**High Energy Density Capacitance Microgenerators**

Igor L. Baginsky and Edward G. Kostsov

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48524

## **1. Introduction**

The problem of continuous production of energy sufficient for modern microcircuits with an almost unlimited service life should be related to searching for power sources in the ambient medium. The comparison of these sources shows that only solar energy and energy of mechanical vibrations of surfaces of various solids can be used for generation of electrical energy in the milliwatt or microwatt range, which is enough for powering these microcirquits.

A typical feature for most modern sources of mechanical vibrations (surfaces of solids) is moderate amplitudes ranging from 0.1 to 2.0 *μ*m; the analysis of the frequency distribution of amplitudes shows that low frequencies (1–100 Hz) have the most power [1]. Examples are vibrations of various building structures: supports, bridges, roadbeds, building walls, etc.

There are numerous recent publications that describe the development of microgenerators of electrical energy, including microelectromechanical systems (MEMS generators) capable of converting mechanical energy from the ambient medium to electrical energy. A new term, "energy harvesting," was accepted [2–11]. MEMS generators can be fabricated in a single technological cycle with fabrication of the basic microcircuit. The problem of powering MEMS devices is recognized as one of the most important issues in modern microelectronics.

Electrostatic energy microgenerators seem to be the most suitable for this task, because fuel or chemical elements need to be refined or renewed, solar or thermo- elements are not suitable for all situations of MEMS operation, electromagnetic generators of energy are ineffective in the range of low-amplitude vibrations and small sizes of the transducers, while piezoelectric generators are ineffective at low frequencies of vibrations [12-14].

Electrostatic generators have been known for a long time. Their operation principle is based on the work of mechanical forces transferring an electrical charge against

electrostatic forces of attraction of unlike charges [15]. Depending on the method of generation and transportation of this charge, generators can be divided into two classes. In the first class, the charge generated by some external action, for instance, by an electric arc or friction, is transferred by a transporter: a belt (Van de Graaf generators) [15] or a disk (friction machines). In the second class, the charged plate of the capacitor moves. Depending on the presence or absence of a built-in charge in this capacitor such devices are classified as either electret [16,17] or capacitance generators , e.g., Toepler or Felichi machines [15].

For electrostatic capacitance machine (Fig.1a) the separation of the plates (vertical, i.e., outof-plane, or lateral, i.e., in-plane) of the capacitor *С(t)* initially charged from the voltage source *V0* up to the value *Q0= CmaxV0* (where *Cmax* is initial maximal value of capacitance) in the conditions of open circuit results in growth of voltage on the capacitor up to the value

$$\mathbf{V}\_{\text{max}} = \mathbf{V}\_{\text{min}} \mathbf{C}\_{\text{max}} / \mathbf{C}\_{\text{min}} \tag{1}$$

Here for the case under consideration *Vmin=V0*. And, respectively, the energy of capacitor is changed from *Wmin = CmaxV02/2* to

$$\mathcal{W}\_{\text{max}} = \frac{\mathcal{C}\_{\text{max}}}{\mathcal{C}\_{\text{min}}} \mathcal{W}\_{\text{min}} = \mathfrak{\eta} \mathcal{W}\_{\text{min}} = \frac{Q\_0^2}{\mathcal{Z} \mathcal{C}\_{\text{min}}} \ , \tag{2}$$

where *η=Cmax/Cmin* is capacitance modulation depth.

The produced electric energy *Wmax-Wmin* is transferred to load *R.* After that, the capacitor plates return to the initial position and are charged by the voltage source, and the energy conversion process is repeated. The power developed by such a generator is *P=(Wmax-Wmin)f* (where *f* is the repetition frequency of conversion cycles), and the efficiency of energy conversion, i.e. the ratio of energy produced by the generator during the conversion period to energy losses during the same period, is *η-1*.

Drawbacks of capacitance machine are the necessity of powering the generator by voltage source *V0* once at each cycle of energy conversion to charge the capacitor *C(t)* and also the need in use of the key, synchronized with the phase of *C(t)* alteration, switching the capacitor to the voltage source V0, in the open circuit, and to the load *R*.

For these reasons, the circuit shown in Fig. 1a had limited applications: only for generation of high voltages (up to several hundreds of kilovolts) in solving special engineering problems.

This type of generators is used in many electrostatic MEMS generators [1,12,18-23]. Their specific power is low not exceeding 1-10 μW/cm2 because of high value of minimum interelectrode gap at low *η*. It is possible to increase the generator power by decreasing the gap between the electrodes, but it results in the rising probability of an electrical breakdown in the gap. Low value of specific power is the main drawback of these generators; another problem is the necessity of using DC voltage sources.

#### High Energy Density Capacitance Microgenerators 63

62 Small-Scale Energy Harvesting

machines [15].

problems.

changed from *Wmin = CmaxV02/2* to

where *η=Cmax/Cmin* is capacitance modulation depth.

to energy losses during the same period, is *η-1*.

problem is the necessity of using DC voltage sources.

electrostatic forces of attraction of unlike charges [15]. Depending on the method of generation and transportation of this charge, generators can be divided into two classes. In the first class, the charge generated by some external action, for instance, by an electric arc or friction, is transferred by a transporter: a belt (Van de Graaf generators) [15] or a disk (friction machines). In the second class, the charged plate of the capacitor moves. Depending on the presence or absence of a built-in charge in this capacitor such devices are classified as either electret [16,17] or capacitance generators , e.g., Toepler or Felichi

For electrostatic capacitance machine (Fig.1a) the separation of the plates (vertical, i.e., outof-plane, or lateral, i.e., in-plane) of the capacitor *С(t)* initially charged from the voltage source *V0* up to the value *Q0= CmaxV0* (where *Cmax* is initial maximal value of capacitance) in the conditions of open circuit results in growth of voltage on the capacitor up to the value

Vmax = Vmin Cmax/Cmin. (1)

Here for the case under consideration *Vmin=V0*. And, respectively, the energy of capacitor is

The produced electric energy *Wmax-Wmin* is transferred to load *R.* After that, the capacitor plates return to the initial position and are charged by the voltage source, and the energy conversion process is repeated. The power developed by such a generator is *P=(Wmax-Wmin)f* (where *f* is the repetition frequency of conversion cycles), and the efficiency of energy conversion, i.e. the ratio of energy produced by the generator during the conversion period

Drawbacks of capacitance machine are the necessity of powering the generator by voltage source *V0* once at each cycle of energy conversion to charge the capacitor *C(t)* and also the need in use of the key, synchronized with the phase of *C(t)* alteration, switching the

For these reasons, the circuit shown in Fig. 1a had limited applications: only for generation of high voltages (up to several hundreds of kilovolts) in solving special engineering

This type of generators is used in many electrostatic MEMS generators [1,12,18-23]. Their specific power is low not exceeding 1-10 μW/cm2 because of high value of minimum interelectrode gap at low *η*. It is possible to increase the generator power by decreasing the gap between the electrodes, but it results in the rising probability of an electrical breakdown in the gap. Low value of specific power is the main drawback of these generators; another

capacitor to the voltage source V0, in the open circuit, and to the load *R*.

max min min

*C Q W WW*

max 0

min min 2

2

*C C* , (2)

**Figure 1.** Various circuits of capacitance electrostatic generators: (**a**) capacitance machine; (**b**) electret current generator; (**c**) electret voltage generator; (**d**) ideal two-capacitor generator; (**e**) two-capacitor generator with loss compensation by a current source; (**f**) two-capacitor generator with loss compensation by a voltage source; *V0* is the voltage source, *I0* is the current source, *R* is the load resistance, and *r* is the leakage resistance.

**Figure 2.** The schematic representation of the designs of capacitance generators, corresponding to the circuits presented on Fig.1: (**a**) – electret current generator, out-of-plane fabrication (see Fig.1b), (**b**) – electret voltage generator, out-of-plane plane fabrication (see Fig.1c), (**c**) – electret current generator, inplane fabrication (see Fig.1b), (**d**) – two-capacitor generator, in-plane fabrication (see Fig.1d). 1 – moving substrate, 2 – stationary substrate, 3 – metal electrode, 4 – electret, 5 – spring, 6 – dielectric layer, 7 – contact 3 (Fig.1c), 8 – contact 1 (Fig.1c).

In electret generators [16,17,24-36] (see the circuits in Fig.1b,c and scematic designs in Fig.2a-c) the dielectric layer with built-in charge is formed on the inner surface of fixed plate of capacitor *C(t)* , and the charge losses are compensated by electrostatic induction of charge on the surface of moving plate, that is a considerable advantage of this way of energy transformation. These generators are further divided into current generators in which the capacitor *C(t)* is directly connected to the load *R* (see Fig.1b and Fig.2a,c) and voltage generators in which the capacitor

*C(t)* is switched just like in capacitance machine but without the voltage source: *V=0*, Fig.1c and Fig.2b. Compared to voltage generators the current generators have the advantage in the simplicity of the circuit of energy transformation. The drawback is lack of the effect of generated voltage amplification, and correspondingly of the output power, proportional to capacitance modulation depth *η*. It should be noted that the "in-plane" constructions of electret generators (with lateral shift of the generator plate) having a comb structure of electrodes are being actively developed now. These devices are simple in production, and the technology earlier developed for smart sensors is used for their fabrication. For these devices the specific power of order 100 μW/cm2 was reached [36]. The further increase of specific power is impeded by large interelectrode gaps used here, of order 20 μm.

The two-capacitor mode of capacitance energy transformation is described in [5,37,38], see Fig.1d and schematic design in Fig.2d. The electric energy is generated by means of capacitance alteration in antiphase of two capacitors (*Ci(t), i=1,2*), initially charged to potential *V0*, under the action of the force on their moving plates. In this case there is no need to feed the capacitors by switching on the voltage source on each cycle of energy transformation, because both capacitors *Ci* alternate in playing this role. The electric power is generated in the load *R* as the current flows from the recharging capacitors. If the generator has initial charge distributed between capacitances *C1* and *C2* then in idealized case with no leakage currents the circuit could operate for unlimitedly long time producing the energy under periodical action of mechanic force.

This approach has been proposed first in [37] as an idea. The evaluations of the efficiency of energy transformation were done in [5] taking into account the compensation of charge losses in the capacitors *Ci* by current source *I0*, Fig.1e. However the total analysis of the generator operation at all possible loads and frequencies of generation, and at various ways of capacitance modulation and compensation of charge losses by connecting the current (Fig.1e) or voltage (Fig.1f) source have not been done.

The present work is aimed at performing the analysis of specific features of operation of electrostatic capacitance generators that do not need the electrical energy sources to compensate for charge leakages permanently, at each cycle of energy transformation. For the sake of generality only electric part of the generator will be analyzed under the modulation of capacitances of generating capacitors both by means of changes of interelectrode gaps and also by lateral shift of capacitors plates. A partial experimental verification of the results of the model proposed will be done.

### **2. High energy density electret generators**

According to the common definition the ability of dielectric (or ferroelectric) to retain the charge produced in the bulk or on the surface of the layer by external action is called the electret effect. So we assume that this charge is constant in the time of generator action.

We will analyze first the vibration mode of this generator when the area of capacitor pates overlapping is constant and the distance between them is varied, or so called out-of-plane vibration mode.

The circuit of the generator considered is shown in Fig. 1, *b,c* and the schematic representation of their designs is represented in Fig.2 a,b. Its structure includes four thin layers: an electrode, a dielectric (ferroelectric) of thickness *d*, having a space charge ρ(x,t), an air gap with a variable thickness *d1*(*t*) changing in time under the action of mechanical forces, and a moving electrode performing oscillatory motions with respect to the dielectric surface. In the general case, the operation principle of such an energy generator is the following. At *t=0* in each layer of the structure including the metallic electrodes the initial distribution of charge and corresponding distribution of electric field *E(x,0)* is set. The value of *E*(*x, t*) is determined by the charge distribution in the dielectric layer and the voltage on the structure *V(t)*; the field in the electrodes is screened by the charge formed in them (Fig. 3).

64 Small-Scale Energy Harvesting

interelectrode gaps used here, of order 20 μm.

the energy under periodical action of mechanic force.

(Fig.1e) or voltage (Fig.1f) source have not been done.

**2. High energy density electret generators** 

of the model proposed will be done.

vibration mode.

*C(t)* is switched just like in capacitance machine but without the voltage source: *V=0*, Fig.1c and Fig.2b. Compared to voltage generators the current generators have the advantage in the simplicity of the circuit of energy transformation. The drawback is lack of the effect of generated voltage amplification, and correspondingly of the output power, proportional to capacitance modulation depth *η*. It should be noted that the "in-plane" constructions of electret generators (with lateral shift of the generator plate) having a comb structure of electrodes are being actively developed now. These devices are simple in production, and the technology earlier developed for smart sensors is used for their fabrication. For these devices the specific power of order 100 μW/cm2 was reached [36]. The further increase of specific power is impeded by large

The two-capacitor mode of capacitance energy transformation is described in [5,37,38], see Fig.1d and schematic design in Fig.2d. The electric energy is generated by means of capacitance alteration in antiphase of two capacitors (*Ci(t), i=1,2*), initially charged to potential *V0*, under the action of the force on their moving plates. In this case there is no need to feed the capacitors by switching on the voltage source on each cycle of energy transformation, because both capacitors *Ci* alternate in playing this role. The electric power is generated in the load *R* as the current flows from the recharging capacitors. If the generator has initial charge distributed between capacitances *C1* and *C2* then in idealized case with no leakage currents the circuit could operate for unlimitedly long time producing

This approach has been proposed first in [37] as an idea. The evaluations of the efficiency of energy transformation were done in [5] taking into account the compensation of charge losses in the capacitors *Ci* by current source *I0*, Fig.1e. However the total analysis of the generator operation at all possible loads and frequencies of generation, and at various ways of capacitance modulation and compensation of charge losses by connecting the current

The present work is aimed at performing the analysis of specific features of operation of electrostatic capacitance generators that do not need the electrical energy sources to compensate for charge leakages permanently, at each cycle of energy transformation. For the sake of generality only electric part of the generator will be analyzed under the modulation of capacitances of generating capacitors both by means of changes of interelectrode gaps and also by lateral shift of capacitors plates. A partial experimental verification of the results

According to the common definition the ability of dielectric (or ferroelectric) to retain the charge produced in the bulk or on the surface of the layer by external action is called the electret effect. So we assume that this charge is constant in the time of generator action.

We will analyze first the vibration mode of this generator when the area of capacitor pates overlapping is constant and the distance between them is varied, or so called out-of-plane

**Figure 3.** The schematic of the field distribution in the structure metal – dielectric with built-in charge – air gap – moving electrode at the initial state (solid lines) and in the phase of maximal plates shift (dashed lines): (**a**) – at the conditions of open circuit (voltage generator, fig.1c), and (**b**) – closed circuit (current generator, fig. 1b).

When the initial state is violated, i.e., the value of *d*1(*t*) is changed by mechanical forces, a redistribution of *E*(*x, t*) occurs in each layer, accompanied by the current flow in the load circuit connecting the electrodes. When the current passes through the resistance *R*, the energy characterizing the energy parameters of the generator is released.

At the moment, the class of generators of electrical energy, which operate on the basis of the above mentioned principle, is well known; they are implemented in practice and are called electret generators [16]. The dielectric used in such generators belongs to a large group of materials that can retain the surface charge for a long time. The electrets [17] differ in the method of generation of this charge and in the form of the distribution of the stored charge and its sign. It can be either uniformly or nonuniformly distributed over the layer thickness and can have either an identical sign (monoelectret) or two different signs (geteroelectret). The electrets used in energy generators are sufficiently thick layers with the minimum thickness of 5–10 μm, and the information on their fabrication by microelectronics technologies is insufficient.

### **2.1. Model. Basic equations that describe the effect of energy generation in electrostatic machines with a dielectric containing an embedded charge**

To describe the general features of operation of the generator considered, we assume that the dielectric contains a charge *ρ*(*x, t*), which can change with time, with a surface density Q (t) <sup>P</sup> ρ(t)x(t) , where ( )*t* is the density of the space charge *ρ*(*x, t*) averaged over the dielectric thickness and *x t*( ) is its centroid.

Analyzing the behavior of the total current in this structure with variations of *d*1(*t*) for given values of *d*, ( )*t* , *x t*( ) we use a system of the classical one-dimensional equations consisting of the expressions for the total current *j*(*t*) and conductivity current *j*c(*x, t*), equation of continuity, Poisson's equation, and expression that determines the potential V(t) between the electrodes at each instant of time. Taking into account that for electrets the space charge is constant for all time of the process, we have:

$$j(t) = \varepsilon \varepsilon\_0 \frac{\partial E(\mathbf{x}, t)}{\partial t} \tag{3}$$

$$\frac{\partial E(\mathbf{x},t)}{\partial \mathbf{x}} = -\frac{\mathbf{p}(\mathbf{x})}{\varepsilon \mathbf{x}\_0} \tag{4}$$

$$\int\_{0}^{d\_1(t) + d} E(\mathbf{x}, t) d\mathbf{x} = -V(t) \quad \text{ }\tag{5}$$

Integrating both parts of equation (4) with respect to the coordinate *x*, taking into account that the field increases in a jumplike manner by a factor of *ε* on the free boundary of the dielectric, i.e.,

$$\varepsilon E(d\_{-}, t) = E(d\_{+}, t) \,, \tag{6}$$

and substituting the formula derived for *E*(*x, t*) into equation (5), we obtain expressions for the field on the boundaries *x* = 0 and *x* = *d*1(*t*) + *d*: *Ec*(*t*) and *EA*(*t*), and, correspondingly, for the specific charge induced on these boundaries:

$$Q\_c(t) = -\varepsilon \varepsilon\_0 E\_c(t) = C(t) \left( V(t) + V\_p \right),\tag{7}$$

$$\mathcal{Q}\_S(t) = \varepsilon\_0 E\_A(t) = -\mathcal{C}(t) \left( V(t) + V\_{\mathcal{P}} \right) \tag{8}$$

$$\text{C}(t) = \frac{\varepsilon \mathfrak{e}\_0}{d + \varepsilon d\_1(t)} \quad \text{s} \tag{9}$$

$$V\_P = \frac{Q\_P}{C\_F} \quad \text{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\textsuperscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxscript{\boxcong}}}}}}}}}}}}}}}}}}} \ \phantom{\overleftarrow{\boxminus{\textbf{\varleft}}}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \end{smallmatrix}} \tag{10}$$

where *CF* = *εε0/d* is the specific capacitance of the dielectric layer.

