Energy harvesting with pairs of variable capacitors without control circuits

Abstract

This work describes several electronic electrostatic generators that can be built using two pairs of complementary variable capacitors, in a way that dispenses control circuits for their operation. In all cases, a basic unstable generator based on an electrostatic charge multiplier is used to bias variable capacitors, and the current generated by these capacitors is then rectified and sent to the load. Experimental versions of the proposed generators were built using variable capacitors made with 3D-printing techniques, moved in a back-and-forth way by a small motor. As these generators operate at high impedance level, always with small currents, they are insensitive to the resistivity of the conductive plastic used to make the variable capacitors. Several new structures are presented, and their properties are evaluated theoretically and experimentally.

1 Introduction

The last few years saw a crescent interest in devices using variable capacitors for low-power energy generation, with the purpose of serving for energy harvesting in autonomous systems. An example of a possible application is in arrays of small sensors that take energy from the ambient, in this case from mechanical vibrations, and connect to other systems by radio signals. The general idea for these generators arose recently from works as [1], where a single variable capacitor extracts energy from the movement of its plates, and is coupled to a DC/DC converter in a way that provides repetitive energy transfer to a load. The variable capacitor could be built using microelectromechanical techniques (MEMS), resulting in a very compact structure. Moreover, the construction with micrometer distances between charged surfaces may allow the exploration of Paschen’s law, which says that electric fields much larger than the limit for large dimensions (about 3 kV/mm) are possible at small (about 10 µm) distances without breakdown of the air insulation, resulting in significant energy densities for these generators. The work [2] showed that it is possible to eliminate the DC/DC converter from the basic generator structure using only capacitors and diodes, in a configuration that works similarly to the classical electrostatic device known as Bennet’s doubler [3]. A similar structure with just one variable capacitor, was studied well before, for high-power rotating generators operating in vacuum [4]. See also [5, 6] for early devices. A MEMS construction of a generator that works in a similar way, exchanging the roles of the fixed and variable capacitors, was proposed in [7]. Another variation, using a variable capacitor made with elastomers coupled to a diode-capacitor voltage multiplier can be found in [8] and related works. Many other different generators are possible, as the structures studied in [9,10,11]. An imitation of the operation of the original Bennet’s doubler using mechanical switches in a MEMS structure was proposed in [12], trying to avoid the losses and leakage introduced by real diodes. The startup problem, common for all these generators, was addressed with the use of RF energy in [13]. Comments about startup of the generators proposed here are included along the text.

The doubler in [2] was proposed used directly as a battery charger, or as a capacitor charger, with energy extracted from the capacitor periodically with a buck DC/DC converter. The battery charger does not exploit the use of high voltages in the variable capacitors to obtain greater energy density. The capacitor-charging version is much better, but requires a low-power DC/DC converter, which may be difficult to implement. Other works, as [14, 15], proposed methods to use high voltages internally in battery chargers without DC/DC converters to increase or to regulate the charging current efficiently. In [16] an intermediate method was proposed, using variable capacitors to make the conversion from high to low voltage efficiently, in a way like the one used in [5], with the doubler used only as high-voltage bias generator after its startup. In this work, that idea is extended, with other structures for the biasing generator and the variable-capacitance power generator studied, considering that Bennet’s doubler is just the simplest form of a family of variable-capacitance electrostatic generators. The generator described in [17] is used in the experiments. The proposed generators don’t require any form of control in their operation, and require only capacitors and diodes in their construction. They extract less energy than versions using DC/DC converters, but their simplicity may be attractive for applications requiring low cost and small size (no bulky inductors). The same structures may also be adapted to provide startup energy for a low-power DC/DC converter, which would afterwards extract energy with greater efficiency.

The next section reviews the experimental device and derives a model for it. The following sections describe some possible generators which it can implement, and their performances are evaluated theoretically and experimentally.

