathrm {max}(C_{t}) (point C).

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 q_{t}=Q_{tM} stays constant until the increasing voltage v_{t}(t)=Q_{tM}/C_{t}(t) reaches the value v_{t}=V_{p0}+V_{s}+2V_{D} (point D), which allows the current to flow through the diodes D_{2},D_{3} . Thus, the voltage v_{t} is kept constant and the decrease of the capacitance reduces the charge to zero (point E). Finally, v_{t} tends to the initial value V_{p0} due to the high impedance of C_{t} and the leakage currents.

Therefore, the diode bridge rectifier implements a rectangular charge-voltage cycle for the variable capacitance C_{t} [25]. The energy released by C_{t} for a single droplet is equal to the area in the q_{t}-v_{t} plane and is equal to \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*}

View SourceRight-click on figure for MathML and additional features.

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*}

View SourceRight-click on figure for MathML and additional features. where I_{L} is the total leakage current and f is the frequency of the falling droplets, the energy stored in the capacitor C_{s} can be estimated as W_{S}=W_{t}-W_{L} , that is \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*}
View SourceRight-click on figure for MathML and additional features.
where W_{M}=1/2\cdot C_{tM}V_{p0}^{2} is the maximum energy that can be converted.

Equation (28) shows that the maximum converted energy depends on the product of the initial voltage on the PTFE, V_{p0} , times the fixed negative charges Q_{P0}=C_{p }V_{p0}\approx C_{tM}V_{p0} . Moreover, the effective converted energy depends on the circuit losses and the voltage on the storage capacitor. The impact of the leakage currents and of the droplet frequency, I_{L}/f , on the extracted energy, still depends on the fixed negative charges Q_{P0}=C_{p }V_{p0}\approx C_{tM}V_{p0} . The optimal voltage on the storage capacitor, which ensures the maximum energy extraction, can be calculated by deriving (28) with respect to V_{s} and equating the result to zero, that is \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*}

View SourceRight-click on figure for MathML and additional features.

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.

SECTION V.

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.

Fig. 5. - (a) Photograph of the experimental setup and (b) is a zoom of (a).
Fig. 5.

(a) Photograph of the experimental setup and (b) is a zoom of (a).

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 R_{L\_{}a}=10 \text {M}\Omega and C_{L\_{}a}=10 \mathrm {pF} , and in a second case when a capacitor C_{E}=4598 \mathrm { pF} was placed in parallel to the probe, leading to an overall load characterized by R_{L\_{}b}=10 \mathrm {M}\Omega and C_{L\_{}b}=4608 \mathrm {pF} . The two waveforms are reported in Fig. 6. By solving system (16) with the measured peak values of the voltage, i.e., V_{L\mathrm {1\_{}}a}=46.55 \mathrm {V} and V_{L\mathrm {1\_{}}b}=0.705 \mathrm {V} , the values of the initial charge Q_{p0}=3.2 \mathrm {nC} and of the PTFE capacitor C_{p}=59 \mathrm {pF} were calculated. From these values, the initial voltage V_{p0}=Q_{p0}{/}C_{p}=54.5 \mathrm {V} was deduced.

Fig. 6. - 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 
${C}_{E}={\text {4598 pF}}$
.
Fig. 6.

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 {C}_{E}={\text {4598 pF}} .

In order to determine the time evolution of C_{t}(t) , since C_{L\_{}b}\gg C_{p} , the load voltage waveform measured for R_{L\_{}b}=10 \mathrm {M}\Omega and C_{L\_{}b}=4608 \mathrm {pF} , was exploited as input to the linear system with the transfer function reported in (23). The calculated output of the linear system, q(t) , allowed us to calculate C_{t}(t) , as shown in Fig. 7. The calculated time waveform was approximated by best fitting with the function in (25) with the coefficients k_{c}=0.97 C_{p} , \tau _{d}=16 \mathrm {ms} , \alpha _{d}=1.6 , \tau _{r}=2 \mu \mathrm {s} , \alpha _{r}=1.4 . The approximated function is also reported in Fig. 7.

Fig. 7. - Time evolution of 
${C}_{t}{({t}{)}}$
, as calculated from laboratory measurements and its analytical approximation. (a) Complete time waveform and (b) zoom of the initial rising.
Fig. 7.

Time evolution of {C}_{t}{({t}{)}} , as calculated from laboratory measurements and its analytical approximation. (a) Complete time waveform and (b) zoom of the initial rising.

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 C_{0} in series with a voltage-controlled voltage source of value v_{x}=[1-f(t)]v_{t} , with f(t) an arbitrary time function, draws a current equal to \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*}

View SourceRight-click on figure for MathML and additional features.