Therefore, according to equations (3) and (8):

66 Small-Scale Energy Harvesting

dielectric, i.e.,

dielectric thickness and *x t*( ) is its centroid.

is constant for all time of the process, we have:

the specific charge induced on these boundaries:

**2.1. Model. Basic equations that describe the effect of energy generation in electrostatic machines with a dielectric containing an embedded charge** 

To describe the general features of operation of the generator considered, we assume that the dielectric contains a charge *ρ*(*x, t*), which can change with time, with a surface density Q (t) <sup>P</sup> ρ(t)x(t) , where ( )*t* is the density of the space charge *ρ*(*x, t*) averaged over the

Analyzing the behavior of the total current in this structure with variations of *d*1(*t*) for given values of *d*, ( )*t* , *x t*( ) we use a system of the classical one-dimensional equations consisting of the expressions for the total current *j*(*t*) and conductivity current *j*c(*x, t*), equation of continuity, Poisson's equation, and expression that determines the potential V(t) between the electrodes at each instant of time. Taking into account that for electrets the space charge

> 0 ( ,) ( ) *Ext j t <sup>t</sup>*

*Ext x* ( ,) ( )

( ,) ()

*E x t dx V t*

Integrating both parts of equation (4) with respect to the coordinate *x*, taking into account that the field increases in a jumplike manner by a factor of *ε* on the free boundary of the

( ,) ( ,) *Ed t Ed t* , (6)

and substituting the formula derived for *E*(*x, t*) into equation (5), we obtain expressions for the field on the boundaries *x* = 0 and *x* = *d*1(*t*) + *d*: *Ec*(*t*) and *EA*(*t*), and, correspondingly, for

> 0 1 ( ) ( )

*d dt*

*P*

*F*

*P*

*<sup>Q</sup> <sup>V</sup>*

*C t*

*x*

( ) 1 0

*dt d*

0

(3)

(4)

, (5)

<sup>0</sup> () () () () *Q t E t Ct Vt V cc P* , (7)

*<sup>S</sup>*() () () () <sup>0</sup> *A P Q t E t Ct Vt V* , (8)

, (9)

*<sup>C</sup>* , (10)

$$j(t) = \frac{dQ\_\mathcal{S}(t)}{dt} = -\frac{d}{dt}\Big(\mathcal{C}(t)(V(t) + V\_p)\Big)\tag{11}$$

When the electrodes are connected via the load *R* and in the case of open circuit (as it is shown in Fig. 1c with the switch in position 3 and with the switch in position 2), the voltage behavior in time is described by following equations:

$$\frac{d}{dt}\Big[\mathcal{C}(t)(V(t) + V\_P)\Big] = -\frac{V(t)}{R} \tag{12}$$

$$\frac{d}{dt}\Big[\mathcal{C}(t)\Big(V(t) + V\_p\Big)\Big] = 0\tag{13}$$

These equations with the corresponding initial conditions describe all possible regimes of operation of the electrostatic generator shown in Fig. 1 *b,c*.

To study specific features of its operation, we choose (without loss of generality) a sine law of variation of the gap size:

$$d\_1(t) = d\_{10}(1 + \alpha + \sin(\alpha t)) \tag{14}$$

*ω=2πf*, *f=1/T*, *T* is the conversion cycle duration.

As has been shown in Introduction two types of generator construction, depending on the method of commutation of the switch (see Fig. 1 *b,c*), are possible in the case of motion of the moving electrode in the field of the space charge (or polarization) *QP* located in the dielectric. They have been called the voltage generator and the current generator.

#### **2.2. Voltage generator in vibration mode**

In such a generator, the output voltage is amplified compared to the case of current generator, which will be analized below in the section 2.3, in the following manner. At the initial state the capacitor electrodes are short-circuited by commutation of the switch to position 1 at the instant when the minimum distance between the surfaces of the moving electrode and dielectric is reached (see Fig. 1c and Fig.2b, the capacitance *C(t)* has the maximum value at this instant). At the beginning of the process of electrodes separation the switch is turned to position 2; at the instant when the maximum value of *d1(t)* is reached, the switch is turned to position 3 and the energy worked out during the cycle is transferred to the load *R*.

Let us analyze the effect of voltage amplification in more detail. Under conditions of electrode motion with a non-closed circuit (*j(t) = 0*) and according to equation (13), we have

$$C(t)\left(V(t) + V\_p\right) = \text{const} \tag{15}$$

then

$$\mathcal{C}\_{\max} \left( V\_{\min} + V\_p \right) = \mathcal{C}\_{\min} \left( V\_{\max} + V\_p \right) \,. \tag{16}$$

Therefore, we obtain

$$V\_{\text{max}} = \frac{C\_{\text{max}}}{C\_{\text{min}}} \left(V\_{\text{min}} + V\_p\right) - V\_p \tag{17}$$

Let the process of vibrations begin from the phase of the maximum convergence of the surfaces; then, in accordance with the initial condition *V* (0) = *Vmin*(0) = 0, after the first displacement of the moving electrode to the maximum distance under the condition of open circuit (switch in position 2), we have

$$V\_{\text{max}} = V\_p \left( \frac{\mathcal{C}\_{\text{max}}}{\mathcal{C}\_{\text{min}}} - 1 \right) \tag{18}$$

In this case, the following amount of energy is produced:

$$W = \frac{\mathcal{C}\_{\text{min}} V\_{\text{max}}^2}{2} = \frac{\mathcal{C}\_{\text{min}} V\_p^2}{2} \left(\frac{\mathcal{C}\_{\text{max}}}{\mathcal{C}\_{\text{min}}} - 1\right)^2 \approx \frac{V\_p^2}{2} \frac{\mathcal{C}\_{\text{max}}^2}{\mathcal{C}\_{\text{min}}} \tag{19}$$

The principle of mechanical energy conversion to electrical energy with electrode motion under the condition of an open circuit is illustrated by the distribution of the electrical field in the structure (Fig. 3a) both in the initial phase of motion *d*1 = *d1min* = *αd10*, and in the phase of the maximum distance of the electrode (anode) *d*1 = *d*1max = 2*d*10 +*d*1min. As the electrode moves, the field in the gap remains constant, the total current equals zero, and the electrical field energy increases in accordance with the increase in the area under the curve *E*2(*x*), which is manifested as an increase in the difference in the potentials *V* between the electrodes (Fig. 4). The energy is transferred to the load *R* when the switch is turned to position 3. This transfer is efficient if *R*  does not exceed the value 1/(2fCS) , where *C* is the cycle-averaged specific capacitance of the structure, *S* is the electrode area, and *f* is the frequency of vibrations.

**Figure 4.** Behavior of *C*(*t*) and *V* (*t*) in a voltage electret generator based on (**a**) – dielectric: *QP* = 10*<sup>−</sup>*<sup>3</sup> C*/*m2, *ε* = 10, and (**b**) – ferroelectric: *ε* = 1000 and *QP* = 10*−*2 C*/*m2. *d* = 1 *μ*m, *d*10 = 0*.*5 *μ*m, *d*1min = 10 nm, *S* = 1 mm2. *R* = 10 MΩ

In contrast to the capacitance machine where the energy transfer to the load is finalized at the end of the cycle by complete discharge of the capacitor *C*(*t*) to *V* (*t*) = 0, the charge induced in the electrodes and screening the field of the space charge in the dielectric flows between the electrodes through the load *R* during the electrode motion in the generator considered here. This process is unsteady and is determined by several constants: instantaneous value of *RC*(*t*)*S*, time of the Debye screening of the charge in the metal (which is the smallest time), and time of motion of the moving electrode during the half-period. At certain times, depending on the relation between these time constants, the total current can change its direction, when the voltage also changes its sign at the instant of the maximum approaching of the surfaces (*d1* (*t*) → *d1*min) (see Fig. 4). To eliminate this effect, the switch (Fig.1c) is turned to position 1 at the beginning of each next cycle; then, the initial voltage *V*min = 0 is recovered, and the process is repeated completely.

68 Small-Scale Energy Harvesting

Therefore, we obtain

1 mm2. *R* = 10 MΩ

circuit (switch in position 2), we have

*CV VCV V* max min *<sup>p</sup>* min max *<sup>p</sup>* . (16)

*<sup>C</sup>* (17)

(18)

(19)

*p p*

 max max min min

max

In this case, the following amount of energy is produced:

structure, *S* is the electrode area, and *f* is the frequency of vibrations.

*<sup>C</sup> V V*

**Figure 4.** Behavior of *C*(*t*) and *V* (*t*) in a voltage electret generator based on (**a**) – dielectric: *QP* = 10*<sup>−</sup>*<sup>3</sup> C*/*m2, *ε* = 10, and (**b**) – ferroelectric: *ε* = 1000 and *QP* = 10*−*2 C*/*m2. *d* = 1 *μ*m, *d*10 = 0*.*5 *μ*m, *d*1min = 10 nm, *S* =

a. b.

*<sup>C</sup> V V VV*

Let the process of vibrations begin from the phase of the maximum convergence of the surfaces; then, in accordance with the initial condition *V* (0) = *Vmin*(0) = 0, after the first displacement of the moving electrode to the maximum distance under the condition of open

max

min 1 *<sup>p</sup>*

2 2 2 2 2 min max min max max

min min 1

*C C*

*C* 

22 2 *C V C Vp p <sup>C</sup> <sup>V</sup> <sup>C</sup> <sup>W</sup>*

The principle of mechanical energy conversion to electrical energy with electrode motion under the condition of an open circuit is illustrated by the distribution of the electrical field in the structure (Fig. 3a) both in the initial phase of motion *d*1 = *d1min* = *αd10*, and in the phase of the maximum distance of the electrode (anode) *d*1 = *d*1max = 2*d*10 +*d*1min. As the electrode moves, the field in the gap remains constant, the total current equals zero, and the electrical field energy increases in accordance with the increase in the area under the curve *E*2(*x*), which is manifested as an increase in the difference in the potentials *V* between the electrodes (Fig. 4). The energy is transferred to the load *R* when the switch is turned to position 3. This transfer is efficient if *R*  does not exceed the value 1/(2fCS) , where *C* is the cycle-averaged specific capacitance of the Despite a principally different method of capacitance recharging, this circuit of energy conversion is similar to the capacitance machine (Fig. 1a), except for the fact that the field of the built-in charge inducing the voltage *VP* serves here as the voltage source.

An universal program that takes into account all parameters of the structure and energy generation modes in the cases of one- and two-capacitor generator was developed for the numerical analysis of the problem. A difference scheme with automatic choice of the time step was used; this scheme ensured solution stability and specified accuracy.

One example of such a solution, which illustrates voltage generation between the electrodes for a particular dielectric with a low value of *ε*, is shown in Fig. 4a. The numbers on the axis *t* characterize the position of the switch (see Fig. 1c). It is seen from the figure that the role of this switch is the recovery of the initial state of the system (*Vmin = 0*) in the phase when *C*(*t*)=*Cmax* in each cycle of energy generation. Such synchronization allows us to obtain the voltage amplification proportional to the capacitance modulation depth in accordance with (18); the power increases thereby in accordance with (19).

If a dielectric with a high value of *ε* is used in the generator, the capacitance modulation *C*max*/C*min increases, but not in proportion to the increase in *ε* (when the air gap modulation depth is constant), because the value of *C*min changes only slightly (it is determined by the maximum value of the air gap *2d*10 + *d*1min). In this case, with a fixed polarization *QP* , the value of *VP* decreases inversely proportional to *ε* (10); the value of *V*max (18) and also the energy generated in one cycle (19) decrease accordingly.

In particular, for the structure parameters used to construct the graphs in Fig. 4a, but with the value of *ε* increased by a factor of 100, the value of *VP* decreases by a factor of 100, but the values of *V*max and *W* decreases only by a factor of 10. Therefore, to reach the output voltage and the generated energy comparable with the case of the classical electret generator (with parameters corresponding to those in Fig. 4a), the polarization in the ferroelectric should be increased by ten times (up to 10*−*2 C*/*m2) (Fig. 4b). Note that such values of polarization are not critical for a number of known ferroelectrics; therefore, it is possible to increase the amount of energy generated during one conversion cycle by the ferroelectricbased generator; this increase is limited only by the voltage of the breakdown in the gap

between the electrodes. Moreover, to increase the energy production, it is possible to use ferroelectrics with low values of *ε* and high values of spontaneous polarization, for instance, lithium niobate and tantalate (*QP* up to 0.5–0.8 C*/*m2 with *ε ≈* 40) [39].

#### **2.3. Current generator in vibration out-of-plane mode**

The operation principle of the current generator is shown in Fig.1b and the example of its design is presented in Fig.2a, this scheme is the simplest among the other possible ones. In the case of structure capacitance modulation, the generator operates without voltage amplification [16].

There are publications on particular cases of the current generator, for instance, electret microphones [17] in which either the load resistance is small or the amplitude of electrode vibrations is small as compared with the air gap thickness. In this case, in accordance with the analysis performed above, we have *V << VP* , and the current in the circuit is described as

$$j(t) = V\_p \frac{d\mathcal{C}(t)}{dt} \tag{20}$$

Let us consider the general solution of the problem of current generator operation with an arbitrary load, using equation (12) with the initial condition *V (0) = 0* and with variation of the gap size in accordance with equation (14).

Equation (12) with provision for (14) is written in dimensionless form as

$$\frac{d\phi(\tau)}{d\tau} = -\left(\frac{1+\alpha'+\sin\tau}{RC\_1\text{So}} - \frac{\cos\tau}{1+\alpha'+\sin\tau}\right)\phi(\tau) + \frac{\cos\tau}{1+\alpha'+\sin\tau} \tag{21}$$

where *ϕ* = *V/VP* , *τ* = *ωt*, *α'* = *α* + *d/εd*10, and

$$\mathbf{C}1 = \varepsilon u / du \,\tag{22}$$

is the specific capacitance at average value of the air gap. Therefore, the problem is determined only by two dimensionless parameters: *α'* and *RC*1*Sω*.

Note that with a sufficiently large air gap modulation depth, the parameter *α'* is inversely proportional to the structure capacitance modulation depth.

One example of the numerical solution of equation (21) is shown in Fig. 5. The initial increase in voltage during the first displacement of the moving electrode is similar to its increase in the voltage generator; it is determined by capacitance modulation and by the value of *VP* (18). In subsequent periods of electrode motion, a quasi-steady screening charge is formed (it is described above), and the voltage amplitude *Vm* decreases; this amplitude becomes sign-variable and tends to *±VP* or to a smaller value, depending on the load resistance *R*. The time constant of the decrease in *Vm* is determined by the value of RCS , where *C ≤ C*1 is the capacitance of the structure *C*(*t*) (see the inset in Fig. 5) averaged over the period of vibrations.

amplification [16].

between the electrodes. Moreover, to increase the energy production, it is possible to use ferroelectrics with low values of *ε* and high values of spontaneous polarization, for instance,

The operation principle of the current generator is shown in Fig.1b and the example of its design is presented in Fig.2a, this scheme is the simplest among the other possible ones. In the case of structure capacitance modulation, the generator operates without voltage

There are publications on particular cases of the current generator, for instance, electret microphones [17] in which either the load resistance is small or the amplitude of electrode vibrations is small as compared with the air gap thickness. In this case, in accordance with the analysis performed above, we have *V << VP* , and the current in the circuit is described as

( ) ( ) *<sup>p</sup>*

Let us consider the general solution of the problem of current generator operation with an arbitrary load, using equation (12) with the initial condition *V (0) = 0* and with variation of

( ) 1 sin cos cos ( ) 1 sin 1 sin

 *C*1 = *ε*0*/d*<sup>10</sup>(22) is the specific capacitance at average value of the air gap. Therefore, the problem is

Note that with a sufficiently large air gap modulation depth, the parameter *α'* is inversely

One example of the numerical solution of equation (21) is shown in Fig. 5. The initial increase in voltage during the first displacement of the moving electrode is similar to its increase in the voltage generator; it is determined by capacitance modulation and by the value of *VP* (18). In subsequent periods of electrode motion, a quasi-steady screening charge is formed (it is described above), and the voltage amplitude *Vm* decreases; this amplitude becomes sign-variable and tends to *±VP* or to a smaller value, depending on the load resistance *R*. The time constant of the decrease in *Vm* is determined by the value of RCS , where *C ≤ C*1 is the capacitance of the structure *C*(*t*) (see the inset in Fig. 5) averaged over

 

Equation (12) with provision for (14) is written in dimensionless form as

1

determined only by two dimensionless parameters: *α'* and *RC*1*Sω*.

proportional to the structure capacitance modulation depth.

*d RC S*

*dC t jt V dt* (20)

, (21)

lithium niobate and tantalate (*QP* up to 0.5–0.8 C*/*m2 with *ε ≈* 40) [39].

**2.3. Current generator in vibration out-of-plane mode** 

the gap size in accordance with equation (14).

*d*

the period of vibrations.

where *ϕ* = *V/VP* , *τ* = *ωt*, *α'* = *α* + *d/εd*10, and

**Figure 5.** Transient of setting a steady state for the current generator at *α'* = 0*.*11 and *ωRC*1*S* = 1*.*11*·*102 (*QP* = 10*−*3 C*/*m2, *ε* = 10, *d* = 1 *μ*m, *d*10 = 1 *μ*m, *d*1min = 10mn, *S* = 1 mm2, *R* = 20 GΩ, and *f* = 100 Hz).