2 A 3D-printed variable-capacitance generator

The generator described in [17], with the structure shown in Fig. 1, comprises two complementary variable capacitors formed by sets of plates, or combs, made with conductive ABS plastic, mounted in a structure that moves the upper combs back and forth, driven by an electric motor, shown in Fig. 2. The original idea was to make a pair of complementary variable capacitors for use in a generator based on Bennet’s doubler, but requiring small displacement (the plate sets move parallel to each other). The device was then split in two sections, reducing the required displacement to about one-half of the displacement of a single device with the same total capacitance. The precision of the 3D printing system used required minimum dimensions of about 1 mm, and so the device was dimensioned with the plates separated by this distance, resulting in an approximately cubic construction, as compact as possible (5.5 × 5.5 × 5.7 cm), with a large capacitance variation, between 15 and 165 pF for each of the four variable capacitors. The alignment of the plates during their movement, quite critical, is ensured by four adjustable rollers that guide the movable assembly, with the two upper sets of plates. These generators are usually operated by vibrating the entire structure along one direction, with the vibration causing the movement of one set of plates in relation to the other. This is possible with the experimental structure with the addition of suitable springs and supports, but for better control in the experiments, the lower combs were kept fixed, with just the upper combs moving, directly connected to the excitation mechanism. The structure does not seem directly suitable for usual MEMS construction, which prefer flat structures, but it may be possible to replicate it with smaller distances in high-precision 3D printing techniques that are in rapid development.

Fig. 1

Basic structure of the variable-capacitance 3D-printed generator

Fig. 2

The experimental generator, with the driving mechanism

In the experiments related in [17], several possible generators were connected to a DC/DC converter, the same described in [2]. It was noticed that, although the operation of the doubler or multiplier was not affected by the resistive plates, the extraction of the accumulated energy by the fast converter was prejudiced by the resistances. Several important parasitic capacitances also appear in the device, affecting its operation. A model for the experimental device was obtained by measurements with a precision RLC meter. It was soon noticed that to model correctly the variable capacitors as simple series RC associations was not possible, considering that connections can be made to the combs at both edges, and that there is a significant resistance from edge to edge of a comb. A consistent model could be obtained by splitting the combs in two, as shown in Fig. 3. The model produces the correct resistances across both terminals of each comb, 22 kΩ for the fixed combs and 23 kΩ for the movable combs, which were printed with a different model. The parallel connection of the pairs of capacitors between the combs result in the observed capacitance values at low frequency, 15–165 pF for C_a and C_b, and 20 pF between adjacent combs. The observed resistances in series with C_a and C_b, 15 kΩ, are also correctly seen, although just at high frequency. In this work, the generators do not use DC/DC converters, and so the resistances have negligible effect. The parasitic capacitances, however, continue to affect the behavior of the devices, as commented in the text of the following sections.

Fig. 3

Model for the experimental generator. a Simplified. b Complete. The nodes are the connections in Fig. 1

3 Generators without control circuits

3.1 Bennet’s doubler as bias generator

This structure, shown in Fig. 4, is similar to one discussed in [16], and may be used as reference for the others. One of the sections is connected as the electronic version of Bennet’s doubler (C_a, C_b, C_e), and the other as a double variable-capacitance generator. The current in the voltage limiter Zener diode Z₁ is routed to the output, adding some extra power.

Fig. 4

Bennet’s doubler (a) and double capacitive converter (b)

The analysis of this circuit was already done in [16], with the approximation of considering the voltage at the output of the doubler, V_b′ at node 4, constant. The average currents through the diodes D_a or D_b were obtained as:

$$\bar{i}_{D} = f\left( {C_{max} (V_{b}^{{\prime }} - V_{d} - V_{o} ) - C_{min} (V_{b}^{{\prime }} + V_{d} )} \right)$$

(1)