Fig. 8. - Emulation of a time-varying capacitor through a series of a constant capacitor and a voltage source.
Fig. 8.

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 C_{T}(t)=C_{0}f(t) .

A first comparison of measurements and simulations was performed by connecting different capacitors C_{E} to the electrical terminals of the droplet generator. These capacitors lie in parallel with the oscilloscope probe impedance characterized by R_{m}=10 \mathrm {M\Omega } and C_{m}=10 \mathrm {pF} , resulting in a load impedance having R_{L}=R_{m} and C_{L}=C_{m}+C_{E} . The measured waveforms and the simulated waveforms are reported in Fig. 9. It is interesting to observe that the simulated circuit well fits the measured waveforms both in the case of the load waveform used for identifying the circuit parameters, i.e., the curve for C_{L}=4608 \mathrm {pF} , and in the case of different values of the load capacitance.

Fig. 9. - Measured and simulated waveforms for different values of the load impedance with 
${R}_{L}=\text {10 M}\omega $
. (a) Full waveforms. (b) Zoom of the initial transients.
Fig. 9.

Measured and simulated waveforms for different values of the load impedance with {R}_{L}=\text {10 M}\omega . (a) Full waveforms. (b) Zoom of the initial transients.

A second set of measurement-simulation comparisons was performed by connecting an additional resistor R_{E} at the electrical terminals of the droplet generator, so that R_{L}=R_{m}//R_{E} and C_{L}=C_{m}+C_{E} . The comparison reported in Fig. 10(a) and in Fig. 10(b) using different time scales, shows that the proposed model is able to predict with good accuracy the system behavior in very different load conditions, not only when the value of R_{L} is large as assumed during the identification procedure, but also for smaller values.

Fig. 10. - Measured and simulated waveforms for different values of the load impedance with 
${R}_{L}=\text {100 k}\omega $
. (a) Full waveforms. (b) Zoom of the initial transients.
Fig. 10.

Measured and simulated waveforms for different values of the load impedance with {R}_{L}=\text {100 k}\omega . (a) Full waveforms. (b) Zoom of the initial transients.

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 R_{L}=R_{m} with a parallel capacitor C_{L}=C_{m}+C_{E} . The prediction error, also reported in the same Figure, shows the good accuracy of the model for practical purposes.

Fig. 11. - Comparison of the energy per droplet measured and predicted by model. Absolute values on the left axis and error on the right axis.
Fig. 11.

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 C_{s}=1 \mu \mathrm {F} . The time evolution of the voltage and charge of the variable capacitance C_{t}(t) obtained by simulations are reported in Fig. 12(a) for different values of the initial voltage V_{s} . The same figure also reports the analytical approximations for the same quantities. The comparison shows that the approximated equations reported in Section IV provide a good estimation of the q_{t}-v_{t} cycle, whose area gives the energy extracted by the droplet generator. In particular, (28), relating the converted energy to the droplet generator parameters and to the rectifier characteristics, can be usefully exploited for design, as shown in Fig. 12(b). There, the converted energy normalized to its maximum value is shown as a function of the storage capacitor voltage and of the ratio between the leakage current and the droplet frequency. It can be observed that, according to (29), the optimal value of V_{s} decreases for increasing values of losses and for decreasing values of frequency. Moreover, the converted energy significantly decreases with the leakage currents. As an example, for I_{L}/f=Q_{p0} the converted energy does not exceed one-third of the theoretical maximum value expected for zero losses.

Fig. 12. - (a) Time evolution of the voltage and charge of the variable capacitance 
${C}_{t}{(}{t} {)}$
 for a single droplet, as obtained by SPICE simulations (continuous lines) and as predicted by theoretical analysis (dashed lines). (b) Normalized energy stored in the storage capacitor, 
${W}_{S}/{W}_{M}$
, as a function of the normalized capacitor voltage, 
${V}_{s}/{V}_{p{0}}$
, and of the normalized leakage losses, 
${(}{I}_{L}/{f}{)}/{Q}_{p{0}}$
.
Fig. 12.

(a) Time evolution of the voltage and charge of the variable capacitance {C}_{t}{(}{t} {)} for a single droplet, as obtained by SPICE simulations (continuous lines) and as predicted by theoretical analysis (dashed lines). (b) Normalized energy stored in the storage capacitor, {W}_{S}/{W}_{M} , as a function of the normalized capacitor voltage, {V}_{s}/{V}_{p{0}} , and of the normalized leakage losses, {(}{I}_{L}/{f}{)}/{Q}_{p{0}} .

SECTION VI.

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

References

References is not available for this document.