The specific features of the behavior of *V* (*t*) in the steady regime are shown in Fig. 6. The change in the gap size *d*1(*t*) accompanied by the corresponding change in the capacitance *C*(*t*) induces variations of the charge on the electrodes in time *QS*(*t*); the maximum of this dependence in the general case can be shifted with respect to the peak value of the capacitance *C*(*t*). This shift of the peaks of *QS*(*t*) and *C*(*t*) as compared with the classical case of the current generator considered in [16] (for which *V* (*t*) = 0 and equation (20) is valid) is caused by the delay in redistribution of the screening charge *QS*(*t*) between the electrodes during current flow through the load *R* because of the finite value of *RC*(*t*)*S*.

**Figure 6.** Example of the numerical solution of the equation that describes current generator operation in a steady regime with sinusoidal variations of the gap size *d*1: *V/VP* (*1*), *d*1*/d*10 (*2*), *C/C*1 (*3*), and *QS/C*1*VP* (*4*); *α'* = 0*.*1 and *ωRC*1*S* = 6*.*25 10*−*4.

The generator considered is qualitatively different both from the capacitance machine and from the voltage generator in one more operation principle: its operation is determined by changes in the conditions of screening of the electric field in metallic electrodes during the

motion of the moving electrode. The electrode recharging current in the circuit of the load *R*  tends to return the system to the equilibrium state with *V* = 0.

**Figure 7.** Frequency-load dependences of the produced power and output voltage for a current electret generator at *α'* = 0*.*1.

If the structure capacitance modulation depth is sufficiently large, the amplitude of the voltage *Vm* produced by the generator tends (in the case of an optimal load) to the limiting value *±VP* , and the power has the maximum value at the frequency

$$\text{fa} = \text{1/RC.S.}\tag{23}$$

as is shown in Fig. 7. Note that the law of current oscillations approaches the sine law as the parameter *RC*1*Sf* increases to values of the order of unity and greater, in contrast to Fig. 6 where *RC*1*Sf <<* 1. Under these conditions, the power produced by the generator is <sup>2</sup> Vm */*2*R*. In the case of generation of the maximum power *P*max at sufficiently large η, the value of *Vm* is close to *VP* ; thus, we obtain

$$P\_{\text{max}} = \frac{1}{2} \frac{V\_p^2}{R\_m} = f \frac{C\_1 S V\_p^2}{2} \,\, , \tag{24}$$

where *Rm* = 1*/*(*C*1*Sf*) is the load resistance at the generation of *P*max. Correspondingly, the energy generated during the conversion period is expressed by

$$\mathcal{W}\_m = \frac{\mathbb{C}\_1 \mathcal{S} \mathcal{V}\_p^2}{2} \tag{25}$$

Thus, in the case of a sufficiently large depth of structure capacitance modulation, the amplitudes of power and the voltage of the current generator are almost independent of the values of *C*max and *C*min; they are determined only by the mean capacitance of the gap *C*1.

Note that if *C*max*/C*min *>* 5 and *C*1*/CF <* 0*.*1 maximal output voltage approaches to *VP*, and the maximum generated energy is given by (25).

#### **2.4. Electret generator operation at lateral displacement of capacitor plates**

The case of lateral displacement of capacitor plates under the operation of electret generator in the current mode (see the scematic design in Fig.2c) should be emphasized particularly, because it is realized in practice, see, e.g., so-called "in-plane gap-closing" constructions [32- 36] and the rotational systems [29]. This operation mode is described by following equations (see equation (12)):

$$\frac{d}{dt}\Big[\text{CS}(t)\big(V(t) + V\_P\big)\Big] = -\frac{V(t)}{R}\Big],\tag{26}$$

where the area of the capacitor plates overlap is described as:

72 Small-Scale Energy Harvesting

generator at *α'* = 0*.*1.

is close to *VP* ; thus, we obtain

motion of the moving electrode. The electrode recharging current in the circuit of the load *R* 

**Figure 7.** Frequency-load dependences of the produced power and output voltage for a current electret

If the structure capacitance modulation depth is sufficiently large, the amplitude of the voltage *Vm* produced by the generator tends (in the case of an optimal load) to the limiting

f1 = 1/RC1S, (23)

as is shown in Fig. 7. Note that the law of current oscillations approaches the sine law as the parameter *RC*1*Sf* increases to values of the order of unity and greater, in contrast to Fig. 6 where *RC*1*Sf <<* 1. Under these conditions, the power produced by the generator is <sup>2</sup> Vm */*2*R*. In the case of generation of the maximum power *P*max at sufficiently large η, the value of *Vm*

> 2 2 1

*C SV*

Thus, in the case of a sufficiently large depth of structure capacitance modulation, the amplitudes of power and the voltage of the current generator are almost independent of the values of *C*max and *C*min; they are determined only by the mean capacitance of the gap *C*1.

Note that if *C*max*/C*min *>* 5 and *C*1*/CF <* 0*.*1 maximal output voltage approaches to *VP*, and the

*P f <sup>R</sup>* , (24)

*W* (25)

*V C SV*

2 2 *p p*

*m*

where *Rm* = 1*/*(*C*1*Sf*) is the load resistance at the generation of *P*max. Correspondingly, the

value *±VP* , and the power has the maximum value at the frequency

max

energy generated during the conversion period is expressed by

maximum generated energy is given by (25).

1

*m*

tends to return the system to the equilibrium state with *V* = 0.

$$\mathbf{S(t)} = \mathbf{S}\_{10}(1 + \beta + \sin\alpha t) \,, \tag{27}$$

and C is a specific capacitance, calculated by eqn. (9), which is constant in this case.

The equations (26) and (27) were solved numerically. The solution is represented in Fig. 8, where Pm is maximum value of power and Pmax value is calculated using (24). This solution is qualitatively different from one for current out-of-plane electret generator, described above, by the presence of the pronounced dependence of output power on the modulation depth of structure capacitance, see Fig.8b, whereas in the previous case this dependence is practically not observed. Point is that in the case of lateral shift of capacitor plates the total value of polarization decreases along with the capacitance whereas in the previous case of vibration mode electret generator, for which the polarization is constant. The polarization decrease reduces in this time interval the influence of parazitic induced charge on the current, thus resulting in the effect of voltage amplification, as in the case of electret generator operation in the voltage vibratory mode. However, because the influence of the parazitic charge is not totally excluded, the value of maximum power Pm now is not proportional to capacitance modulation depth η (as for the case of capacitance machine) but Pm depends on η according to the logarithmic law, see Fig. 8b.

**Figure 8.** Electret generator with lateral shift of capacitor plates: (**a**) - frequency-load dependences of the output power for η = 2 (*1*), 5 (*2*), 20 (*3*), 100 (*4*) and (**b**) –the dependence of maximum power on capacitance modulation depth.

The drawback of this type of generator is the residual influence of parasitic induced charge mentioned above resulting in the reduction of the efficiency of the generation per unit area at equal parameters compared to out-of-plane generator in a current mode. In particular, the maximal output power of this generator becomes larger than that of its out-of-plane analog only for η>5.6, see Fig.8b. Inability to reach the small values of interelectrode gaps at high enough areas of generator plates should be marked as another drawback. These defects do not permit to reach high values of specific powers at η increase.

Note, that under transition in voltage generator mode the effects similar to those discussed in section **2.2** are observed. Therefore, in this mode the value of *Pm/Pmax* will be proportional to capacitance modulation depth, and output power could be increased considerably.

## **3. Analysis of possible versions of implementation of two-capacitor generator circuits**

Various versions of implementation of two-capacitor generators are based on the ideal generator circuit shown in Fig. 1d. They differ only in the method of compensation of charge losses caused by leakage currents in the generator capacitors (Fig. 1e,f).

Such compensation can be provided by an external source of electric energy. In the first case, a current source, i.e., a source that provides constant current and has a sufficiently high (in the ideal case, infinite) internal resistance, is connected to one of the generator capacitors (see Fig. 1e).

In the case of compensation of losses from the voltage source (whose internal resistance is low), it is necessary to use a switch connecting the source to one of the capacitors for a certain time sufficient for capacitor recharging (Fig. 1f).

Moreover, charge losses can be compensated by using an additional low-power generator connected to the input of the basic generator and consuming a minor portion of mechanical energy of the system, for instance, by using the electret effect [16].

It is also possible to compensate charge losses by organizing a feedback for transferring some part of energy from the generator output to its input for charging one of the capacitors, as shown in [10].

#### **3.1. Compensation of charge losses with the use of a current source**

The circuit based on this principle is shown in Fig. 1e. Operation of this generator is described by the system of differential equations

$$I(t) = (V\_2(t) - V\_1(t)) / R;$$

$$I(t) = -I\_0 + \frac{d(V\_1(t)C\_1(t))}{dt} + \sigma\_1(t)V\_1(t);\tag{28}$$

$$I(t) = -\frac{d(V\_2(t)C\_2(t))}{dt} - \sigma\_2(t)V\_2(t);$$

where *V*1(*t*) and *V*2(*t*) are the drops of voltage on capacitors *C*1 and *C*2, respectively; *σ*1(*t*) and *σ*2(*t*) are the conductivities arising owing to leakages in these capacitors (in the general case, they are time-dependent).

Two operation modes of two-capacitor generators are possible: with lateral shift of the plates (with variations of the electrode overlapping area, a particular case is the rotor-type generator) and with vertical out-of plane vibrations of the plates (with variations of the interelectrode gap, vibrational mode). To obtain the maximum efficiency of energy generation, the capacitor plates move in the opposite phases in both cases.

#### *3.1.1. Two-capacitor generator with lateral shift of the capacitors plates*

This operation mode is demonstrated schematically in Fig.2d. In the case with capacitances changes in the opposite phases and with lateral shift of the capacitors plates, their total capacitance is constant:

$$\text{Cu}(\text{t}) \star \text{Cu}(\text{t}) \star \text{Cu} \tag{29}$$

In a particular case, when the charge leakage is proportional to the electrode overlapping area, i.e., with similar changes in the conductivities, we have

$$\text{cov(t)} \langle \text{C1(t)} \text{w} \text{(t)} \langle \text{C2(t)} \rangle \tag{30}$$

In the case of constant leakages, we have

$$\mathbf{r}\_{\mathbf{O1}} = \mathbf{O2} = \mathbf{1/r}\_{\mathbf{A}} \tag{31}$$

and for both cases

74 Small-Scale Energy Harvesting

**generator circuits** 

(see Fig. 1e).

capacitors, as shown in [10].

The drawback of this type of generator is the residual influence of parasitic induced charge mentioned above resulting in the reduction of the efficiency of the generation per unit area at equal parameters compared to out-of-plane generator in a current mode. In particular, the maximal output power of this generator becomes larger than that of its out-of-plane analog only for η>5.6, see Fig.8b. Inability to reach the small values of interelectrode gaps at high enough areas of generator plates should be marked as another drawback. These defects do

Note, that under transition in voltage generator mode the effects similar to those discussed in section **2.2** are observed. Therefore, in this mode the value of *Pm/Pmax* will be proportional

Various versions of implementation of two-capacitor generators are based on the ideal generator circuit shown in Fig. 1d. They differ only in the method of compensation of charge

Such compensation can be provided by an external source of electric energy. In the first case, a current source, i.e., a source that provides constant current and has a sufficiently high (in the ideal case, infinite) internal resistance, is connected to one of the generator capacitors

In the case of compensation of losses from the voltage source (whose internal resistance is low), it is necessary to use a switch connecting the source to one of the capacitors for a

Moreover, charge losses can be compensated by using an additional low-power generator connected to the input of the basic generator and consuming a minor portion of mechanical

It is also possible to compensate charge losses by organizing a feedback for transferring some part of energy from the generator output to its input for charging one of the

The circuit based on this principle is shown in Fig. 1e. Operation of this generator is

1 1 0 1 1

*dV tC t It I tV t dt dV tC t I t tV t dt*

( ( ) ( )) ( ) ( ) ( );

( ( ) ( )) ( ) ( ) ( );

2 2

(28)

to capacitance modulation depth, and output power could be increased considerably.

**3. Analysis of possible versions of implementation of two-capacitor** 

losses caused by leakage currents in the generator capacitors (Fig. 1e,f).

certain time sufficient for capacitor recharging (Fig. 1f).

described by the system of differential equations

energy of the system, for instance, by using the electret effect [16].

**3.1. Compensation of charge losses with the use of a current source** 

2 1

( ) ( ( ) ( )) / ;

*It V t V t R*

2 2

not permit to reach high values of specific powers at η increase.

$$\begin{aligned} \, \, \text{c}\mathbf{u}(\mathbf{t}) \mathbf{+} \text{c}\mathbf{z}(\mathbf{t}) \mathbf{=} \mathbf{0} \end{aligned} \tag{32}$$

System (28) was solved numerically. In the dimensionless form system (28) was formulated in [5], and its solution is determined only by two dimensionless parameters characterizing the load properties of the system (*fRC*0) and the charge losses due to leakage currents (*fC*0*/σ*0). By solving this system, we determined the voltages *V*1 and *V*2 on the capacitors *C*<sup>1</sup> and *C*2, and also the corresponding charges *Q1(t)* and *Q2(t)*, and the total charge QΣ(t), the current *I*(*t*) flowing through the load resistance *R* and the power released in this load resistance *P*, averaged over the time of the cycle of energy transformation *T*.

The system of equations that describes operation of the two-capacitor generator was analyzed in [5], where numerical solutions were obtained for the energy generation efficiency for various methods of excitation of shift vibrations of electrode grates. The output power generated by the generator, however, was not analyzed, and no analytical estimates were obtained.

Let us estimate the value of the maximum energy generated by this generator during one conversion cycle and, correspondingly, the power. As the first approximation, we consider the ideal two-capacitor generator (see Fig. 1d), which ensures minor leakages; therefore,

recharging of the capacitors (e.g., from a current source) is not needed. Let the generator capacitors be initially charged to a voltage V0. Taking into account equation (29), we obtain

$$\text{Q}\mathfrak{w}\text{CoV}\mathfrak{o}\mathfrak{m}\text{ const},\tag{33}$$

where *Q*0 = *Q*Σ(0) is the total initial charge accumulated on the capacitors. As there are no charge leakages in this case, the charge is retained during the entire time of generator operation.

Under the conditions described above, there is an initial segment of current relaxation with the characteristic time constant of the order of *RC*0; during this time a dynamically equilibrium mode of generation is established owing to charge redistribution on the capacitors. The behavior of voltages on the capacitors depends in this case on the initial phases of *C*1(*t*) and *C*2(*t*). Other conditions are also possible, for instance, a gradual smooth increase in the amplitude and frequency of capacitance oscillations, which is closer to reality. We do not analyze this case in detail here, because the same dynamically equilibrium mode is established for all initial conditions.

Taking into account equations (1), (29) and (33) and also that *Q0= CminVmax+CmaxVmin*, it is easy to get:

$$\frac{V\_{\text{min}}}{V\_0} = \frac{1 + 1/\eta}{2} \,\text{\,\,\,}\tag{34}$$

As both capacitors participate here in energy conversion, we can easily show that the energy *W2* produced during one conversion cycle is

$$\frac{\mathcal{W}\_2}{\mathcal{W}\_0} = \frac{P}{P\_0} = \frac{\eta^2 - 1}{2\eta} \quad \text{{} \tag{35}}$$

$$\mathcal{W}\_0 = \frac{\mathcal{C}\_0 V\_0^{\;\;\;\;\;\prime}}{\mathcal{Z}} \tag{36}$$

is initial energy accumulated in the capacitors and *P* is the power of the two-capacitor generator and *P0 = W0f*. A comparison with one-capacitor generator (Fig.1a, energy W1) results in the following expression

$$\frac{\mathcal{W}\_2}{\mathcal{W}\_1} = \frac{1}{2} \left( \frac{1}{\eta} + 1 \right)^2 \tag{37}$$

For *η>>1* we have

$$\mathcal{W}\_2 \mid \mathcal{W}\_1 \approx 1 / 2 \tag{38}$$

Thus, at identical initial voltages, the power provided by single-capacitor generator is twice as high as the maximum power of the two-capacitor generator. However, for the case of one-capacitor generator the charge is completely consumed in each cycle of energy conversion, and it should be renewed, which makes this method of energy conversion more difficult in many cases. The power of the two-capacitor generator (with identical initial voltages) is lower because of the non-optimum incomplete charging of the capacitors Ci under the initial conditions mentioned above.

76 Small-Scale Energy Harvesting

operation.

to get:

recharging of the capacitors (e.g., from a current source) is not needed. Let the generator capacitors be initially charged to a voltage V0. Taking into account equation (29), we obtain

Q0=C0V0= const, (33)

where *Q*0 = *Q*Σ(0) is the total initial charge accumulated on the capacitors. As there are no charge leakages in this case, the charge is retained during the entire time of generator

Under the conditions described above, there is an initial segment of current relaxation with the characteristic time constant of the order of *RC*0; during this time a dynamically equilibrium mode of generation is established owing to charge redistribution on the capacitors. The behavior of voltages on the capacitors depends in this case on the initial phases of *C*1(*t*) and *C*2(*t*). Other conditions are also possible, for instance, a gradual smooth increase in the amplitude and frequency of capacitance oscillations, which is closer to reality. We do not analyze this case in detail here, because the same dynamically

Taking into account equations (1), (29) and (33) and also that *Q0= CminVmax+CmaxVmin*, it is easy

As both capacitors participate here in energy conversion, we can easily show that the energy

is initial energy accumulated in the capacitors and *P* is the power of the two-capacitor generator and *P0 = W0f*. A comparison with one-capacitor generator (Fig.1a, energy W1)

> 1 1 <sup>1</sup> 2

Thus, at identical initial voltages, the power provided by single-capacitor generator is twice as high as the maximum power of the two-capacitor generator. However, for the case of

 

1 1/ 2

2

2 0 0 <sup>0</sup> 2

1 2

2

, (34)

, (35)

*C V <sup>W</sup>* (36)

(37)

2 1 *W W*/ 1/2 (38)

min 0

*V V*

> 2 0 0

*W P W P*

> 2 1

*W W*

equilibrium mode is established for all initial conditions.

*W2* produced during one conversion cycle is

results in the following expression

For *η>>1* we have

Solving system (28) with condition (29) numerically in the absence of charge leakages, we found the voltages on the capacitors Vi and then determined the current in the load resistance, the charge on the capacitors, and the generator power *P*. The dependence of *P/P0* on *fRC0*, where *P0 = fW0*, is shown in Fig. 9.