V_d is a diode drop voltage in D_a–D_d, f is the vibration frequency, C_min is the minimum value of C_c or C_d and C_max the maximum. V_o is the output voltage, considered fixed, as in a battery. Using the same notation in [16], V_b′ is the Zener voltage V_b added to the output voltage V_o, for this case and all the others. The current through the Zener diode was also calculated in [16], with the same approximation and assuming complementary variation of C_a and C_b, with C_a small at phase 1, C_min, and large at phase 2 of the movement, C_max, and V_d1 the voltage drop in D₁–D₃ (larger than V_d in the experiment) as:

$$\bar{i}_{Z} = f\frac{{\left( {(C_{min} (V_{d1} - V_{b}^{{\prime }} ) - C_{max} V_{b}^{{\prime }} )(C_{min}^{2} V_{b}^{{\prime }} + C_{min} C_{max} (V_{d1} + V_{b}^{{\prime }} ) + C_{max}^{2} (V_{d1} - V_{b}^{{\prime }} ))} \right)}}{{(C_{min} + C_{max} )(C_{min} V_{b}^{{\prime }} + C_{max} (V_{b}^{{\prime }} - V_{d1} ))}}$$

(2)

The structure was tested using as load a parallel RC association of 5 MΩ and 0.1 µF, with the device vibrated with a period of 140 ms (f = 7.14 Hz). The output voltage was measured with a 10 MΩ oscilloscope probe over a 10 MΩ resistor, resulting in the 5 MΩ load. The bias voltage was measured with a high-impedance electrostatic voltmeter, producing 1 V for each 100 V at the oscilloscope, with significant attenuation of frequencies above a few Hz. C_e was not used. Figure 5 shows the resulting output and bias voltages during the startup of the generator. The assembly was carefully grounded and shielded from interferences and discharged before the tests, and so the generator was not starting easily by itself, requiring a starting voltage to be applied to the output of the doubler. A momentary touch with 10 V was used (observable at 2.5 s). The bias voltage then rises to ~ 220 V with the output reaching ~ 3.3 V, or ~ 2.4 V with Z₁ grounded. The corresponding output currents are then 0.66 and 0.48 µA respectively. Ideally, the bias voltage would reach a peak value of 243.3 V, with two 120-V Zener diodes in series used and 3.3 V at V_o, but the Zeners are not so precise and may conduct significantly a bit before their nominal voltages at low currents, and there is some ripple at the bias voltage, filtered by the electrostatic voltmeter. With V_o = 3.3 V and V_b′ = 220 V, V_d = 0.6 V and V_d1 = 3 V (high-voltage rectifiers, used due to their low capacitance and leakage), (1) and (2) predict 2i_D+ i_z = 0.462 + 0.213 = 0.675 µA, and 2i_D = 0.464 µA with V_o = 2.4 V. The maximum predicted output power is 2.2 µW, all in good agreement with the experiment.

Fig. 5

Bennet`s doubler and double capacitive converter. Outputs using or not the Zener current, and bias voltage (100 V/div)

Another possibility is to simplify the bias generator using a doubler with C_b fixed. The other section of the variable capacitor can then be used as another variable-capacitance generator, increasing the output current. This structure, Fig. 6, is a combination of structures presented in [16] too. Figure 7 shows the experimental results, with the fixed capacitor C_f = 220 pF replacing C_b. The startup is much slower, and the output voltage reaches ~ 3.9 V, with 0.78 µA. (1) can still be used, but (2) is different with fixed C_b [16]:

$$\bar{i}_{zf} = f\frac{{V_{b}^{{\prime }} \left( {\frac{{C_{max} }}{{C_{min} }} - 2} \right) - V_{d1} \left( {1 + \frac{{C_{max} }}{{C_{min} }}} \right)}}{{\frac{1}{{C_{min} }} + \frac{1}{{C_{f} }}}}$$

(3)

Fig. 6

Bennet’s doubler with fixed C_b (C_f) (a) and triple capacitive converter (b)

Fig. 7

Bennet’s doubler with fixed C_f and triple capacitive converter. Outputs using or not the Zener current and the current through C_b, and the bias voltage

Equations (1) and (3) predict 3i_D+ i_zf = 0.691 + 0.195 = 0.886 µA. The curves in Fig. 7 show that the current through the diodes agrees well with the prediction, but the current through the Zener diode is smaller. The reasons for the reduction, verifiable by simulation of the circuit, are the capacitances between the lower combs, which add ~ 20 pF in parallel with D₃, and the large ripple in the voltage at the output of the doubler due to the asymmetrical capacitive loading caused by C_c and C_b+ C_d. The last effect can be reduced by the use of a large C_e, not used in the experimental setup.

