Diode-capacitor voltage divider for electrostatic microelectromechanical generator

. The simulation results of the voltage divider operation based on switchable capacitors with diode switching for electrostatic microelectromechanical (MEM) generators are presented. It has been established that for the divider with diode-switched capacitors, the voltage dividing coefficient is not remain the same when the load resistance changes, and for higher load resistances, the load voltage asymptotically approaches the value of the power source. Analytical expressions have been obtained for estimating the parameters of the divider with diode switching, which allow evaluating its characteristics at the preliminary stage of the MEM generator design. It has been shown that the simulation of the diode-capacitor voltage divider operation has to be carried out taking into account the dynamic volt-ampere characteristics of diodes.


Introduction
Over the past 20 years, the energy harvesting from environmental sources to power lowpower electronic devices, such as intelligent wearable systems, biomechanical implants and wireless sensor network nodes, has attracted much attention of academic and industrial communities [1][2][3][4][5][6].
Kinetic energy of surrounding vibrations, including human activity motion, is one of the most common energy sources in the environment.Therefore, considerable efforts are being made to develop high-efficient energy harvesters for converting mechanical kinetic energy into electrical one.Currently, autonomous power sources based on electrostatic microelectromechanical (MEM) energy converters (MEMC) located directly at the location of autonomous devices, such as nodes of wireless sensor networks, are being actively developed and investigated.
The operation of electrostatic MEMC is based on a change of the variable capacitor capacitance under the influence of external mechanical oscillations (vibrations) [4,7].In order to transfer the electrical energy obtained using the MEMC to the load, the converter is combined with the corresponding electrical circuits (conditioning or interface circuits) [8][9][10][11][12][13].Such a system as a whole is called as microelectromechanical generator (MEMG).
The MEMG parameters significantly depend both on the design and characteristics of the MEMC and on the conditioning circuit.Therefore, the choice of the interface circuit during the design stage of the MEMG is one of the most important tasks.
The development and improvement of MEMC and microgenerators based on them are primarily aimed at increasing of the generated electrical power.The energy that can be stored in the capacitor and extracted from it is proportional to the square of the voltage across the capacitor.So, in order to increase the generated electrical power, electrostatic MEMG are very often developed with the highest possible operating voltage.
In order to increase the operating voltage, MEMC with electrets charged to several hundred volts and/or special circuits exploiting voltage multipliers are used to generate a high bias voltage from an initially low voltage provided, for example, by a battery or ambient noise [14][15][16][17].
However, most of the modern electronic devices operate with supply voltages of less than 10 V.Moreover, the trend towards miniaturization of electronic circuits requires further reduction of their supply voltage.Therefore, there is a need to develop microgenerators with a reduced output voltage that, in the case of electrostatic MEMG, requires the conversion of a high AC voltage to a low DC voltage.Such a transformation can be carried out by means of a transformer, which will significantly worsen the weight and size parameters of the MEMG.As a result, the most promising way seems to be the transformer-free conversion of alternating voltage to constant voltage with simultaneous combination of various elements functions of the rectifier block circuit.
There are publications on transformer-free voltage dividers in literature.However, they assume that the primary voltage source has infinite power (infinite power bus), but electrostatic MEMG develops significantly less power.In addition, in power electronics, when working with large capacities, losses in converter control systems are not so significant.Therefore, to achieve good converter performance, individual control of each transistor is used, which is also not acceptable in microgenerators.
This paper presents the results of study on the development of a voltage divider using switchable capacitors for electrostatic MEM generators with a reduced output voltage.A feature of the operation of such DC-DC systems with voltage reduction is that their structure changes twice during the switching period.Due to this, during one part of the switching period, the capacitors are charged from the primary source, and during the other part they are discharged.

Model
The operation of transformer-free low-voltage rectifiers with voltage division on capacitors is based on the charge of N series-connected capacitors Ci from a high-voltage source and the subsequent discharge of these capacitors to the load, but already connected in parallel.
When N capacitors are sequentially charged, the voltage of the primary source at C1 = C2 = … = CN is divided almost equally between them.As a result, the voltage across one of the capacitor is N times less than the amplitude of the primary source voltage.During switching of the capacitors for discharge to the load, the output voltage of the rectifier will be equal to the voltage across one of the capacitor, and the load current will be N times greater than the current consumed from the primary source.
Thus, capacitors with elements of restructuring the charge circuit into a discharge circuit and vice versa represent the basis of the energy storage and transfer scheme with simultaneous voltage division.
As a result of the analysis of the known circuits of dividers on switchable capacitors [18][19][20][21][22][23][24], the most promising for use in electrostatic microgenerators is the voltage division circuit with diode switching, shown in Fig. 1.It contains N switchable capacitors Ci, (N-1) charging diodes Dch and 2(N-1) discharge diodes Ddis.Such a voltage divider can be used not only in electrostatic MEM generators, but also in other systems where it is necessary to reduce the voltage.