The dependence of the maximum power *Pmax/P0* on the modulation factor *η* (Fig. 10, curve 1) for the ideal generator is almost linear in the interval η >> 1 and is adequately described by (35) and curve 3 in Fig. 10. At small values of η, 1 < η < 3, there are significant deviations from equation (35), because *Vmax/Vmin < η* in this case and equation (1) is invalid.

With our method of normalization used here the curves in Fig. 9 depend only on the capacitance modulation factor. The curves in Fig. 10 are independent on the absolute parameters of the model, i.e., they have a universal character and describe all possible solutions of system (28) for the ideal generator case. From this viewpoint, we called them the "characteristic" curves.

**Figure 9.** Characteristic curves of the generated power for the ideal generator: *η* = 2 (*1*), 10 (*2*), and 100 (*3*).

**Figure 10.** Characteristic curves of the maximum generated power *Pmax/P0* versus the capacitance modulation factor for the following operation modes: ideal generator with lateral shift of the plates (*1*), lateral shift of the plates with modulation of charge leakages synchronized with capacitance modulation (*2*), lateral shift of the plates with constant leakages (*4*), out-of-plane antiphase vibrations of the plates (vibration generator) (*5*), lateral shift of the plates with recharging the capacitor *C1* from a voltage source under conditions *C1(0) = Cmax* (*6*), out-of-plane antiphase vibrations of the plates for the ideal generator at *C1(0) = Cmax* (*7*) and at *C1(0)=C2(0)* (*8*), and analytical estimate for the ideal generator from equation (13) - (*3*).

An example of the numerical solution of (28) illustrating the operation of the two-capacitor generator in the absence of charge losses and with a set of parameters corresponding to its peak power is shown in Fig. 11.

Sinusoidal antiphase oscillations of the capacitors *C1* and *C2* lead to anharmonic oscillations of the charges *Qi* potentials *Vi,* and also of the current *I* in the load resistance *R*. The greater the amplitude of oscillations of the values of *Vi*, *Q*i, and *I*, the greater the generator power or its normalized value *P/P0* determined as *W/W0*, i.e., the ratio of the energy produced by the generator during the period of oscillations to the initial energy accumulated on the capacitors:

**Figure 11.** Time evolution of the capacitance (*1*), voltage (*2*), and charge (*3*) for one of the generator capacitors at the vicinity of optimum power (*η* = 10 and *fRC*0 = 1*.*9).

High Energy Density Capacitance Microgenerators 79

$$\begin{aligned} \bigvee\_{\mathbf{P}\_0} &= \bigvee\_{\mathbf{W}\_0} \quad \text{and} \quad \bigvee\_{\mathbf{P}\_0} \quad \end{aligned} \tag{39}$$

$$\mathcal{W} = \int\_{t\_1}^{t\_1 + T} I^2(t) R dt \quad \text{and} \tag{40}$$

*P = Wf* is the power.

78 Small-Scale Energy Harvesting

(13) - (*3*).

peak power is shown in Fig. 11.

**Figure 10.** Characteristic curves of the maximum generated power *Pmax/P0* versus the capacitance modulation factor for the following operation modes: ideal generator with lateral shift of the plates (*1*), lateral shift of the plates with modulation of charge leakages synchronized with capacitance modulation (*2*), lateral shift of the plates with constant leakages (*4*), out-of-plane antiphase vibrations of the plates (vibration generator) (*5*), lateral shift of the plates with recharging the capacitor *C1* from a voltage source under conditions *C1(0) = Cmax* (*6*), out-of-plane antiphase vibrations of the plates for the ideal generator at *C1(0) = Cmax* (*7*) and at *C1(0)=C2(0)* (*8*), and analytical estimate for the ideal generator from equation

An example of the numerical solution of (28) illustrating the operation of the two-capacitor generator in the absence of charge losses and with a set of parameters corresponding to its

Sinusoidal antiphase oscillations of the capacitors *C1* and *C2* lead to anharmonic oscillations of the charges *Qi* potentials *Vi,* and also of the current *I* in the load resistance *R*. The greater the amplitude of oscillations of the values of *Vi*, *Q*i, and *I*, the greater the generator power or its normalized value *P/P0* determined as *W/W0*, i.e., the ratio of the energy produced by the generator during the period of oscillations to the initial energy accumulated on the capacitors:

**Figure 11.** Time evolution of the capacitance (*1*), voltage (*2*), and charge (*3*) for one of the generator

capacitors at the vicinity of optimum power (*η* = 10 and *fRC*0 = 1*.*9).

Note that the results of the numerical analysis support the above-formulated approximation (1), from which it follows that the charges *Qi* at the maximum and minimum values of the capacitance are equal, i.e., *C*min*V*max = *C*max*V*min (see Fig. 11, curve *3*).

At sufficiently high frequencies (*f >>* 1*/RC*0), the capacitors *C*1 and *C*2 do not have enough time to exchange the charge during one cycle of energy conversion, which reduces the charge modulation factor on each capacitor in the dynamically equilibrium mode. Therefore, the energy *W* generated during the cycle becomes smaller than the limiting value *W*2. On the other hand, at low frequencies (*f <<* 1*/RC*0), energy conversion is also ineffective, because the charge passes to the second capacitor under these conditions faster than the capacitance of the generating capacitor reaches the minimum value. The charge on the capacitors "tracks" the changes in the capacitance. Thus, a typical feature of two-capacitor generators is the optimum of the normalized power *P/P*0 in the frequency range *f* ~ 1*/RC*0, which is consistent with the results of the numerical analysis (see Fig.9). Under the optimum generation conditions, as the capacitance of the generating capacitor (in which energy conversion occurs in the time interval considered) decreases, a significant portion of the charge flows to the other capacitor, thus, recovering the state corresponding to the beginning of generation on this capacitor. As the capacitance modulation factor *η* increases, the generator power *P/P*<sup>0</sup> also increases, and its peak is shifted toward higher frequencies.

Let us consider the operation of the two-capacitor generator taking into account the charge losses due to leakage currents and its compensation from an external current source. In the equivalent circuit shown in Fig. 1e, the charge losses are shown as conductivities *σi*(*t*) connected in parallel to the capacitors *Ci*(*t*). A d. c. current source *I0* is used for compensation of these losses. The operation of such a generator is described by the system of differential equations (28) with initial conditions corresponding to the steady state of the system with the current *I*0 flowing in the circuit.

Solving this system numerically, we determined the dimensionless values of the potentials *y*(*x*) and *z*(*x*) on the capacitors *Ci* and then the quantities characterizing the generator operation.

Let us first analyze the case with capacitance and conductivity modulation in accordance with an identical sine law, i.e., when the conditions (29) and (30) are satisfied.

Such modulation of conductivities is typical for real situation when leakages are proportional to the area of overlapping of the capacitors plates. Subtracting the corresponding components of the third equation of system (28) from the left and right sides of the second equation of the same system and taking into account equations (29,30,32) we obtain the expression of total charge:

$$\frac{dQ\_{\Sigma}(t)}{dt} = -Q\_{\Sigma}(t)\frac{\sigma\_{0}}{C\_{0}} + I\_{0} \tag{41}$$

Using the initial condition that describes the charge of the capacitors from the current source *I0* in the steady state: *QΣ(0)=I0C0/σ0* we can easily show that equation (41) has only one unique solution (Fig.12, curve 1)

**Figure 12.** Time evolution of the total charge of the generator capacitances at *fRC*0 = 9*.*6 and *η* = 10: modulation of leakages synchronized with modulation of the capacitances, equation (18)), - (*1*), and constant leakages (*2*) (*frC*0 = 9*.*6 102). The inset shows a zoomed-in fragment of curve *2*.

$$Q\_{\Sigma}(t) = Q\_{\Sigma}(0) = \frac{I\_0 C\_0}{\sigma\_0} \tag{42}$$

Then, all estimates of the maximum energy produced by the ideal generator during the period of energy conversion and the estimates of the generator power are valid at a certain effective value of the initial voltage

$$V\_0^\* = I\_0 \, / \, \sigma\_0 \tag{43}$$

As an example, Fig. 10 shows the characteristic curve 2 of *Pmax/P0* as a function of *η*, which completely coincides with curve 1 for the ideal generator. In the general case, the value of *V0\** is not equal to the real value of the initial voltage *V(0)*, because the value of *V(0)* depends on the initial phase of oscillations, i.e., on particular values of the conductivities *σ1* and *σ<sup>2</sup>* at the time *t = 0*.

The second case also observed in practice is the case with constant leakages:

$$
\sigma\_1(t) = \sigma\_2(t) = \sigma\_0 / \,\,\,\,\,\,\,\,\tag{44}
$$

A significant difference of this solution from the case of negligibly small leakage currents and also from the case of proportionality of the conductivity *σi* to the electrode overlapping area considered above is the initial decrease in the total charge *QΣ* in time (see Fig. 12, curve 2) and its low-amplitude oscillations (see the inset in Fig. 12) in accordance with the period of changes in the capacitances *Ci*: *T = 2π/ω.* This effect is explained by the increase in the leakage currents in each period owing to the increase in the potentials *Vi* on the capacitances, which leads to a certain decrease in the charges *Qi* (later on, the charges *Qi* again increase when the potentials *Vi* pass through their minimum values owing to recharging from the source *I0*). At the initial stage of the process, the leakage currents averaged over the cycle of generation are greater than the source current *I0*, and the discharge of capacitors takes place. For this mode, the steady-state value of the charge *QΣ(∞)* cannot be estimated analytically; therefore, the decay of the charge in time was analyzed numerically: it grows with increasing of both *η* and the absolute value of the capacitance. In most realistic cases, however, the decrease in the total charge *k = QΣ(0)/QΣ(∞)* is not more than a factor of 2.

80 Small-Scale Energy Harvesting

obtain the expression of total charge:

unique solution (Fig.12, curve 1)

effective value of the initial voltage

*V0\**

the time *t = 0*.

of the second equation of the same system and taking into account equations (29,30,32) we

( ) ( ) *dQ t Qt I dt <sup>C</sup>*

Using the initial condition that describes the charge of the capacitors from the current source *I0* in the steady state: *QΣ(0)=I0C0/σ0* we can easily show that equation (41) has only one

**Figure 12.** Time evolution of the total charge of the generator capacitances at *fRC*0 = 9*.*6 and *η* = 10: modulation of leakages synchronized with modulation of the capacitances, equation (18)), - (*1*), and

Then, all estimates of the maximum energy produced by the ideal generator during the period of energy conversion and the estimates of the generator power are valid at a certain

As an example, Fig. 10 shows the characteristic curve 2 of *Pmax/P0* as a function of *η*, which completely coincides with curve 1 for the ideal generator. In the general case, the value of

A significant difference of this solution from the case of negligibly small leakage currents and also from the case of proportionality of the conductivity *σi* to the electrode overlapping area

 is not equal to the real value of the initial voltage *V(0)*, because the value of *V(0)* depends on the initial phase of oscillations, i.e., on particular values of the conductivities *σ1* and *σ<sup>2</sup>* at

\*

The second case also observed in practice is the case with constant leakages:

0 0 0

( ) (0) *I C Qt Q* (42)

0 00 *V I* / (43)

<sup>120</sup> ( ) ( ) /2 *t t* (44)

constant leakages (*2*) (*frC*0 = 9*.*6 102). The inset shows a zoomed-in fragment of curve *2*.

0 0 0

(41)

Assuming that the total charge decreases by a factor of k and taking into account the expression for the initial charge (33), we can easily obtain expressions for steady-state values of the minimum voltage on the capacitors *Vmin(∞)* and the energy produced by the generator *W(∞)*. These formulas are completely identical to the expressions for the ideal generator (34- 36) with *V*0 being replaced by V ( ) I /(k ) 0 0 *.* 

Thus, a typical feature of the two-capacitor generator with constant charge leakages is an additional decrease in the generated voltage by a factor of k and, correspondingly an additional decrease in power by a factor of k2, as compared with the above-described case where the leakages are proportional to the electrode overlapping area (30). The dependence of the power *P/P0* on *fRC0* is qualitatively similar to the corresponding curves for the ideal generator with the only difference that it is additionally affected by the leakage currents (parameter *frC0*): it increases linearly with decreasing leakages at a constant current *I0*. In our scales, the value of *P/P0* is almost independent of *frC0* if the leakages are sufficiently small: *r << R* (Fig. 13). As for the ideal generator, we see that *Pmax/P0* increases with increasing capacitance modulation factor η and then reaches a "plateau" (see Fig. 10, curve 4). The frequency dependence of the power is qualitatively similar to the corresponding characteristic of the ideal generator.

**Figure 13.** Characteristic curves of the generator power for different leakage resistances *r*: *frC*0 = 960 (*1*), 96 (*2*), and 9.6 (*3*); *η* = 10.

The decrease in the total charge (see Fig. 12) and the generated power, which is determined by the charge redistribution after generator actuation at the beginning of modulation of the capacitances *Ci*, has an exponential character. The time needed for the generator to reach a steady-state mode is inversely proportional to the conductivity of the leakage currents. As energy generation is efficient only at *r >> R* [5], and the time constant of the decrease in power τ has the order of *rC0/2*, this time can be sufficiently large (more than 102–103 s). Note that the quasi-steady mode of generator operation is not reached under real conditions (e.g., in the regime of harvesting the energy of microvibrations of environment) where the modulation frequencies change in a shorter time than τ; therefore, the maximum of the generated power lies between curves 1 and 4 in Fig. 10.

#### *3.1.2. Two capacitor generator in vibration out-of-plane mode*

If the two-capacitor generator works at the interelectrode gap modulation mode, when the electrode overlapping area remains constant, called "mode of vibrations", then the condition of the constant total capacitance of the capacitors in time (29) is not satisfied. In this case, the gaps of two capacitors are modulated in opposition in accordance with a sinusoidal law, while the capacitance of each capacitor is inversely proportional to the gap value (Fig. 14, curves 1 and 1 ; the quantity C0 has the meaning of a capacitance averaged over the period of vibrations). Therefore, the capacitance of each capacitor is close to the minimum value during the major part of the period of vibrations *T*. The greater the capacitance modulation factor *η*, the more pronounced this effect: when the plates of one capacitor become separated (curve 1, motion toward decreasing *C1(t)*) and the voltage on this capacitor increases accordingly (curve 2), the charge from this capacitor flows to the second capacitor whose capacitance *C2(t)* is still low (curve 1 ). Thus, in contrast to the lateral shift of the plates, the charge overflow is not matched with the motion of the plates of the second capacitor: the peak of *V1(t)* occurs earlier than the peak of *C2(t)*, i.e., the charge from the first capacitor flows to the second capacitor mainly during the time when its capacitance is close to the minimum value.

The absence of "synchronization" of the charge exchange between the capacitors in the generator in the mode of vibrations reduces the generator power (see Fig. 10, curve 5, P0 is determined by equations (36) and (39), in which *C0* is replaced by C0 ), which is manifested as a decrease in the ratio *Vmax/V0* (see Fig. 14, curve 2). Because of the lack of synchronization, there appears a second peak (with a lower amplitude) on the curve *V*1(*t*) after the peak on the curve *C*2(*t*), which also decreases the efficiency of generation in this mode.

Speaking about the generator in the vibration mode, we should emphasize the ideal generator mode (Fig. 10, curves 7, 8). In contrast to the mode of the lateral shift of the electrodes, the generated energy here depends appreciably on the initial values of the capacitances *Ci*. In particular, if the plates of one capacitor are located at the minimum distance at the beginning of the vibration process, then the maximum of energy generated in one cycle normalized to its initial value (Fig.10, curve 7) is greater than the value typical for the generator whose operation principle is based on the shift of the plates (Fig.10, curve 1). In

#### High Energy Density Capacitance Microgenerators 83

82 Small-Scale Energy Harvesting

generated power lies between curves 1 and 4 in Fig. 10.

curves 1 and 1 ; the quantity C0

minimum value.

*3.1.2. Two capacitor generator in vibration out-of-plane mode* 

The decrease in the total charge (see Fig. 12) and the generated power, which is determined by the charge redistribution after generator actuation at the beginning of modulation of the capacitances *Ci*, has an exponential character. The time needed for the generator to reach a steady-state mode is inversely proportional to the conductivity of the leakage currents. As energy generation is efficient only at *r >> R* [5], and the time constant of the decrease in power τ has the order of *rC0/2*, this time can be sufficiently large (more than 102–103 s). Note that the quasi-steady mode of generator operation is not reached under real conditions (e.g., in the regime of harvesting the energy of microvibrations of environment) where the modulation frequencies change in a shorter time than τ; therefore, the maximum of the

If the two-capacitor generator works at the interelectrode gap modulation mode, when the electrode overlapping area remains constant, called "mode of vibrations", then the condition of the constant total capacitance of the capacitors in time (29) is not satisfied. In this case, the gaps of two capacitors are modulated in opposition in accordance with a sinusoidal law, while the capacitance of each capacitor is inversely proportional to the gap value (Fig. 14,

of vibrations). Therefore, the capacitance of each capacitor is close to the minimum value during the major part of the period of vibrations *T*. The greater the capacitance modulation factor *η*, the more pronounced this effect: when the plates of one capacitor become separated (curve 1, motion toward decreasing *C1(t)*) and the voltage on this capacitor increases accordingly (curve 2), the charge from this capacitor flows to the second capacitor whose capacitance *C2(t)* is still low (curve 1 ). Thus, in contrast to the lateral shift of the plates, the charge overflow is not matched with the motion of the plates of the second capacitor: the peak of *V1(t)* occurs earlier than the peak of *C2(t)*, i.e., the charge from the first capacitor flows to the second capacitor mainly during the time when its capacitance is close to the

The absence of "synchronization" of the charge exchange between the capacitors in the generator in the mode of vibrations reduces the generator power (see Fig. 10, curve 5, P0 is

as a decrease in the ratio *Vmax/V0* (see Fig. 14, curve 2). Because of the lack of synchronization, there appears a second peak (with a lower amplitude) on the curve *V*1(*t*) after the peak on

Speaking about the generator in the vibration mode, we should emphasize the ideal generator mode (Fig. 10, curves 7, 8). In contrast to the mode of the lateral shift of the electrodes, the generated energy here depends appreciably on the initial values of the capacitances *Ci*. In particular, if the plates of one capacitor are located at the minimum distance at the beginning of the vibration process, then the maximum of energy generated in one cycle normalized to its initial value (Fig.10, curve 7) is greater than the value typical for the generator whose operation principle is based on the shift of the plates (Fig.10, curve 1). In

determined by equations (36) and (39), in which *C0* is replaced by C0

the curve *C*2(*t*), which also decreases the efficiency of generation in this mode.

has the meaning of a capacitance averaged over the period

), which is manifested

**Figure 14.** Time evolution of the capacitances of the first (*1*) and second (*1'* ) capacitors and the voltage (*2*) for the first capacitor of the generator in the optimum power region. Mode of out-of-plane antiphase vibrations, *η* = 10, and *fRC*0 = 0*.*854.

this case, the ultimate power for two capacitor generators is reached, because it is possible to ensure the minimum possible gaps between the electrodes of the generator capacitors and, therefore, the maximum possible values of the capacitance. However, if the beginning of the vibration process does not coincide with the instant when the maximum capacitance of one of the capacitors is reached, then the energy generation efficiency drastically decreases (Fig.10, curve 8). This behavior of the generated power is explained by the magnitude of the charge trapped at the beginning of the process: if one of the capacitances has the maximum value at the beginning of the process (Fig.10, curve 7), then the initial charge also reaches the maximum value. Under different initial conditions, the smaller charge is trapped first (intermediate curves between 7 and 8, Fig.10).