3.2 Inverted Bennet’s doubler

An interesting possibility, not considered before, is to invert the polarity of the complete Bennet’s doubler, inverting its diodes, and then to take its output at the other terminal of C_e. The current through the capacitor C_a that would go to ground can then be routed to the rectifier connected to C_c, without affecting significantly the operation of the doubler. The added current is practically identical to the current through the Zener diode, because in the normal doubler when the Zener conducts its current comes from C_a. The result is a significant improvement over the last configuration, and the most powerful generator of all proposed here. From Fig. 8, V_o reaches ~ 4.6 V, with 0.920 µA of output current, with the bias voltage V_b′ at ~ 240 V, almost without ripple. Equations (1) and (2) predict 2i_d+ 2i_z = 0.502 + 0.465 = 0.967 µA, slightly greater than the observed. Note in Fig. 8 that if the currents through the Zener or through C_a are not used the drops in V_o are identical, verifying that the currents contributed by them are really similar. C_e was not used, and would make negligible difference, due to the low ripple over Z₁. This structure could be used without the capacitive converters around C_c and C_d too, requiring only a single pair of variable capacitors, using both sections, and generating two times the current of a version using only the current through the Zener diode routed to the output.

Fig. 8

Inverted Bennet’s doubler and double capacitive converter. Output with and without the currents through the Zener and C_a, and the bias voltage

3.3 Symmetrical generator with synchronous movement

Structures where the multiplier has a symmetrical structure were reported in [17], with energy extraction by a DC/DC converter. They can be used biasing capacitive converters too, starting with the structure shown in Fig. 9. The multiplier around C_b and C_d, which vary synchronously, is similar to the one described in [6]. It produces opposite voltages at nodes 2 and 5 that can be used as bias for the capacitive converters using the two other variable capacitors C_a and C_c. The current through the positive limiter Z₁ can be added to the output. The current through the negative limiter Z₂ cannot be used, unless a negative output is also desired. The current from one of the capacitive converters could be added to it too, with the diodes inverted, resulting a completely symmetrical structure. Figure 10 shows V_o ≈ 3.5 V, with 0.7 µA. The positive bias voltage reaches V_b′ ≈ 220 V in average. The currents through the capacitive converters can be calculated by (1), since the voltage variations over C_a and C_c are identical. The current through Z₁ comes from the capacitive converter formed by C_b, D₃ and D₄, with negative bias − V_b and output V_b′ = V_b+ V_o, and can be obtained approximately from a variation of (1) too, as (4), where the variable capacitor is C_b. The formulas give 2i_d+ i_z = 0.462 + 0.230 = 0.692 µA, practically identical to the experimental result.

$$\bar{i}_{Z} = f\left( {C_{max} (V_{b} - V_{d1} ) - C_{min} (V_{b}^{{\prime }} + V_{b} + V_{d1} )} \right)$$

(4)

Fig. 9

Symmetrical multiplier with synchronous movement combined with double capacitive converter

Fig. 10

Symmetrical multiplier with synchronous movement and double capacitive converter. Output with and without the current through the positive limiter and the bias voltage