Results and Discussion
Using the model of ideal diodes and assuming that the capacitances of all capacitors Ci are the same, it can be shown that with periodic working of the switches with a period T, the dependence of the maximum voltage across RL (voltage on RL at the beginning of the discharge cycle of capacitors Ci to the RC load chain) on the number of the divider operation cycles can be represented as where During the operation cycle, the load voltage will change from the maximum VL,max,m to the minimum where Δτ is the time interval for connecting the divider to the RC load chain,  Fig. 3 shows the dependences of the maximum voltage on the load and on one (each) of the capacitor divider Ci in steady-state mode versus the load resistance, calculated with the same circuit parameters as for Fig. 2. It should be noted that the maximum voltage across the capacitors Ci is reached when the divider is connected to the capacitor CS (Sw2 is closed, Sw1 and Sw3 are opened), but the maximum voltage across the load resistance is reached at the time when the divider is connected to the load (Sw3 is closed, and Sw2 is opened).
It can be seen from Fig. 3 that the maximum load voltage of this divider approaches the value V0/N only at high load resistances.This is due to the fact that at low RL, the output voltage and the charge accumulated in CL decrease by so much at the end of the conversion cycle that they cannot be fully restored to V0/N when CS is connected (the charge on CS is limited by the value of V0CS).As a result, an equilibrium between the charge given to RL and the charge taken by the divider from CS occurs at the load voltage less than V0/N, which can be seen from Fig. 4, where the dependencies shown in Fig. 2 are recalculated at RL = 10 5 Ω.In this case, the voltage on the load during the operation cycle of the divider decreases almost to zero, and every time by the beginning of a new cycle, the divider is discharged to the same voltage.In this case, the accumulation of charge does not occur, but the amount of charge taken from the CS increases.
To verify the expressions obtained, the corresponding calculations were carried out in the OrCAD PSpice software package using static current-voltage characteristics of the diodes.Fig. 5 (1, 2, 5, 6) shows the dependences of the maximum and minimum voltages on the load resistance in steady-state mode versus the load resistance, calculated using expressions (1), ( 2   One can see from Fig. 5 that the dependences obtained using analytical and numerical calculations are qualitatively consistent.However, the values obtained using the OrCAD PSpice software package are less than the corresponding values obtained using analytical expressions.We suppose that this discrepancy is due to the use of the ideal diode model in analytical evaluations, where the diodes open at zero voltage.Assuming that the diodes open at voltage VD, the expressions for VL,max and VL,min in steady-state mode take the form The corresponding dependences calculated using ( 3) and ( 4) at VD = 0.1 V are shown in Fig. 5 (3 and 4).It can be seen that in this case, the quantitative agreement between analytical and numerical calculations is quite good.
Thus, it can be noted that the idea found in literature about the operation of this type of dividers agrees quite well with the results of calculations using static current-voltage characteristics of the diodes.At the same time, during the preliminary design stage, it is better to use analytical expressions which take into account the non-zero opening voltage VD of the diodes.
It should be also noted that during the charge phase of the divider capacitors, simultaneously with the charge of the main capacitors Ci, the capacitances of the diodes connected in the opposite direction are charged too.Moreover, these CD capacitances are charged to voltages which are several times higher than the voltages across Ci.As a result, when the divider is connected to the load at the first moment, the voltage at each cascade of the divider turns out to be greater than V0/N (i.e.greater than the voltage on Ci).
In order to analyze the effect of the diode capacitance on the parameters of the divider, the corresponding calculations were performed in the OrCAD PSpice software package, taking into account the dynamic current-voltage characteristics of the diodes.
Similar dependencies for the voltage divider with four cascades calculated at V0 = 12 V are shown in Fig. 7.It can be seen that with taking into account the capacitances of the diodes, these dependences at high load resistances asymptotically approach the voltage V0, but not the voltage V0/N, i.e. the divider does not work as predicted in literature.However, for load resistances of less than 300 MΩ, the dependences correspond to calculations carried out using static diode models.
Assuming that if the divider is connected to the load at high load resistances, when a small charge is taken from the divider, the capacitances of the discharging diodes Ddis are not fully discharged, the diodes Ddis do not open, and the expressions for VL,max and VL,min in steady-state mode can be represented as L,max  At the same time, at low load resistances, when a significant charge is taken from the divider during connection to the load, the expressions for VL,max and VL,min will not change in steady-state mode, and will correspond to expressions (3) and ( 4).
It is obvious that in this case there is a fairly good quantitative agreement between analytical and numerical calculations.Thus, expressions (3)-( 6) can be used at the stage of preliminary design of MEM generators with diode-capacitor voltage dividers.Moreover, for estimation the magnitude of the critical load resistance, at which it is necessary to switch from using expressions (3) and (4) to using ( 5) and ( 6), we can suggest the following expression: During the derivation of the equation (7), it was assumed that the divider is connected to the load for a very short time Δτ.