#### **3.2. Compensation of charge losses with the use of a voltage source**

The charge losses are compensated with the use of a voltage source by connecting the source for a short time to one capacitor only (see Fig. 1f). The charge losses on the second capacitor are compensated owing to the current flowing through the load resistance *R* in the process of generation with modulation of the capacitances. Breaux [37] considered another method: the charging of both capacitors. In this case, however, highly accurate synchronization of two switches is needed because even a small delay in commutation of switches leads to significant reduction of the generation efficiency.

The recharging voltage pulse should be applied at *V1(t) <V0* and it should be finished at *C1(t) = Cmax*. An example of the optimum synchronization of the switch connecting the source *V0* in agreement with the capacitance modulation periods for the case of the lateral antiphase shift of the moving electrodes of the capacitors is shown in Fig. 15. Here the period of charging pulses *Tch* was less than time constant of charge losses, no pronounced charge decay at *t< Tch* was observed, see Fig.15b.

When a series of recharging pulses is applied to compensate charge losses one can see the additional effect of charge growth by a factor of two compared to the case of ideal generator (Fig.15b) and corresponding growth of *Vmin* value up to *V*min = *V*0 (Fig.15a).

**Figure 15.** Example of solving the problem of compensation of charge losses by recharging the capacitor from a voltage source. Mode of the lateral shift of the capacitor plates: (**a**) voltage on the first capacitor; (**b**) total charge. Recharging pulse duration 10 μs, amplitude 1 V, supply period 100 ms, number of cycles per the period *N =* 10, frequency 100 Hz, *Cmin* = 87.6 pF, *η* = 10, *R* = 2 108 Ω, and *r* = 1012 Ω.

Therefore, in this case the maximum energy transferred to the load resistance is greater than the energy of ideal generator by a factor of 4, and of single-capacitor generator (2) by a factor of 2; in the limit (at *η>>*1) it tends to the value *C*max*V02 η* (see Fig. 10, curve *6*).

Thus, the use of a switch performing synchronous recharging of the capacitor *C1*, in addition to compensation of charge losses, increases the charge to the limiting value *2CmaxV0*, which involves an increase in the generated energy up to values exceeding the energy generated by the ideal generator by a factor of 4. Note that similar features are also observed for the two-capacitor generator in vibration out-of-plane mode, leading to an even more dramatic increase in the generator power in this case (cf. curves 5 and 6 in Fig. 10).

### **4. Experimental studies of two-capacitor rotational generator**

To prove the possibility of electric energy generation under the action of mechanical forces with highly efficient utilization of the charge injected into the generator, we performed experimental studies using a macroscopic model of two-capacitor generator consisting of two stator plates and rotor plate located exactly between them. Each plate was metalized and divided into 12 sectors in such a way as to provide the central plate with the two series of the connected capacitors modulated in antiphase when central plate is rotating. The area of the electrodes was 25 cm2. The moving electrode was fixed on the shaft of a d. c. motor rotated with a frequency of the order of 1–50 Hz, therefore, the capacitor modulation frequency was 10–600 Hz. All plates were insulated from the body and shaft of the motor with the use of insulators having a high resistance (above 100 GΩ). The gap between the plates was 100–200 μm, the capacitance *C0* had the order of 250–350 pF, and the modulation factor was *η* = 1*.*8*−*3*.*5; these parameters were determined by independent measurements. The current through the load resistance *I(t)* and the voltage on it *V (t)* were measured by a digital oscilloscope using the matching circuit.

Ω.

**Figure 15.** Example of solving the problem of compensation of charge losses by recharging the capacitor from a voltage source. Mode of the lateral shift of the capacitor plates: (**a**) voltage on the first capacitor; (**b**) total charge. Recharging pulse duration 10 μs, amplitude 1 V, supply period 100 ms, number of cycles per the period *N =* 10, frequency 100 Hz, *Cmin* = 87.6 pF, *η* = 10, *R* = 2 108 Ω, and *r* = 1012

of 2; in the limit (at *η>>*1) it tends to the value *C*max*V02 η* (see Fig. 10, curve *6*).

increase in the generator power in this case (cf. curves 5 and 6 in Fig. 10).

digital oscilloscope using the matching circuit.

**4. Experimental studies of two-capacitor rotational generator** 

Therefore, in this case the maximum energy transferred to the load resistance is greater than the energy of ideal generator by a factor of 4, and of single-capacitor generator (2) by a factor

a. b.

Thus, the use of a switch performing synchronous recharging of the capacitor *C1*, in addition to compensation of charge losses, increases the charge to the limiting value *2CmaxV0*, which involves an increase in the generated energy up to values exceeding the energy generated by the ideal generator by a factor of 4. Note that similar features are also observed for the two-capacitor generator in vibration out-of-plane mode, leading to an even more dramatic

To prove the possibility of electric energy generation under the action of mechanical forces with highly efficient utilization of the charge injected into the generator, we performed experimental studies using a macroscopic model of two-capacitor generator consisting of two stator plates and rotor plate located exactly between them. Each plate was metalized and divided into 12 sectors in such a way as to provide the central plate with the two series of the connected capacitors modulated in antiphase when central plate is rotating. The area of the electrodes was 25 cm2. The moving electrode was fixed on the shaft of a d. c. motor rotated with a frequency of the order of 1–50 Hz, therefore, the capacitor modulation frequency was 10–600 Hz. All plates were insulated from the body and shaft of the motor with the use of insulators having a high resistance (above 100 GΩ). The gap between the plates was 100–200 μm, the capacitance *C0* had the order of 250–350 pF, and the modulation factor was *η* = 1*.*8*−*3*.*5; these parameters were determined by independent measurements. The current through the load resistance *I(t)* and the voltage on it *V (t)* were measured by a

**Figure 16.** Experimental studies of two-capacitor generator: (**a**) - relaxation of voltage oscillations in the load resistance (*C*0 = 300 pF, *η* = 1*.*8, *R* = 10 MΩ, and *f* = 400 Hz), (**b**) - characteristic loading curves of the ideal generator: *f* = 400 Hz (*1*), *f* = 100 Hz (*C*0 = 300 pF and *η* = 1*.*8) (*2*), and calculation by the proposed model (*3*).

The oscillogram characterizing energy generation is shown in Fig. 16a. A charge of 2 10−8 C was initially injected at the time *t* = 0 into this structure by a short pulse of voltage equal to 80 V. After that, the flow of the current of up to 8 μA (acting value) through the load resistance *R* = 10 MΩ in the process of rotation of the moving electrode with the effective frequency of 400 Hz is determined by this charge. The initial time of voltage redistribution is of order *RC0* = 3·10−3 s, after that dynamically equilibrium mode of generation is established. Therefore, the measured initial amplitude of the voltage on the load resistance *R* (e.g., 40 V, based on the data in Fig. 16a) was used to calculate the power of the ideal generator in this mode on the basis of the proposed experimental model. Initial charging of the capacitors ensures energy generation for a long time (up to 1000 s). During this time, more than 4 105 cycles of energy conversion with the use of this charge take place, and the Joule energy released on the load resistance is much greater than the energy spent on initial charging of the generator capacitors. Based on the time constant of charge decay, we can easily estimate the leakage resistance; for the circuit considered, it is approximately 1012 Ω. The power developed by the generator with a 80-V starting voltage was 1 mW.

To confirm the main result of the developed generator model, i.e., the universal character of the dependences of the generated power on the load resistance R, we studied the specific features of energy generation at the initial stage of the process when the leakages could be neglected (in this period, the current amplitude is close to its value for the ideal generator). The loading curves plotted in the *RfC0–P/P0* coordinates for different modulation frequencies of the generator capacitances were found to be almost coincident, i.e., to have a universal character and also to agree well with the model described above (Fig. 16b). Moderate disagreement is explained by a small difference in the modulation factors of two capacitors.

## **5. Peculiarities of microgenerators operation with vibrations of moving electrode in submicron gap above the surface of ferroelectric-metal structure**

A technology of mutual shifting (vibration) of the surfaces of the microcircuit components in the submicron range was developed during the last 5–7 years in modern microelectronics, namely, in its most intensely developing direction: MEMS, e.g., gyroscopes, generators, frequency stabilizers, high-frequency filters. Recently we have demonstrated the possibility of use the nanogaps in electromechanical energy conversion for applications in micromotors and actuators [40-42], which is promising for use in inverse mechanic-to-electricity energy transformation also.

The characteristic feature of operation of electrostatic capacitance microgenerators, the capacitors of that having the structure consisting of substrate – metal - thin ferroelectric layer – moving electrode, is the possibility of creation of high electric field densities in the submicron gap between the surfaces of ferroelectric and moving electrode. Thus, high energy of electric field is stored, which is transformed then into the current. In this structure thin ferroelectric with high ε (more than 1000) plays a role of damping layer to supress a breakdown in the air gap, because the breakdown is controlled here by breakdown field strength of the ferroelectric (more than 107 V/m). Because the field strength distribution in the layers of the structure is inversly proportional to their ε ratio then the major portion of voltage is applied to the gap and the field in ferroelectric is much less than its breakdown value, even at high *V*. Therefore, the breakdown does not occure at high field strength in the gap reaching 1010 V/m at the gaps of order 10-100 nm and votages of 100 V, according to our experimental data.

The expressions for electric field strength in the gap *E1* and in the ferroelectric film *E*, and for the energy stored in the structure are the following:

$$E\_1 = \frac{V}{d \;/\; \varepsilon + d\_1}, \quad E = E\_1 \;/\; \varepsilon, \quad W = \frac{\varepsilon\_0 V^2}{2(d \;/\; \varepsilon + d\_1)}\;. \tag{45}$$

At *d1>>d/ε*, the following expressions are always true in the field of parameters listed above:

$$E\_1 \approx \frac{V}{d\_1}, \quad E \approx \frac{V}{\varepsilon d\_1}, \quad W \approx \frac{\varepsilon\_0 V^2}{2d\_1} = \frac{\varepsilon\_0 V}{2} \\ E\_1 = \frac{\varepsilon\_0 d\_1 E\_1^2}{2}. \tag{46}$$

Thus the maximum energy generated per one sycle could reach 1-4 J/m2 during the operation of microgenerators at submicron gaps. At low frequencies of order 10-100 Hz the output power could reach up to 10-40 mW/cm2. This estimate is true for all types of capacitance generators discribed above, in spite of a number of differencies in various types of their implementation.

### **6. Discussion and conclusions**

1. The analysis of general laws of operation of microgenerators based on the use of multilayer structure consisting of electrode – thin dielectric - air gap – moving electrode has been performed taking into account the oscillatory motions with modulation of both the electrodes overlap area (including a rotational motion) and interelectrode gap.

86 Small-Scale Energy Harvesting

**structure** 

**5. Peculiarities of microgenerators operation with vibrations of moving electrode in submicron gap above the surface of ferroelectric-metal** 

A technology of mutual shifting (vibration) of the surfaces of the microcircuit components in the submicron range was developed during the last 5–7 years in modern microelectronics, namely, in its most intensely developing direction: MEMS, e.g., gyroscopes, generators, frequency stabilizers, high-frequency filters. Recently we have demonstrated the possibility of use the nanogaps in electromechanical energy conversion for applications in micromotors and actuators [40-42], which is promising for use in inverse mechanic-to-electricity energy transformation also. The characteristic feature of operation of electrostatic capacitance microgenerators, the capacitors of that having the structure consisting of substrate – metal - thin ferroelectric layer – moving electrode, is the possibility of creation of high electric field densities in the submicron gap between the surfaces of ferroelectric and moving electrode. Thus, high energy of electric field is stored, which is transformed then into the current. In this structure thin ferroelectric with high ε (more than 1000) plays a role of damping layer to supress a breakdown in the air gap, because the breakdown is controlled here by breakdown field strength of the ferroelectric (more than 107 V/m). Because the field strength distribution in the layers of the structure is inversly proportional to their ε ratio then the major portion of voltage is applied to the gap and the field in ferroelectric is much less than its breakdown value, even at high *V*. Therefore, the breakdown does not occure at high field strength in the gap reaching 1010 V/m at the gaps of

order 10-100 nm and votages of 100 V, according to our experimental data.

1 , /

> *<sup>V</sup> <sup>E</sup> <sup>d</sup>* ,

*d d* <sup>1</sup> *E E* / ,

1

the energy stored in the structure are the following:

1

1

of their implementation.

**6. Discussion and conclusions** 

*<sup>V</sup> <sup>E</sup>*

1 *<sup>V</sup> <sup>E</sup> <sup>d</sup>* ,

The expressions for electric field strength in the gap *E1* and in the ferroelectric film *E*, and for

At *d1>>d/ε*, the following expressions are always true in the field of parameters listed above:

Thus the maximum energy generated per one sycle could reach 1-4 J/m2 during the operation of microgenerators at submicron gaps. At low frequencies of order 10-100 Hz the output power could reach up to 10-40 mW/cm2. This estimate is true for all types of capacitance generators discribed above, in spite of a number of differencies in various types

1. The analysis of general laws of operation of microgenerators based on the use of multilayer structure consisting of electrode – thin dielectric - air gap – moving electrode has

*d*

2 0 <sup>1</sup> 2( / )

. (45)

. (46)

*d d*

2 2 0 0 011 1 <sup>1</sup> 22 2 *V V dE W E*

*<sup>V</sup> <sup>W</sup>*

2. It was shown that the use of the submicron gaps in these microgenerators gives rise to considerable increase of output power. To achieve high energy output of these microgenerators it is necessary to have high values of maximal capacitance of generating element *Cmax* and electric field strength in the interelectrode gap. These generators can develop a power of 40 mW/cm2 and more for the range of low-frequency vibrations characteristic of the vibrations of environment without the use of voltage sources.

 The closest manufactured analogs of such generators are electret in-plane devices (with lateral shift of moving electrode). They have rather large sizes (about 20\*20 mm2) and small specific power (of order 100 μW/cm2) despite the fact that the value of power was considerably increased by means of multiple overlapping of strips in interdigitated comb structure (see [5]) of the generator with high displacements (of about 1 mm) in resonance mode of operation [36]. It should be noted that these devices have large interelectrode gaps (more than 20 μm), which prevent the essential decrease of the sizes of generators to reach the values needed for the microelectronics. Therefore, we believe that the only alternative to solve the problem is the development of out-of-plane (vibration) constructions of generators.

3. The mathematical model of the generators was developed, and the numerical solutions describing the process of generation were derived. The universal type of these solutions was confirmed, and the analytical description of the output maximum power in dependence of capacitance modulation depth was carried out.

 The main parameters controlling the efficiency of electret generator operation were determined to be the ratio of charge built in dielectric layer to its geometric capacitance *Vp=Qp/CF* and the value of mean capacitance of air gap *C1*. It was shown that for twocapacitor capacitance generators these parameters are the values of maximum capacitance *Cmax* and capacitance modulation depth *η*.

4. Unlike the one-capacitor prototype it was shown that for these generators it is not necessary to recuperate the charge in each cycle of power generation. For the electret generator there is no need at all to turn on the source to compensate charge losses. However, unlike the electret generator working in the current mode, to enlarge the output power it is necessary to use the switch synchronized with the certain phase of oscillations to short-circuit the plates of capacitor to eliminate the parasitic induced charge.

 There is no need to renew the charge at each cycle of power generation for twocapacitor generator in which the charge serves as "working medium" for production the electric power. It is necessary to compensate only small charge losses arising due to leakage currents in the capacitors.

It was determined that for the charge recuperation there is no need to connect the charge sources to each capacitor, it is enough to connect the source only to one of the capacitors. It

can be the current source (connected permanently) or the voltage source (connected for a short period of time in a certain phase of capacitance alteration and periodically after the high enough number of energy transformation cycles). In this case one can use the feedback circuit utilizing a small part of output energy for the recuperation of the initial charge.

### **Author details**

Igor L. Baginsky and Edward G. Kostsov *Institute of Automation and Electrometry, Russian Academy of Sciences, Russia* 

### **7. References**


**Author details** 

**7. References** 

409–425.

Electron. E87: 549–555.

A115:.523–529.

No.1: 89-102.

160.

661.

234–242.

Igor L. Baginsky and Edward G. Kostsov

can be the current source (connected permanently) or the voltage source (connected for a short period of time in a certain phase of capacitance alteration and periodically after the high enough number of energy transformation cycles). In this case one can use the feedback circuit utilizing a small part of output energy for the recuperation of the initial charge.