3.4 Symmetrical generator with complementary movement

A similar generator can be obtained with complementary movement of the two pairs of variable capacitors, by interchanging the connections of one of the pairs of capacitors, as C_c and C_d, as shown in Fig. 11. This generator works similarly to the classical multiplier known as “Wilson’s machine” [18], with diodes instead of contacts. It results in very fast startup because it multiplies its previous charges by a larger factor, which can reach a maximum of ~ 2.618 instead of the 2 of the previous structures [19]. The fast rise, with about 1.7 of multiplication factor per cycle, can be observed in the bias voltage in Fig. 12. Assuming constant voltages over the limiting Zeners and constant voltage V_o, in steady state its average output current is ideally identical to the case with synchronous capacitance variations, but the different switching produces significant ripple in the bias voltage, causing a drop in the output current. Figure 12 shows V_o ≈ 3.2 V with 0.64 µA and V_b′ ≈ 210 V. The expressions (1) and (4) result in 2i_d+ i_z = 0.441 + 0.231 = 0.672 µA. Note from Figs. 10 and 12 that there are some differences, as a greater ripple at the high-voltage limiter and at the output, because diodes D_a and D_b conduct at the same time. The capacitances between the lower combs prejudice this generator too, as in the other cases, causing most of the observed current loss. Note that the structures with synchronous and complementary movement are just limit cases, as the generator works with both sets of variable capacitors operating with any phase difference, or even at different frequencies.

Fig. 11

Symmetrical multiplier with complementary movement combined with double capacitive converter

Fig. 12

Symmetrical multiplier with complementary movement and double capacitive converter. Output with and without the current through the positive limiter and the bias voltage

3.5 Symmetrical unipolar generator

The structures of the basic symmetrical generators, the central sections in Figs. 9 and 11 can be considered as a pair of capacitive converters feeding each other. A possibility is then to invert the polarity of one of the converters, by the inversion of the polarity of two diodes, as in the structure shown in Fig. 13. Differently from the other symmetrical generators, it generates two bias voltages, at nodes 2 and 5, with the same polarity, positive in this case. The currents through the two limiter Zeners can then be routed to the output, what could, in principle, significantly improve the output power. The experimental result of the structure, however, was not so good. There is large ripple at the bias voltages, with a timing that significantly reduces the current through the outer capacitive converters. The currents through the Zeners cannot be predicted by a version of (1) with V_o = V_b′, because the input and output voltages of the converter would be identical, if assumed constant, and the calculated current would be completely incorrect, negative. The exact analysis is complicated, but a significant current flows through them, because they continue to limit the voltage growth of the unstable generator. The output reached only ~ 2 V, with 0.4 µA (Fig. 14). It is counterproductive to try to hold the bias voltages stable by adding capacitances, because they reduce the effective variation of C_a and C_d. This generator requires their variation to operate, what does not happen in the other symmetrical structures. In this structure, C_b and C_c can even be fixed, but then the generated current is then even smaller. The unipolar structure does not work with synchronous capacitance variations.

Fig. 13

Symmetrical unipolar multiplier combined with double capacitive converter

Fig. 14

Symmetrical unipolar multiplier and double capacitive converter. Output with and without the current through the positive limiter and the bias voltage

The simulations in Figs. 15 and 16 show waveforms obtained assuming models as in Fig. 3(b) for the generator, for the structure in Fig. 11 and for the unipolar structure of Fig. 13. It can be noticed, by comparison, that the reason for the low output of the latter are the relatively low bias voltages for the capacitive converters, e₂ and e₅, when the rectifier diodes conduct, resulting in smaller currents in them. This structure, with its double unipolar output, was exploited and extended in [20] for the generation of more powerful generators.

Fig. 15

Simulated waveforms for the generator in Fig. 11. Voltages at nodes 2 and 5, e₂ and e₅, and currents through the diodes D_a, D_b, and Z₁, j_Da, j_Db, and j_Z, along with the capacitances C_a and C_b (the maximum capacitance corresponds to one division)

Fig. 16

Simulated waveforms for the unipolar multiplier, Fig. 13, showing the same waveforms of Fig. 15

4 Ideal startup behavior of the generators

The ideal behavior of the generators in the startup can be studied by considering them composed of parallel plates that form capacitors, and tracking what happens with charges in the plates. This is easy if C_min is considered negligible.