Conclusions
In general, the analysis of the operation of the diode-capacitor voltage divider showed that: -the voltage division coefficient of this circuit is not remain the same when the load resistance changes, and at high load resistances, the voltage value across the load asymptotically approaches V0, but not V0/N.As a result, when the circuit parameters are changed, it is necessary to evaluate the division coefficient again each time; -during modeling of such divider operation, it is necessary to take into account voltage drops across open diodes and the capacitance of the diodes; -simulation of the diode-capacitor voltage divider operation using software systems should be carried out taking into account the dynamic current-voltage characteristics of the diodes; -the obtained analytical expressions describing the operation of the diode-capacitor voltage divider at low and high load resistances, taking into account voltage drops on open diodes and capacitance of diodes, enable to calculate the parameters of the divider with sufficient accuracy for practical applications.Therefore they can be used at the stage of preliminary design of MEM generators with diode-capacitor voltage dividers.

Fig. 1 .
Fig. 1.Electrical circuit of the transformerless diode-capacitor voltage divider.Two cycles can be distinguished during the operation of this voltage divider.At the first cycle of operation (the switches Sw1 and Sw3 are closed, the switch Sw2 is opened), the capacitor CS is charged from the primary source V0 to the value of V0, and at the same time, N parallel-connected capacitors Ci are discharged through 2(N-1) discharging diodes Ddis to the RC load chain.At the second cycle (Sw1 and Sw3 are opened, Sw2 is closed), the capacitor CS is discharged to N series-connected capacitors Ci through the (N-1) charging diode Dch, and, thus, Ci is charged.Then the push-pull cycle of the divider is repeated.Such a voltage divider can be used not only in electrostatic MEM generators, but also in other systems where it is necessary to reduce the voltage.

E3S
Fig. 2 shows time dependences of the load resistance voltage and the voltage across one (each) of the divider capacitor Ci calculated at V0 = 9 V, CL = Ci = 1 nF, T = 1 ms, Δτ = 0.5T, CS = 2 nF, N = 3 and RL = 10 7 Ω.It can be seen that the steady state occurs with time.Fig. 2 also clearly shows two cycles of the divider operation.During the first cycle (Sw1 and Sw3 are closed, Sw2 is opened) N parallel-connected capacitors Ci and CL are discharged to the RC load chain (Fig. 1) with a time constant τ1 = RL(NCi+CL).During the second cycle (Sw1 and Sw3 are opened, Sw2 is closed), the capacitor CS instantly recharges N capacitors Ci which are already connected in series.After that, the voltage on each capacitor Ci remains constant until the end of the cycle.At the same time, the capacitor CL continues to discharge through RL, but faster, with a time constant τ2 = RLCL.Then the push-pull cycle of the divider is repeated taking into account the redistribution of charges in the previous cycle.

Fig. 2 .
Fig. 2. Time dependences of the voltage: 1 -on the load and 2 -on one (each) of the divider capacitor.

Fig. 3 .
Fig.3shows the dependences of the maximum voltage on the load and on one (each) of the capacitor divider Ci in steady-state mode versus the load resistance, calculated with the same circuit parameters as for Fig.2.It should be noted that the maximum voltage across the capacitors Ci is reached when the divider is connected to the capacitor CS (Sw2 is closed, Sw1 and Sw3 are opened), but the maximum voltage across the load resistance is reached at the time when the divider is connected to the load (Sw3 is closed, and Sw2 is opened).It can be seen from Fig.3that the maximum load voltage of this divider approaches the value V0/N only at high load resistances.This is due to the fact that at low RL, the output voltage and the charge accumulated in CL decrease by so much at the end of the conversion cycle that they cannot be fully restored to V0/N when CS is connected (the charge on CS is limited by the value of V0CS).As a result, an equilibrium between the charge given to RL and the charge taken by the divider from CS occurs at the load voltage less than V0/N, which can be seen from Fig.4, where the dependencies shown in Fig.2are recalculated at RL = 10 5 Ω.In this case, the voltage on the load during the operation cycle of the divider decreases almost to zero, and every time by the beginning of a new cycle, the divider is discharged to the same voltage.In this case, the accumulation of charge does not occur, but the amount of charge taken from the CS increases.To verify the expressions obtained, the corresponding calculations were carried out in the OrCAD PSpice software package using static current-voltage characteristics of the diodes.Fig.5(1,2,5,6) shows the dependences of the maximum and minimum voltages on the load resistance in steady-state mode versus the load resistance, calculated using expressions (1),(2) and in the OrCAD PSpice software package, at V0 = 9 V, CL = Ci = 1 nF, T = 20 ms, Δτ = 2 ms, CS = 1 nF and N = 3.

Fig. 4 .
Fig. 4. Time dependences of the voltage calculated at RL = 10 5 Ohm: 1 -on the load and 2 -on one (each) of the divider capacitor Ci.

Fig. 6 .
Fig.6.Dependences of the maximal and minimal voltages across the load resistance in the steady state versus the load resistance, calculated taking into account CD of the diodes: 1 and 3 -the results of VL,max calculations using (3), (5) and in PSpice; 2 and 4 are the results of VL,min calculations using (4), (6) and in PSpice, respectively.

Fig. 7 .
Fig. 7. Dependences of the maximal and minimal voltages across the load resistance in the steady state versus the load resistance, calculated taking into account CD of the diodes at V0 = 12 V and N = 4: 1 and 3 are the results of VL,max calculations using (3), (5) and in PSpice; 2 and 4 are the results of VL,min calculations using (4), (6) and in PSpice, respectively.