[1] Roundy S, Wright P K, Rabaey J (2003) A study of low level vibrations as a power source

[2] Stephen N G (2006) On energy harvesting from ambient vibration. J. Sound Vib*.* 293:

[3] Miyazaki M., Tanaka H, Ono G, Nagano T, Ohkubo N, Kawahara T (2004) Electricenergy generation using variable-capacitive resonator for power-free-LSI. IEICE Trans.

[4] Mitcheson P D, Miao P, Stark B H, Yeatman E M, Holmes A S, Green T C (2004) MEMS electrostatic micropower generator for low frequency operation. Sensors Actuators.

[5] Baginsky I L, Kostsov E G (2002) The possibility of creating a microelectronic electrostatic energy generator. Optoelectronics, Instrumentation and Data Processing.

[6] El-Hami M, Glynne-Jones P, White N M, et al. (2001) Design and fabrication of a new vibration based electromechanical power generator. Sensors Actuators. A92: 335–342. [7] Meninger S, Mur-Miranda J, Lang J, et al. (2001) Vibration to electric energy conversion.

[8] Roundy S, Wright P K (2004) A piezoelectric vibration based generator for wireless

[9] Du Toit N E, Wardle B L, Kim S-G (2005) Design considerations for MEMS-scale piezoelectric mechanical vibration energy harvesters. Integrated Ferroelectrics. 71: 121–

[10] Chen C-T, Islam R A, Priya S (2006) Electric energy generator, ultrasonics, ferroelectrics and frequency control. IEEE Trans. Ultrasonics, Ferroelectrics Freq. Control, 53: 656–

[11] Dragunov V P, Kostsov E G (2009) Specific features of operation of electrostatic microgenerators of energy. Optoelectronics, Instrumentation and Data Processing. 45:

[12] Beeby S P, Tudor M J, White N M (2006) Energy harvesting vibration sources for

*Institute of Automation and Electrometry, Russian Academy of Sciences, Russia* 

for wireless sensor nodes. Computer Communs. 26: 1131-1144.

IEEE Trans. Very Large Scale Integration (VLSI) Syst.. 9: 64–76.

mycrosystems applications. Meas. Sci. Technol. 17: R175-R195.

electronics. Smart Mater. Struct.. 13: 1131–1142.


**Chapter 5** 

## **Electrostatic Conversion for Vibration Energy Harvesting**

S. Boisseau, G. Despesse and B. Ahmed Seddik

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51360

### **1. Introduction**

90 Small-Scale Energy Harvesting

U.S.A.: 223-226.

Trans. 6: 101–111.

Processing. 47: 100 – 120.

Microeng. 18 104006 – 104014.

Micromech. Microeng. 18: 104011-104017.

Materials. Oxford: Oxford University Press.

ceramics. J. Micromech. Microeng. 14: 1569–1575.

ferroelectric materials. Ferroelectrics. 351: 66–78.

Micromech. Microeng. 13: 190–200.

[31] Ma W, Zhu R, Rufer L, Zohar Yi, Wong M (2007) An integrated floating-electrode

[32] Mahmoud M A, El-Saadany E F, Mansour R R (Nov. 29 - Dec. 1, 2006) Planar electret based electrostatic micro-generator. The Sixth International Workshop on Micro and Nanotechnology for Power Generation and Energy Conversion Applications. Berkeley,

[33] Lo H, Tai Y-Ch (2008) Parylene-based electret power generators. J. Micromech.

[34] Suzuki Y (2011) Recent progress in MEMS electret generator for energy harvesting. IEEJ

[36] Sakane Y, Suzuki Y, Kasagi N (2008) The development of a high-performance perfluorinated polymer electret and its application to micro power generation. J.

[38] Baginsky I L, Kamyshlov V F, Kostsov E G (2011) Specific features of operation of a two-capacitor electrostatic generator. Optoelectronics, Instrumentation and Data

[39] Lines M, Glass A (1977) Principles and Applications of Ferroelectrics and Related

[40] Baginsky I L, Kostsov E G (2003) High-energy capacitive electrostatic micromotors. J.

[41] Baginsky I L, Kostsov E G (2004) Electrostatic micromotor based on ferroelectric

[42] Baginsky I L, Kostsov E G (2007) High energy output MEMS based on thin layers of

[35] Lo H-W, Tai Y-Ch (2009) Electret power generator. US patent 0174281 A1.

[37] Breaux O P (1978) Electrostatic energy conversion system. US patent 4127804.

electric microgenerator. J. Microelectromech. Sys. 16: 29-37.

"Everything will become a sensor"; this is a global trend to increase the amount of information collected from equipment, buildings, environments… enabling us to interact with our surroundings, to forecast failures or to better understand some phenomena. Many sectors are involved: automotive, aerospace, industry, housing. Few examples of sensors and fields are overviewed in Figure 1.

Force sensor (CEA-Leti)

**Figure 1.** Millions sensors in our surroundings

Unfortunately, it is difficult to deploy many more sensors with today's solutions, for two main reasons:

1. Cables are becoming difficult and costly to be drawn (inside walls, on rotating parts)

© 2012 Boisseau et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Boisseau et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2. Battery replacements in wireless sensor networks (WSN) are a burden that may cost a lot in large factories (hundreds or thousands sensor nodes).

As a consequence, industrialists, engineers and researchers are looking for developing autonomous WSN able to work for years without any human intervention. One way to proceed consists in using a green and theoretically unlimited source: ambient energy [1].

### **1.1. Ambiant energy & applications**

Four main ambient energy sources are present in our environment: mechanical energy (vibrations, deformations), thermal energy (temperature gradients or variations), radiant energy (sun, infrared, RF) and chemical energy (chemistry, biochemistry).

These sources are characterized by different power densities (Figure 2). Energy Harvesting (EH) from outside sun is clearly the most powerful (even if values given in Figure 2 have to be weighted by conversion efficiencies of photovoltaic cells that rarely exceed 20%). Unfortunately, solar energy harvesting is not possible in dark areas (near or inside machines, in warehouses). And similarly, it is not possible to harvest energy from thermal gradients where there is no thermal gradient or to harvest vibrations where there is no vibration.

As a consequence, the source of ambient energy must be chosen according to the local environment of the WSN's node: no universal ambient energy source exists.

**Figure 2.** Ambient sources power densities before conversion

Figure 2 also shows that 10-100µW of available power is a good order of magnitude for a 1cm² or a 1cm³ energy harvester. Obviously, 10-100µW is not a great amount of power; yet it can be enough for many applications and especially WSN.

### **1.2. Autonomous wireless sensor networks & needs**

A simple vision of autonomous WSN' nodes is presented in Figure 3(a). Actually, autonomous WSN' nodes can be represented as 4 boxes devices: (i) "sensors" box, (ii) "microcontroller (µC)" box, (iii) "radio" box and (iv) "power" box. To power this device by EH, it is necessary to adopt a "global system vision" aimed at reducing power consumption of sensors, µC and radio.

Actually, significant progress has already been accomplished by microcontrollers & RF chips manufacturers (Atmel, Microchip, Texas Instruments…) both for working and standby modes. An example of a typical sensor node's power consumption is given in Figure 3(b). 4 typical values can be highlighted:


92 Small-Scale Energy Harvesting

vibration.

**1.1. Ambiant energy & applications** 

2. Battery replacements in wireless sensor networks (WSN) are a burden that may cost a

As a consequence, industrialists, engineers and researchers are looking for developing autonomous WSN able to work for years without any human intervention. One way to proceed consists in using a green and theoretically unlimited source: ambient energy [1].

Four main ambient energy sources are present in our environment: mechanical energy (vibrations, deformations), thermal energy (temperature gradients or variations), radiant

These sources are characterized by different power densities (Figure 2). Energy Harvesting (EH) from outside sun is clearly the most powerful (even if values given in Figure 2 have to be weighted by conversion efficiencies of photovoltaic cells that rarely exceed 20%). Unfortunately, solar energy harvesting is not possible in dark areas (near or inside machines, in warehouses). And similarly, it is not possible to harvest energy from thermal gradients where there is no thermal gradient or to harvest vibrations where there is no

As a consequence, the source of ambient energy must be chosen according to the local

Figure 2 also shows that 10-100µW of available power is a good order of magnitude for a 1cm² or a 1cm³ energy harvester. Obviously, 10-100µW is not a great amount of power; yet it

A simple vision of autonomous WSN' nodes is presented in Figure 3(a). Actually, autonomous WSN' nodes can be represented as 4 boxes devices: (i) "sensors" box, (ii)

lot in large factories (hundreds or thousands sensor nodes).

energy (sun, infrared, RF) and chemical energy (chemistry, biochemistry).

environment of the WSN's node: no universal ambient energy source exists.

**Figure 2.** Ambient sources power densities before conversion

can be enough for many applications and especially WSN.

**1.2. Autonomous wireless sensor networks & needs** 


**Figure 3.** (a) Autonomous WSN node and (b) sensor node's power consumption

Then, the energy harvester has to scavenge at least 5µW to compensate the standby mode's power consumption, and a bit more to accumulate energy (50-500µJ) in a storage that is used to supply the following measurement cycle.

Today's small scale EH devices (except PV cells in some cases) cannot supply autonomous WSN in a continuous active mode (500µW-1mW power consumption vs 10-100µW for EH output power). Fortunately, thanks to an ultra-low power consumption in standby mode, EH-powered autonomous WSN can be developed by adopting an intermittent operation mode as presented in Figure 4. Energy is stored in a buffer (a) (capacitor, battery) and used to perform a measurement cycle as soon as enough energy is stored in the buffer (b & c). System then goes back to standby mode (d) waiting for a new measurement cycle.

Therefore, it is possible to power any application thanks to EH, even the most consumptive one; the main challenge is to adapt the measurement cycle frequency to the continuously harvested power.

As a consequence, Energy Harvesting can become a viable supply source for Wireless Sensor Networks of the future.

**Figure 4.** WSN measurement cycle

This chapter focuses on Vibration Energy Harvesting that can become an interesting power source for WSN in industrial environments with low light or no light at all. We will specifically concentrate on electrostatic devices, based on capacitive architectures, that are not as well-known as piezoelectric or electromagnetic devices, but that can present many advantages compared to them.

The next paragraph introduces the general concept of Vibration Energy Harvesters (VEH) and of electrostatic devices.

### **2. Vibration energy harvesting & electrostatic devices**

Vibration Energy Harvesting is a concept that began to take off in the 2000's with the growth of MEMS devices. Since then, this concept has spread and conquered macroscopic devices as well.

### **2.1. Vibration energy harvesters – Overview**

The concept of Vibration Energy Harvesting is to convert vibrations in an electrical power. Actually, turning ambient vibrations into electricity is a two steps conversion (Figure 5(a)). Vibrations are firstly converted in a relative motion between two elements, thanks to a mass-spring system, that is then converted into electricity thanks to a mechanical-toelectrical converter (piezoelectric material, magnet-coil, or variable capacitor). As ambient vibrations are generally low in amplitude, the use of a mass-spring system generates a phenomenon of resonance, amplifying the relative movement amplitude of the mobile mass compared to the vibrations amplitude, increasing the harvested power (Figure 5(b)).

Figure 5(c) represents the equivalent model of Vibration Energy Harvesters. A mass (m) is suspended in a frame by a spring (k) and damped by forces (felec and fmec). When a vibration occurs *y*(*t*) *Y* sin(*t*) , it induces a relative motion of the mobile mass *x*(*t*) *X* sin(*t* ) compared to the frame. A part of the kinetic energy of the moving mass is converted into electricity (modeled by an electromechanical force felec), while an other part is lost in friction forces (modeled by fmec).

94 Small-Scale Energy Harvesting

buffer

buffer

**Figure 4.** WSN measurement cycle

advantages compared to them.

and of electrostatic devices.

µC

µC

sensor

T°, P, A (b)

sensor

**2. Vibration energy harvesting & electrostatic devices** 

**2.1. Vibration energy harvesters – Overview** 

T°, P, A (d)

This chapter focuses on Vibration Energy Harvesting that can become an interesting power source for WSN in industrial environments with low light or no light at all. We will specifically concentrate on electrostatic devices, based on capacitive architectures, that are not as well-known as piezoelectric or electromagnetic devices, but that can present many

The next paragraph introduces the general concept of Vibration Energy Harvesters (VEH)

Vibration Energy Harvesting is a concept that began to take off in the 2000's with the growth of MEMS devices. Since then, this concept has spread and conquered macroscopic devices as

The concept of Vibration Energy Harvesting is to convert vibrations in an electrical power. Actually, turning ambient vibrations into electricity is a two steps conversion (Figure 5(a)). Vibrations are firstly converted in a relative motion between two elements, thanks to a mass-spring system, that is then converted into electricity thanks to a mechanical-toelectrical converter (piezoelectric material, magnet-coil, or variable capacitor). As ambient vibrations are generally low in amplitude, the use of a mass-spring system generates a phenomenon of resonance, amplifying the relative movement amplitude of the mobile mass

compared to the vibrations amplitude, increasing the harvested power (Figure 5(b)).

Figure 5(c) represents the equivalent model of Vibration Energy Harvesters. A mass (m) is suspended in a frame by a spring (k) and damped by forces (felec and fmec). When a vibration

emitter

ambient energy

> ambient energy

buffer

buffer

µC

µC

sensor

T°, P, A .

sensor

T°, P, A

emitter

emitter

emitter

(a)

(c)

well.

ambient energy

ambient energy

**Figure 5.** Vibration Energy Harvesters (a) concept (b) resonance phenomenon and (c) model

Newton's second law gives the differential equation that rules the moving mass's relative movement (equation 1). Generally, the mechanical friction force can be modeled as a viscous force *fmec bm x* . Then, the equation of movement can be simplified by using the natural angular frequency <sup>0</sup> *k m* and the mechanical quality factor *<sup>m</sup> <sup>m</sup> <sup>Q</sup> <sup>m</sup> <sup>b</sup>* <sup>0</sup> .

$$m\ddot{\mathbf{x}} + f\_{mca} + k\mathbf{x} + f\_{elec} = \mathbf{-}m\ddot{\mathbf{y}} \implies \ddot{\mathbf{x}} + \frac{a\mathbf{o}\_0}{Q\_m}\dot{\mathbf{x}} + a\mathbf{o}\_0^2 \mathbf{x} + \frac{f\_{elec}}{m} = \mathbf{-}\ddot{\mathbf{y}} \tag{1}$$

Then, when the electromechanical and the friction forces can be modeled by viscous forces, *f b x elec <sup>e</sup>* and *fmec bm x* , where *be* and *bm* are respectively electrical and mechanical damping coefficients, William and Yates [2] have proven that the maximum output power of a resonant energy harvester submitted to an ambient vibration is reached when the natural angular frequency (<sup>0</sup> ) of the mass-spring system is equal to the angular frequency of ambient vibrations ( ) and when the damping rate <sup>0</sup> *<sup>e</sup> be* 2*m* of the electrostatic force *elec f* is equal to the damping rate <sup>0</sup> *<sup>m</sup> bm* 2*m* of the mechanical friction force *mec f* . This maximum output power *PW&Y* can be simply expressed with (2), when *<sup>e</sup> <sup>m</sup>* 1 2*Qm* .

$$P\_{W\&Y} = \frac{mY^2 \alpha\_0^{-3} Q\_m}{8} \tag{2}$$

But obviously, to induce this electromechanical force, it is necessary to develop a mechanical-to-electrical converter to extract a part of mechanical energy from the mass and to turn it into electricity.

#### **2.2. Converters & electrostatic devices – Overview**

Three main converters enable to turn mechanical energy into electricity: piezoelectric devices, electromagnetic devices and electrostatic devices (Table 1).


**Table 1.** Mechanical-to-electrical converters for small-scale devices

Obviously, each of these converters presents both advantages and drawbacks depending on the application (amplitudes of vibrations, frequencies…).

### **2.3. Advantages & Drawbacks of Electrostatic Devices**

A summary of advantages and drawbacks of electrostatic devices is presented in Table 2. In most cases, piezoelectric and electrostatic devices are more appropriate for small scale energy harvesters (<1-10 cm³) while electromagnetic converters are better for larger devices.

This chapter is focused on electrostatic vibration energy harvesters. These VEH are welladapted for size reduction, increasing electric fields, capacitances and therefore converters' power density capabilities. They also offer the possibility to decouple the mechanical structure and the converter (which is not possible with piezoelectric devices). Finally, they can be a solution to increase the market of EH-powered WSN by giving the possibility to develop "low-cost" devices as they do not need any magnet or any piezoelectric material that can be quite expensive.

The next paragraph is aimed at presenting the conversion principles of electrostatic devices. It covers both standard (electret-free) and electret-based electrostatic converters.

### **2.4. Conversion principles**

Electrostatic converters are capacitive structures made of two plates separated by air, vacuum or any dielectric materials. A relative movement between the two plates generates a capacitance variation and then electric charges. These devices can be divided into two categories:



**Table 2.** Advantages and drawbacks of converters

#### *2.4.1. Electret-free electrostatic converters*

These first electrostatic devices are passive structures that require an energy cycle to convert mechanical energy into electricity. Many energy cycles enable such a conversion, but the most commonly-used are charge-constrained and voltage-constrained cycles (Figure 6). They both start when the converter's capacitance is maximal. At this point, a charge is injected into the capacitor thanks to an external source, to polarize it. Charge-constrained and voltage-constrained cycles are presented in the following sub-sections.

1. Charge-constrained Cycle

96 Small-Scale Energy Harvesting

and a magnet.

Use of piezoelectric

piezoelectric electrode

electrode

stress/strain

that can be quite expensive.

**2.4. Conversion principles** 

categories:

charges when they are under stress/strain.

R

**Table 1.** Mechanical-to-electrical converters for small-scale devices

the application (amplitudes of vibrations, frequencies…).

**2.3. Advantages & Drawbacks of Electrostatic Devices** 

relative motion between two plates.