The operation of the generator based on Bennet’s doubler, Fig. 4, is shown in Fig. 17, where a pair of movable plates 2 and 5 move synchronously, parallel to the fixed plates 1, 3, 4, and 6. They represent the combs of the experimental generator, with just one plate in each, in horizontal position. One of the sections makes the unstable bias generator, and the other the two capacitive converters, which act as a fixed capacitive load with capacitance equal to C_max+ C_min. Approximating C_min = 0, in position (a), equal charges ± Q are in C_a, plates 4–5, and in C_c, plates 1–2 of the load capacitors. In position (b), an inverted copy of the charge in plate 5 is attracted from ground to plate 6, and the charge that was in plate 4 accumulates in plate 2. In the return to position (a), the accumulated charge + 3Q in plates 2 and 6 is equally split between plates 2 and 4, with complementary copies in plates 1 and 5. The generator then multiplies its charge by 1.5 at each cycle. With smaller load capacitances, the charges there would be smaller, and the multiplication factor would tend to 2. From the analysis in [2], the ideal multiplication factor is:

$$z = \frac{{C_{dmax} + 2C_{max} }}{{C_{dmax} + C_{max} }}$$

(5)

Fig. 17

Operation of the generator based on Bennet’s doubler. The marked diodes conduct during the transitions, connecting the plates

Note that C_c is not important. Because C_min is not so negligible and other capacitances and losses, the experimental multiplication factor seen in Fig. 5 is approximately 1.25. The version with fixed C_b, C_f in Fig. 6, is slower and more complicated due to the irregular load. Its multiplication factor can be predicted as in [2], even with the load capacitance not constant, resulting in 1.33 if C_f = C_max, with the load capacitance varying between 2C_max when C_a = 0 and C_max when C_a = C_max. In the general case with C_min = 0 it would be:

$$z = \frac{{C_{f} + 3C_{max} }}{{C_{f} + 2C_{max} }}$$

(6)

With the values used, (6) predicts z = 1.3. The curves in Fig. 7 show approximately 1.12, because of the other stray capacitances not considered. The expression (6) apparently predicts that the doubler would work even with C_f = 0. This happens because it was assumed that the minimum value of C_a is zero. This would leave node 5 in Fig. 4 disconnected in that condition, with infinite negative voltage. A better expression, with C_min considered but C_max≫ C_min, is obtained as (7), which predicts z = 1.26 for the experiment. The maximum z can be obtained as 1.5, from (6) or (7).

$$z = \frac{{C_{f}^{2} + 3C_{f} C_{max} + 2C_{max} C_{min} }}{{(C_{f} + C_{min} )(C_{f} + 2C_{max} )}}$$

(7)

The inverted doubler of Fig. 18 has the same startup behavior of the regular version, as can be seen in Figs. 5 and 8.

Fig. 18

Inverted Bennet’s doubler (a) and double capacitive converter (b)

The operation of the unipolar generator in Fig. 13 is shown in Fig. 19. The output rectifiers would be connected to the ground connections of plates 3 and 4, capturing the current through the grounded capacitors. Supposing an initial charge + Q in plate 2, it splits by electrostatic induction an initially neutral charge in plates 1–5, pushing + Q to plate 5, while plate 6 (supposed previously negative) is discharged. When the plates move, plate 5 splits the + Q charge in plates 2–6, pushing + 2Q to plate 2. When the plates return to the first position, the charge in plate 5 is increased to + 3Q by the + 2Q charge repelled from plate 1. It is easy to verify that cycle-by-cycle the charge in plate 2 grows in the sequence + Q, + 2Q, + 5Q, + 13Q, etc., while the charge in plate 5 grows in the sequence + Q, + 3Q, + 8Q, + 21Q, etc. The multiplication factor for the charges from cycle to cycle increase from a minimum of 2 for plate 2 and decrease from a maximum of 3 for plate 5. After some cycles, they tend to ~ 2.618 [19]. As there is no extra load capacitance to be considered, in this and in the other two symmetrical generators, the predicted ideal multiplication factor is the maximum, and the startup is very fast, although not as fast as the ideal analysis predicts, due to unaccounted stray capacitances and losses.