Piezoelectric devices: they use piezoelectric materials that present the ability to generate

 Electromagnetic devices: they are based on electromagnetic induction and ruled by Lenz's law. An electromotive force is generated from a relative motion between a coil

Electrostatic devices: they use a variable capacitor structure to generate charges from a

**Piezoelectric converters Electromagnetic converters Electrostatic converters** 

S N

magnet

movement

materials Use of Lenz's law Use of a variable capacitor

<sup>i</sup> coil

Obviously, each of these converters presents both advantages and drawbacks depending on

A summary of advantages and drawbacks of electrostatic devices is presented in Table 2. In most cases, piezoelectric and electrostatic devices are more appropriate for small scale energy harvesters (<1-10 cm³) while electromagnetic converters are better for larger devices. This chapter is focused on electrostatic vibration energy harvesters. These VEH are welladapted for size reduction, increasing electric fields, capacitances and therefore converters' power density capabilities. They also offer the possibility to decouple the mechanical structure and the converter (which is not possible with piezoelectric devices). Finally, they can be a solution to increase the market of EH-powered WSN by giving the possibility to develop "low-cost" devices as they do not need any magnet or any piezoelectric material

The next paragraph is aimed at presenting the conversion principles of electrostatic devices.

Electrostatic converters are capacitive structures made of two plates separated by air, vacuum or any dielectric materials. A relative movement between the two plates generates a capacitance variation and then electric charges. These devices can be divided into two

It covers both standard (electret-free) and electret-based electrostatic converters.

R

structure

electrode

movement electrode

R

i

The charge-constrained cycle (Figure 7) is the easiest one to implement on electrostatic devices. The cycle starts when the structure reaches its maximum capacitance Cmax (Q1). In this position, the structure is charged thanks to an external polarization source: an electric charge Qcst is stored in the capacitor under a given voltage Umin. The device is then let in open circuit (Q2). The structure moves mechanically to a position where its capacitance is minimal (Q3). As the charge Qcst is kept constant while the capacitance C decreases, the voltage across the capacitor U increases. When the capacitance reaches its minimum (Cmin) (or the voltage its maximum (Umax)), electric charges are removed from the structure (Q4).

**Figure 6.** Standard energy conversion cycles for electret-free electrostatic devices

**Figure 7.** Charge-constrained cycle

The total amount of energy converted at each cycle is presented in equation (3).

$$E\_{Q=ct\epsilon} = \frac{1}{2} Q\_{cst}^2 (\frac{1}{C\_{\text{min}}} - \frac{1}{C\_{\text{max}}}) \tag{3}$$

#### 2. Voltage-Constrained Cycle

The voltage-constrained cycle (Figure 8) also starts when the capacitance of the electrostatic converter is maximal. The capacitor is polarized at a voltage Ucst using an external supply source (battery, charged capacitor…) (V1). This voltage will be maintained throughout the conversion cycle thanks to an electronic circuit. Since the voltage is constant and the capacitance decreases, the charge of the capacitor increases, generating a current that is scavenged and stored (V2). When the capacitance reaches its minimum value, the charge Q still presents in the capacitor is completely collected and stored (V3).

**Figure 8.** Voltage-constrained cycle

The total amount of energy converted at each cycle is presented in equation (4).

$$E\_{\rm LI=cte} = \mathcal{U}\_{\rm cst} \,^2(\mathcal{C}\_{\rm max} \cdot \mathcal{C}\_{\rm min}) \tag{4}$$

In order to maximize the electrostatics structures' efficiency, a high voltage polarization source (>100V) is required. Obviously, this is a major drawback of these devices as it implies that an external supply source (battery, charged capacitor) is required to polarize the capacitor at the beginning of the cycle or at least at the first cycle (as one part of the energy harvested at the end of a cycle can be reinjected into the capacitor to start the next cycle).

One solution to this issue consists in using electrets, electrically charged dielectrics, that are able to polarize electrostatic energy harvesters throughout their lives, avoiding energy cycles and enabling a direct mechanical-to-electrical conversion. Electrostatic energy harvesters developed today tend to use them increasingly.

### *2.4.2. Electret-based electrostatic converter*

Electret-based electrostatic converters are quite similar to electret-free electrostatic converters. The main difference relies on the electret layers that are added on one (or two) plate(s) of the variable capacitor, polarizing it.

1. Electrets

98 Small-Scale Energy Harvesting

(Q1)

(V1)

*Q*

*Cmax*

V1

V3

**Figure 6.** Standard energy conversion cycles for electret-free electrostatic devices

**+ + + + + + + + + + --------------**

still presents in the capacitor is completely collected and stored (V3).

Q Cmax

**+ + + + + + + + + + --------------** Ucst

**Figure 8.** Voltage-constrained cycle

The total amount of energy converted at each cycle is presented in equation (3).

Q2 Q3

*Umin*

*Ucst*

Qcst Cmax

(Q3)


The voltage-constrained cycle (Figure 8) also starts when the capacitance of the electrostatic converter is maximal. The capacitor is polarized at a voltage Ucst using an external supply source (battery, charged capacitor…) (V1). This voltage will be maintained throughout the conversion cycle thanks to an electronic circuit. Since the voltage is constant and the capacitance decreases, the charge of the capacitor increases, generating a current that is scavenged and stored (V2). When the capacitance reaches its minimum value, the charge Q

(V2) Cmin

i

**---- ----**

The total amount of energy converted at each cycle is presented in equation (4).

**+ + + + + + + + + + --------** Ucst


In order to maximize the electrostatics structures' efficiency, a high voltage polarization source (>100V) is required. Obviously, this is a major drawback of these devices as it implies

min max

Q4

V2

**V-constrained cycle**

*Cmin*

Conversion

Charge is collected

**+ + + + + + + + + + --------------** Umax

*Qmax*

Charge injection

*Qcst*

0

Qcst Cmax

(Q2)

**+ + + + + + + + + + --------------** Umin

**Figure 7.** Charge-constrained cycle

2. Voltage-Constrained Cycle

Q1

*Umax U*

**Q-constrained cycle**

Qcst Cmin

1 11 ( ) <sup>2</sup> *Q cte cst E Q C C* (3)

(Q4)

Qcst Cmin

**+ + +** <sup>R</sup>

i

**---**

(V3) Cmin

i

**----**

Ucst

max min ( ) *U cte cst E UC C* (4)

**+ + + +**

Electrets are dielectric materials that are in a quasi-permanent electric polarization state (electric charges or dipole polarization). They are electrostatic dipoles, equivalent to permanent magnets (but in electrostatic) that can keep charges for years. The word **electret** comes from "**electr**icity magn**et**" and was chosen by Oliver Heaviside in 1885.

2. Definition and electret types

Electret's polarization can be obtained by dipole orientation (Figure 9(a)) or by charge injection (Figure 9(b)) leading to two different categories of electrets:


**Figure 9.** Standard electrets for electret-based electrostatic converters (a) dipole orientation and (b) charge injection

In the past, electrets were essentially obtained thanks to dipole orientation, from Carnauba wax for example [3]. Today, real-charge electrets are the most commonly used and especially in vibration energy harvesters because they are easy to manufacture with standard processes.

Indeed, oriented-dipole electrets and real-charge electrets are obtained from very different processes, leading to different behaviors.

#### a. Fabrication Processes

The first step to make oriented-dipole electrets is a heating of a dielectric layer above its melting temperature. Then, an electric field is maintained throughout the dielectric layer when it is cooling down. This enables to orient dielectric layer's dipoles in the electric field's direction. Solidification enables to keep dipoles in their position. This manufacturing process is similar to the one of magnets.

As for real-charge electrets, they are obtained by injecting an excess of charges in a dielectric layer. Various processes can be used: electron beam, corona discharge, ion or electron guns.

Here, we focus on charge injection by a triode corona discharge, which is probably the quickest way to charge dielectrics. Corona discharge (Figure 10) consists in a point-gridplane structure whose point is submitted to a strong electric field: this leads to the creation of a plasma, made of ions that are projected onto the surface of the sample to charge and whose charges are transferred to the dielectric layer's surface.

**Figure 10.** Corona discharge device (a) principle and (b) photo (CEA-LETI)

Corona discharge may be positive or negative according to the sign of the point's voltage. Positive and negative corona discharges have different behaviors as the plasma and the charges generated are different. Obviously positive corona discharges will lead to positively-charged electrets and negative corona discharges to negatively-charged electrets that have also different behaviors (stability, position of the charges in the dielectric). The grid is used to control the electret's surface voltage Vs that results from the charges injected. Actually, when the electret's surface voltage Vs reaches the grid voltage Vg, there is no potential difference between the grid and the sample any longer and therefore, no charge circulation anymore. So, at the end of the corona charging, the electret's surface voltage is equal to the grid voltage.

b. Equivalent model of electrets

Charge injection or dipole orientation leads to a surface potential Vs on the electret (Gauss's law). It is generally assumed that charges are concentrated on the electret's surface and therefore, the surface potential can be simply expressed by: 0 / *V d <sup>s</sup>* , with ε the electret's dielectric permittivity, σ its surface charge density and d its thickness.

The equivalent model of an electret layer (Figure 11(a) and (b)) is then a capacitor *C S* / *d* <sup>0</sup> in series with a voltage source whose value is equal to the surface voltage of the electret Vs (Figure 11(c)).

**Figure 11.** (a) electret layer (b) parameters and (c) equivalent model

#### c. Charge stability and measurement

100 Small-Scale Energy Harvesting

(a)

equal to the grid voltage.

b. Equivalent model of electrets

a. Fabrication Processes

process is similar to the one of magnets.

plasma ions

whose charges are transferred to the dielectric layer's surface.

Charge transfer σ

Vg(100V)

**Figure 10.** Corona discharge device (a) principle and (b) photo (CEA-LETI)

(b)

Corona discharge may be positive or negative according to the sign of the point's voltage. Positive and negative corona discharges have different behaviors as the plasma and the charges generated are different. Obviously positive corona discharges will lead to positively-charged electrets and negative corona discharges to negatively-charged electrets that have also different behaviors (stability, position of the charges in the dielectric). The grid is used to control the electret's surface voltage Vs that results from the charges injected. Actually, when the electret's surface voltage Vs reaches the grid voltage Vg, there is no potential difference between the grid and the sample any longer and therefore, no charge circulation anymore. So, at the end of the corona charging, the electret's surface voltage is

Charge injection or dipole orientation leads to a surface potential Vs on the electret (Gauss's law). It is generally assumed that charges are concentrated on the electret's surface and

> 

, with ε the

therefore, the surface potential can be simply expressed by: 0 / *V d <sup>s</sup>*

electret's dielectric permittivity, σ its surface charge density and d its thickness.

electrode sample Vp(10kV)

grid

point


The first step to make oriented-dipole electrets is a heating of a dielectric layer above its melting temperature. Then, an electric field is maintained throughout the dielectric layer when it is cooling down. This enables to orient dielectric layer's dipoles in the electric field's direction. Solidification enables to keep dipoles in their position. This manufacturing

As for real-charge electrets, they are obtained by injecting an excess of charges in a dielectric layer. Various processes can be used: electron beam, corona discharge, ion or electron guns. Here, we focus on charge injection by a triode corona discharge, which is probably the quickest way to charge dielectrics. Corona discharge (Figure 10) consists in a point-gridplane structure whose point is submitted to a strong electric field: this leads to the creation of a plasma, made of ions that are projected onto the surface of the sample to charge and

> Nevertheless and unfortunately, dielectrics are not perfect insulators. As a consequence, some charge conduction phenomena may appear in electrets, and implanted charges can move inside the material or can be compensated by other charges or environmental conditions, and finally disappear. Charge stability is a key parameter for electrets as the electret-based converter's lifetime is directly linked to the one of the electret. Therefore, it is primordial to choose stable electrets to develop electret-based vibration energy harvesters.

> Many measurement methods have been developed to determine the quantity of charges stored into electrets and their positions. These methods are really interesting to understand what happens in the material but are complicated to implement and to exploit. Yet, for vibration energy harvesters, the most important data is the surface potential decay (SPD), that is to say, the electret's surface voltage as a function of the time after charging.

> In fact, the surface voltage can be easily measured thanks to an electrostatic voltmeter (Figure 12(a)). This method is really interesting as it enables to make the measurement of the surface voltage without any contact and therefore without interfering with the charges injected into the electret.

> Figure 12(b) presents some examples of electrets' SPDs (good, fair and poor stability). Electret stability depends of course of the dielectric material used to make the electret: dielectrics that have high losses (high tan(δ)) are not good electrets and may lose their charges in some minutes, while materials such as Teflon or silicon dioxide (SiO2) are known as stable electrets. But, other parameters, such as the initial surface voltage or the environmental conditions (temperature, humidity) have an important impact as well. Generally, for a given material, the higher the initial surface voltage is, the lower the stability becomes. For environmental conditions, high temperatures and high humidity tend to damage the electret stability.

**Figure 12.** (a) Electrostatic voltmeter (Trek® 347) and (b) examples of Surface Potential Decays (SPDs)

The electret behaviors of many materials have been tested. The next sub-section gives some examples of well-known and stable electrets.

d. Well-known electrets

Teflon [4-7], SiO2 [8-9] and CYTOP [10-14] are clearly the most well-known and the most used electrets in electret-based electrostatic converters. Of course, many more electrets can be found in the state of the art. Table 3 presents some properties of these electrets. It is for example interesting to note that SiO2-based electrets have the highest surface charge densities. As for the stability, it is quite complicated to provide a value. Actually, it greatly depends on the storage, the humidity, the temperature, the initial conditions, the thickness… Yet, the examples given below show a stability Vs90% (90% of the initial surface voltage) generally higher than 2-3 years.


**Table 3.** Well-known electrets from the state of the art

Added in capacitive structures, electrets enable a simple mechanical-to-electrical energy conversion.

#### 3. Conversion principle

The conversion principle of electret-based electrostatic converters is quite similar to electretfree electrostatic converters and is tightly linked to variations of capacitance. But contrary to them, the electret-based conversion does not need any initial electrical energy to work; a structure deformation induces directly an output voltage, just like a piezoelectric material.

#### a. Principle

102 Small-Scale Energy Harvesting

(a) (b)

d. Well-known electrets

examples of well-known and stable electrets.

voltage) generally higher than 2-3 years.

**Electret Deposition** 

**Teflon** 

**SiO2-based electrets** 

**Parylene (C/HT)** 

conversion.

**method** 

Thermal oxidization of silicon wafers (+ LPCVD Si3N4)

> PVD-like deposition method

**Table 3.** Well-known electrets from the state of the art

**(PTFE/FEP/PFA)** Films are glued Some 100

0% 20% 40% 60% 80% 100%

**Figure 12.** (a) Electrostatic voltmeter (Trek® 347) and (b) examples of Surface Potential Decays (SPDs)

The electret behaviors of many materials have been tested. The next sub-section gives some

Teflon [4-7], SiO2 [8-9] and CYTOP [10-14] are clearly the most well-known and the most used electrets in electret-based electrostatic converters. Of course, many more electrets can be found in the state of the art. Table 3 presents some properties of these electrets. It is for example interesting to note that SiO2-based electrets have the highest surface charge densities. As for the stability, it is quite complicated to provide a value. Actually, it greatly depends on the storage, the humidity, the temperature, the initial conditions, the thickness… Yet, the examples given below show a stability Vs90% (90% of the initial surface

> **Maximum thickness**

> > µm

Some µm

**CYTOP** Spin-coating 20µm 110 2 1-2 **Teflon AF** Spin-coating 20µm 200 1.9 0.1-0.25

Added in capacitive structures, electrets enable a simple mechanical-to-electrical energy

**normalized surface voltage** 

**(V/V0)**

0 200 400 600 800 1000

**time (days)**

good stability fair stability poor stability

**Dielectric Strength (V/µm)** 

(<3µm) 500 4 5-10

Some 10µm 270 3 0.5-1

**ε**

100-140 2.1 0.1-0.25

**Standard surface charge density (mC/m²)** 

Electret-based converters are electrostatic converters, and are therefore based on a capacitive structure made of two plates (electrode and counter-electrode (Figure 13)). The electret induces charges on electrodes and counter-electrodes to respect Gauss's law. Therefore, Qi, the charge on the electret is equal to the sum of Q1 and Q2, where Q1 is the total amount of charges on the electrode and Q2 the total amount of charges on the counter-electrode (Qi=Q1+Q2). A relative movement of the counter-electrode compared to the electret and the electrode induces a change in the capacitor geometry (e.g. the counter-electrode moves away from the electret, changing the air gap and then the electret's influence on the counterelectrode) and leads to a reorganization of charges between the electrode and the counterelectrode through load R (Figure 14). This results in a current circulation through R and one part of the mechanical energy (relative movement) is then turned into electricity.

**Figure 13.** Electret-based electrostatic conversion – Concept

**Figure 14.** Electret-based electrostatic conversion – Charge circulation

The equivalent model of electret-based electrostatic converters is presented below.

b. Equivalent model and Equations

The equivalent model of electret-based electrostatic converters is quite simple as it consists in a voltage source in series with a variable capacitor. This model has been confronted to experimental data and corresponds perfectly to experimental results (see section 3.2.4).

Figure 15 presents an electret-based electrostatic converter connected to a resistive load R. As the capacitances of electret-based converters are quite low (often lower than 100pF), it is important to take parasitic capacitances into account. They can be modeled by a capacitor in parallel with the electret-based converter. And actually, only 10pF of parasitic capacitances may have a deep impact on the electret-based converter's output voltages and output powers.

**Figure 15.** Electrical equivalent model of electret-based electrostatic converters

In the case of a simple resistive load placed at the terminals of the electret-based converter, the differential equation that rules the system is presented in equation (5).

$$\frac{dQ\_2}{dt} = \frac{V\_s}{R} - \frac{Q\_2}{C(t)R} \tag{5}$$

Taking parasitic capacitances into account, this model is modified into (6) [15].

$$\frac{d\,\mathrm{Q}\_2}{dt} = \frac{1}{\left(1 + \frac{\mathrm{C}\_{par}}{\mathrm{C}(t)}\right)} \left(\frac{V\_s}{R} \cdot \mathrm{Q}\_2 \left(\frac{1}{RC(t)} \cdot \frac{\mathrm{C}\_{par}}{\mathrm{C}(t)^2} \frac{d\mathrm{C}(t)}{dt}\right)\right) \tag{6}$$

Actually, electret-based converter's output powers (P) are directly linked to the electret's surface voltage Vs and the capacitance variation dC/dt when submitted to vibrations [16]. And, as a first approximation:

$$P \propto V\_s^{-2} \frac{d\mathbb{C}}{dt} \tag{7}$$

As a consequence, as the electret-based converter's output powers is linked to the electret's surface voltage Vs and its lifetime to the electret's lifetime, we confirm that Surface Potential Decays (SPDs) are the most appropriate way to characterize electrets for an application in energy harvesting.