Fig. 19

Operation of the unipolar generator

The generator with complementary movement in Fig. 11 differs just by the polarities of two diodes. Its idealized behavior is as in Fig. 20. Supposing initial charges + Q and − Q in plates 2 and 5, opposite charges are attracted from the ground to plates 1 and 4. When the upper plates move, the charges in plates 1 and 5 accumulate to − 2Q, attracting + 2Q from the ground to plate 6. In the following cycles, the charge in plate 2 grows in the sequence + Q, + 3Q, + 8Q, + 21Q, etc., while the charge in plate 5 grows as − Q, − 2Q, − 5Q, − 13Q, etc. The device multiplies the charges with the same maximum multiplication factors of the unipolar generator [19], but generating two outputs with opposite polarities. Indeed, Fig. 14 shows a rise of the bias voltage similar to the case in Fig. 12, the fastest among the studied structures.

Fig. 20

Operation of the generator with complementary movement, similar to Fig. 19, but generating opposite polarities

The generator with synchronous movement in Fig. 9 is just a symmetrical doubler, as illustrated in Fig. 21. With initial charges + Q and − Q in plates 2 and 5, when the plates move to position b they attract opposite charges to plates 3 and 6, and when returning accumulate these charges in plates 5 and 2, doubling the charges.

Fig. 21

Operation of the generator with synchronous movement, also generating two opposite outputs, but with slower charge rise

The presence of the grounded plates 3 and 4 is not essential for the operation of the bipolar generators. Fixed capacitances to ground from plates 2 and 5 would be enough, at the expense of slower startup due to smaller multiplication factor, which can be calculated as in [19]. In the case of the unipolar generator, however, these plates are necessary for the charge-splitting operations unique to this structure, using electrostatic repulsion of equal charges instead of attraction of opposite charges, and must exist. C_b and C_c may be fixed in this structure, but also causing smaller multiplication factor.

5 Efficiency

The proposed generators are not the most efficient. They waste significant power in the high-voltage Zener diodes of the bias generators. The most powerful generator, the one of Fig. 18, wastes i_zV_b = 0.233 µA × (240–4.6) V = 54.8 µW in the Zener diode, with just 4.2 µW of effective output power. The waste can be reduced by decreasing the size of the bias generating section, using the efficient capacitive converters to generate a greater fraction of the output current, but this would also increase the startup times of the devices. It is also possible to use something as in [15], stopping the operation of the bias generator when the desired bias is reached, but this requires a control system. Examples of the same 3D-printed generator coupled to a DC/DC converter are presented in [17], generating more than 10 µW with a maximum of 140 V at the input of the converter, even with losses in the converter and in the resistance of the conductive plastic. The generators described here use 240 V of bias, but produce at most less than half this power. The reason for the large difference is that with a DC/DC converter the generated power depends on the square of the input voltage of the converter, independently of the output voltage, while for the capacitive converters it is proportional to the product of the bias and output voltages for low output voltages, the product of (1) and V_o. For larger output voltage the power varies as an inverted parabolic function, reaching a maximum (8) when the output voltage is at (9). For the converters used, with 15–165 pF capacitances, V_b = 240 V, f = 7.14 Hz, and V_d = 0.6 V, P_max = 153 µW, a large gain, but with the optimum V_o = 1196.4 V, impractically high. If larger load resistances were used in the examples, the output power would grow almost in the same proportion even for output voltages in the tens of volts.

$$P_{max} = f\frac{{(C_{max} (V_{b} - V_{d} ) - C_{min} (V_{d} + V_{b} ))^{2} }}{{4C_{min} }}$$

(8)

$$V_{o\,optimum} = \frac{{C_{max} (V_{b} - V_{d} ) - C_{min} (V_{d} + V_{b} )}}{{2C_{min} }}$$

(9)

There is loss in the diodes of the capacitive converters, significant if the output voltage is low, as in the examples. The power loss there can be estimated by multiplying (1) by V_d, and is the same in both diodes, which have the same average current. The effect in the output power is small, however. For a single converter with the parameters above and V_o = 3.3 V, the diodes consume 0.15 µW each. The output power with V_d = 0 would be 0.847 µW, while with V_d = 0.6 V it drops just 0.3% to 0.845 µW. The structures are attractive due to their simplicity, and are insensitive to the high resistivity of the plastic used to make the variable capacitors and to diode voltage drops, but are deficient in the output power for low output voltage.

6 Comparison with similar works

The described generators can just be compared with others in the literature when no DC/DC converter is used, and the exponential voltage growth of the unstable circuit section is stopped by some mean. Two cases were selected. The work [14] proposes just inserting a Zener diode in series with the output of the Bennet’s doubler circuit, resulting in an equivalent to Fig. 4 without the capacitive converters. The output current would be identical to (2). Using the same parameters used here, Fig. 5 indicates that the output voltage would be of just ~ 1 V, the difference of the two curves, with most of the extracted power dissipated in the Zener as discussed above. In [15] an interesting concept is proposed, of stopping the operation of the bias generator when the desired bias voltage is reached. This eliminates the loss in the Zener diode, but requires a control circuit. Similar operations could be applied to all the bias generators described in this work, eliminating the large losses in Zener diodes, but also the currents through them contributing to the output. In cases where low-energy mechanical systems are used to operate the variable capacitors, the reduced loss may redirect more mechanical power to the capacitive generators, turning the approach attractive.

7 Conclusions

A study was made about possible structures for electrostatic generators suitable for energy harvesting applications, without requirement of a control system, using as experimental verification a generator made with 3D printing techniques. In the described possibilities, the two double variable capacitors of the experimental generator were connected in several ways, resulting in structures using biased variable-capacitance generators where the bias voltages are generated by unstable generators, all made with variable capacitors and diodes. In some of the structures, the variable capacitors of the bias generator were also used for output energy generation, as in Fig. 18 and in the symmetrical generators, particularly in the case of the unipolar generator. The experimental generators use back-and-forth movement, but the same ideas can be applied to rotating movement. The movement does not have to be regular, but the capacitances must vary sufficiently, especially at the bias generator to start the voltage rise and keep it. The variation must be fast enough to overcome leakage in the diodes and insulators, what in the experimental setup was not a problem, but at high temperature may be a problem due to the increased reverse current of diodes. Table 1 lists a summary of the properties of the generators. The minimum capacitance variations come from [2, 19]. The maximum startup multiplication factor assumes loading as used, which reduces the multiplication factor of the structures based on Bennet’s doubler, but does not affect the symmetrical structures.

Table 1 Properties of the proposed generators

The exact analysis of the proposed structures is substantially complex, but it was shown that the assumption of fixed bias and output voltages works well for all structures, except for the unipolar symmetrical generator, which had its behavior verified only by simulation. The experimental generators were observed to require an initial excitation to operate, what was not observed in [16], with the bigger experimental generator used, which probably acted as a more effective antenna for ambient electrical interference. An explicit antenna connected to a high-impedance node of the structure (as node 6 of the symmetrical generators) was found to be effective for excitation of the small generator, however. See [13] for an example of excitation of similar generators using antennas to provide startup energy. The described structures are, in principle, compatible with MEMS construction. They require two complementary variable capacitors, which can be built in several ways. The structure used for the 3D-printed generator, convenient for that realization, does not appear directly suitable for MEMS realization because it is not planar, but Figs. 17, 19, 20 and 21 suggest another possibility using flat plates moving above others with variable superposition, maybe with several levels. Realizations using elastic membranes for the variable capacitors [8] are surely possible too, with adaptations to allow complementary pairs of variable capacitors.