#### *2.4.3. Capacitors and capacitances' models*

Whether it is electret-free or electret-based conversion, electrostatic converters are based on a variable capacitive structure. This subsection is focused on the main capacitor shapes employed in electrostatic converters and on their models.

#### 1. Main capacitor shapes

104 Small-Scale Energy Harvesting

powers.

Figure 15 presents an electret-based electrostatic converter connected to a resistive load R. As the capacitances of electret-based converters are quite low (often lower than 100pF), it is important to take parasitic capacitances into account. They can be modeled by a capacitor in parallel with the electret-based converter. And actually, only 10pF of parasitic capacitances may have a deep impact on the electret-based converter's output voltages and output

> Electret energy harvester

In the case of a simple resistive load placed at the terminals of the electret-based converter,

*Q1*

*Q2*

*C(t) Uc*

*Vs*

**Figure 15.** Electrical equivalent model of electret-based electrostatic converters

the differential equation that rules the system is presented in equation (5).

Taking parasitic capacitances into account, this model is modified into (6) [15].

( )

*C t*

And, as a first approximation:

*2.4.3. Capacitors and capacitances' models* 

employed in electrostatic converters and on their models.

energy harvesting.

*par*


( ) ( ) <sup>1</sup>

Actually, electret-based converter's output powers (P) are directly linked to the electret's surface voltage Vs and the capacitance variation dC/dt when submitted to vibrations [16].

> 2 *s dC P V*

As a consequence, as the electret-based converter's output powers is linked to the electret's surface voltage Vs and its lifetime to the electret's lifetime, we confirm that Surface Potential Decays (SPDs) are the most appropriate way to characterize electrets for an application in

Whether it is electret-free or electret-based conversion, electrostatic converters are based on a variable capacitive structure. This subsection is focused on the main capacitor shapes

*dQ <sup>V</sup> <sup>C</sup> dC t <sup>Q</sup> dt <sup>C</sup> R RC t dt C t*

*C(t)* + + + + +

R

*i*

2 2 1 1 ( )

*par s*

*Cpar*

*i1 i2*

*U*

*<sup>2</sup> <sup>s</sup>* <sup>2</sup> *dQ Q <sup>V</sup> <sup>=</sup> dt R C(t)R* (5)

*dt* (7)

(6)

Most of the electrostatic converters' shapes are derived from accelerometers. Actually, it is possible to count four main capacitor shapes for electrostatic converters (Figure 16).

(a) in-plane gap closing converter: interdigitated comb structure with a variable air gap between fingers and movement in the plane

(b) in-plane overlap converter: interdigitated comb structure with a variable overlap of the fingers and movement in the plane

(c) out-of-plane gap closing converter: planar structure with a variable air gap between plates and perpendicular movement to the plane

(d) in-plane converter with variable surface: planar structure with a variable overlap of the plates and movement in the plane. There is a great interest of developing patterned versions with bumps and trenches facing.

(full-plate and patterned version)

**Figure 16.** Basic capacitor shapes for electrostatic converters

Obviously, these basic shapes can be adapted to electret-free and electret-based electrostatic converters. As capacitances and electrostatic forces are heavily dependent on the capacitor's shape and on its dimensions, it is interesting to know the capacitance and the electrostatic forces generated by each of these structures to design an electrostatic converter. Capacitances values and electrostatic forces for each shape are presented in the next subsection.

2. Capacitances values and electrostatic forces

Capacitances values are all deduced from the simple plane capacitor model. In this subsection, the capacitance is computed with an electret layer. To get the capacitances for electret-free electrostatic converters, one has just to take d=0 (where d is the electret thickness).

**Figure 17.** Capacitance of the simple plane capacitor

The total capacitance of the electrostatic converter presented in Figure 17 corresponds to two capacitances (C1 and C2) in series.

$$C(t) = \frac{C\_1(t)C\_2}{C\_1(t) + C\_2} = \frac{\varepsilon\_0 S(t)}{\mathcal{g}(t) + \bigvee\_{\mathcal{E}}} \tag{8}$$

The electrostatic force felec induced by this capacitor can be expressed by:

$$F\_{elec} = \frac{d}{d\mathbf{x}} \left( \mathcal{W}\_{elec} \right) = \frac{d}{d\mathbf{x}} \left( \frac{1}{2} \mathbf{C}(\mathbf{x}) \mathcal{U}\_c(\mathbf{x})^2 \right) = \frac{d}{d\mathbf{x}} \left( \frac{1}{2} \frac{\mathcal{Q}\_c^2(\mathbf{x})}{\mathbf{C}(\mathbf{x})} \right) \tag{9}$$

With Welec the total amount of electrostatic energy stored in C, Qc the charge on C, Uc(x) the voltage across C and x the relative movement of the upper plate compared to the lower plate.

Capacitances and electrostatic forces of the four capacitor shapes are obtained by integrating equations (8) and (9).

a. In-plane gap-closing converter

In-plane gap-closing converters are interdigitated comb devices with a variable air gap between fingers as presented in Figure 18.

**Figure 18.** (a) In-plane gap-closing converters, (b) zoom on one finger with Cmin position, (c) Cmax positions

The capacitance of the converter corresponds to the two capacitors Cp1 and Cp2 in parallel and is expressed in equation (10).

Electrostatic Conversion for Vibration Energy Harvesting 107

$$\mathbf{C}(\mathbf{x}) = \frac{2\mathbf{N}\varepsilon\_0 \mathbf{S} \left(\mathbf{g}\_0 + \mathbf{d}\bigvee\_{\mathcal{E}}\right)}{\left(\mathbf{g}\_0 + \mathbf{d}\bigvee\_{\mathcal{E}}\right)^2 - \mathbf{x}^2} \tag{10}$$

Where N is the number of fingers of the whole electrostatic converter, and S the facing surface.

#### b. In-plane overlap converter

106 Small-Scale Energy Harvesting

d

capacitances (C1 and C2) in series.

plate.

positions

equations (8) and (9).

a. In-plane gap-closing converter

(a) (b)

and is expressed in equation (10).

between fingers as presented in Figure 18.

ε

**Figure 17.** Capacitance of the simple plane capacitor

Counter-electrode

(8)

(9)

Electrode Electret

The total capacitance of the electrostatic converter presented in Figure 17 corresponds to two

1 2 0

<sup>2</sup> 1 1 <sup>2</sup> () () <sup>2</sup> 2 ()

*c*

1 2 ( ) ( ) ( ) ( ) ( ) *C tC S t C t Ct C d g t*

*dd d Q x F W C xU x dx dx dx C x* 

With Welec the total amount of electrostatic energy stored in C, Qc the charge on C, Uc(x) the voltage across C and x the relative movement of the upper plate compared to the lower

Capacitances and electrostatic forces of the four capacitor shapes are obtained by integrating

In-plane gap-closing converters are interdigitated comb devices with a variable air gap

x(t)

(c)

g S <sup>0</sup>

**Figure 18.** (a) In-plane gap-closing converters, (b) zoom on one finger with Cmin position, (c) Cmax

The capacitance of the converter corresponds to the two capacitors Cp1 and Cp2 in parallel

Cp2

Cp1

The electrostatic force felec induced by this capacitor can be expressed by:

*elec elec c*

g(t)

ε<sup>0</sup> C1(t) C2

S(t)

In-plane overlap converters are interdigitated comb structure with a variable overlap of the fingers as presented in Figure 19. The whole structure must be separated into two variable capacitors Cc1 and Cc2 as the increase of Cc1's capacitance leads to a decrease of Cc2's capacitance and vice-versa.

**Figure 19.** (a) In-plane overlap converters, (b) zoom on two fingers (c) Cmax position for Cc1 and Cmin position for Cc2 and (d) Cmin position for Cc1 and Cmax position for Cc2.

The capacitance of Cc1 and Cc2 are expressed in equation (11).

$$\mathbf{C}\_{c1}(\mathbf{x}) = \frac{\varepsilon\_0 N w}{g\_0 + d} \Big|\_{\mathcal{E}} (l\_0 - \mathbf{x}) \quad \text{and} \quad \mathbf{C}\_{c2}(\mathbf{x}) = \frac{\varepsilon\_0 N w}{g\_0 + d} \Big|\_{\mathcal{E}} (l\_0 + \mathbf{x}) \tag{11}$$

Where N is the number of fingers, l0 the facing length of Cc1 and Cc2 at the equilibrium position and w the thickness of the fingers (third dimension).

c. Out-of-plane gap closing converter

In this configuration, the counter-electrode moves above the electrode, inducing a variation of the gap between the electret and the counter-electrode. The initial gap between the counter-electrode and the electret is g0 and the surface is denoted by S (Figure 20(a)). Cmax and Cmin positions are presented in Figure 20(b, c).

**Figure 20.** (a) Out-of-plane gap closing, (b) Cmax position and (c) Cmin position

The capacitance of the converter is expressed in equation (12).

$$\mathbf{C}(\mathbf{x}) = \frac{\mathbf{c}\_0 \mathbf{S}}{\mathbf{g}\_0 + \bigvee\_{\mathcal{E}} - \mathbf{x}} \tag{12}$$

#### d. In-plane converter with variable surface

In in-plane converters with variable surface, it is a change in the capacitor's area that is exploited, as presented in Figure 21(a).

**Figure 21.** (a) In-plane with variable surface, (b) Cmax position and (c) Cmin position

The capacitance of the converter is expressed in equation (13).

$$C(\mathbf{x}) = \frac{\varepsilon\_0 w(l\_0 - \mathbf{x})}{\mathcal{S} + \bigvee\_{\mathcal{E}}} \tag{13}$$

Where w is the converter's thickness (third dimension), l0 the facing length between the plates at t=0.

So as to increase the capacitance variation for a given relative displacement x, it is interesting to pattern the capacitive structure, as presented in Figure 22.

**Figure 22.** (a) In-plane converter with variable patterned surface, (b) Cmax position and (c) Cmin position

To develop efficient vibration energy harvesters able to work with low vibrations, it is necessary to use micro-patterned capacitive structure (small e and b). With such dimensions, fringe effects must be taken into account and it was proven [17] that the capacitance of the energy harvester can be simply modeled by a sine function presented in equation (14).

Electrostatic Conversion for Vibration Energy Harvesting 109

$$\mathcal{C}(\mathbf{x}) = \frac{\mathcal{C}\_{\text{max}} + \mathcal{C}\_{\text{min}}}{2} + \left(\frac{\mathcal{C}\_{\text{max}} - \mathcal{C}\_{\text{min}}}{2}\right) \times \cos\left(\frac{2\pi\chi}{e+b}\right) \tag{14}$$

Where Cmax and Cmin are the maximal and the minimal capacitances of the energy harvester and computed by finite elements.

108 Small-Scale Energy Harvesting

Relative motion

d ε ε0 0 x(t)

**Figure 20.** (a) Out-of-plane gap closing, (b) Cmax position and (c) Cmin position

The capacitance of the converter is expressed in equation (12).

Relative motion

The capacitance of the converter is expressed in equation (13).

**Figure 21.** (a) In-plane with variable surface, (b) Cmax position and (c) Cmin position

interesting to pattern the capacitive structure, as presented in Figure 22.

Relative motion

(b) (c)

0

(12)

(13)

0 ( ) *<sup>S</sup> C x <sup>d</sup> <sup>g</sup> <sup>x</sup>* 

In in-plane converters with variable surface, it is a change in the capacitor's area that is

(b) (c)

0 0 ( ) ( ) *wl x C x <sup>d</sup> <sup>g</sup>* 

Where w is the converter's thickness (third dimension), l0 the facing length between the

So as to increase the capacitance variation for a given relative displacement x, it is

**Figure 22.** (a) In-plane converter with variable patterned surface, (b) Cmax position and (c) Cmin position

To develop efficient vibration energy harvesters able to work with low vibrations, it is necessary to use micro-patterned capacitive structure (small e and b). With such dimensions, fringe effects must be taken into account and it was proven [17] that the capacitance of the energy harvester can be simply modeled by a sine function presented in equation (14).

 

(b) (c)

C1 g(x)=g (x) 0-x

d. In-plane converter with variable surface

exploited, as presented in Figure 21(a).

C1 g (x)

l0-x

C2

S

C2

(a)

(a)

plates at t=0.

0 x(t)

d

g

b e

(a)

0 x(t)

d ε ε0 Electrostatic forces are deduced from the derivation of the electrostatic energy stored into the capacitor, as presented in equation (9). Concerning electret-based devices, the electrostatic force cannot be easily expressed as both capacitor's charge and voltage change when the geometry varies. Table 4 overviews the electrostatic forces for the various converters and their operation modes.


**Table 4.** Electrostatic forces according to the converter and its operation mode

These electrostatic converters are then coupled to mass-spring systems to become vibration energy harvesters.

#### **3. Electrostatic Vibration Energy Harvesters (eVEH)**

As presented in section 2.1, harvesting vibrations requires two conversion steps: a mechanical-to-mechanical converter made of a mass-spring resonator that turns ambient vibrations into a relative movement between two elements (presented in 2.1) and a mechanical-to-electrical converter using, in our case, a capacitive architecture (presented in 2.2) that converts this relative movement into electricity. Section 3 is aimed at presenting complete devices that gather these two converters. It is firstly focused on electret-free electrostatic devices before presenting electret-based devices.

### **3.1. Electret-free Electrostatic Vibration Energy Harvesters (eVEH)**

### *3.1.1. Devices*

The first MEMS electrostatic comb based VEH was developed at the MIT by Meninger et al. in 2001 [18]. This device used an in-plane overlap electrostatic converter. Operating cycles are described and it is proven that the voltage-constrained cycle enables to maximize output power (if the power management electronic is limited in voltage). Yet, for the prototype, a charge-constrained cycle was adopted to simplify the power management circuit even if it drives to a lower output power.

Electrostatic devices can be particularly suitable for Vibration energy harvesting at low frequencies (<100Hz). In 2002, Tashiro et al. [19] developed a pacemaker capable of harvesting power from heartbeats. The output power of this prototype installed on the heart of a goat was 58µW.

In 2003, Roundy [20] proved that the best structure for electrostatic devices was the in-plane gap closing and would be able to harvest up to 100µW/cm³ with ambient vibrations (2.25m/s²@120Hz). Roundy et al. then developed an in-plane gap closing structure able to harvest 1.4nJ/cycle.

In 2005, Despesse et al. developed a macroscopic device (Figure 23(a)) able to work on low vibration frequencies and able to harvest 1mW for a vibration of 0.2G@50Hz [21]. This prototype has the highest power density of eVEH ever reached. Some other MEMS devices were then developed by Basset et al [22] (Figure 23(b)) and Hoffmann et al. [23].

**Figure 23.** Electrostatic vibration energy harvesters from (a) Despesse et al. [21] and (b) Basset et al. [22].

### *3.1.2. State of the art – Overview*

An overview of electret-free electrostatic vibration energy harvesters is presented in Table 5.

Many prototypes of electret-free electrostatic vibration energy harvesters have been developed and validated. Currently, the tendency is to couple these devices to electrets. The next subsection is focused on them.

### **3.2. Electret-Based Electrostatic VEH**

Electret-based devices were developed to enable a direct vibration-to-electricity conversion (without cycles of charges and discharges) and to simplify the power management circuits.


**Table 5.** Electret-free electrostatic vibration energy harvesters from the state of the art

#### *3.2.1. History*

110 Small-Scale Energy Harvesting

drives to a lower output power.

of a goat was 58µW.

harvest 1.4nJ/cycle.

[22].

*3.1.1. Devices* 

**3.1. Electret-free Electrostatic Vibration Energy Harvesters (eVEH)** 

The first MEMS electrostatic comb based VEH was developed at the MIT by Meninger et al. in 2001 [18]. This device used an in-plane overlap electrostatic converter. Operating cycles are described and it is proven that the voltage-constrained cycle enables to maximize output power (if the power management electronic is limited in voltage). Yet, for the prototype, a charge-constrained cycle was adopted to simplify the power management circuit even if it

Electrostatic devices can be particularly suitable for Vibration energy harvesting at low frequencies (<100Hz). In 2002, Tashiro et al. [19] developed a pacemaker capable of harvesting power from heartbeats. The output power of this prototype installed on the heart

In 2003, Roundy [20] proved that the best structure for electrostatic devices was the in-plane gap closing and would be able to harvest up to 100µW/cm³ with ambient vibrations (2.25m/s²@120Hz). Roundy et al. then developed an in-plane gap closing structure able to

In 2005, Despesse et al. developed a macroscopic device (Figure 23(a)) able to work on low vibration frequencies and able to harvest 1mW for a vibration of 0.2G@50Hz [21]. This prototype has the highest power density of eVEH ever reached. Some other MEMS devices

**Figure 23.** Electrostatic vibration energy harvesters from (a) Despesse et al. [21] and (b) Basset et al.

An overview of electret-free electrostatic vibration energy harvesters is presented in Table 5. Many prototypes of electret-free electrostatic vibration energy harvesters have been developed and validated. Currently, the tendency is to couple these devices to electrets. The

Electret-based devices were developed to enable a direct vibration-to-electricity conversion (without cycles of charges and discharges) and to simplify the power management circuits.

were then developed by Basset et al [22] (Figure 23(b)) and Hoffmann et al. [23].

(a) (b)

*3.1.2. State of the art – Overview* 

next subsection is focused on them.

**3.2. Electret-Based Electrostatic VEH** 

The idea of using electrets in electrostatic devices to make generators goes back to about 40 years ago. In fact, the first functional electret-based generator was developed in 1978 by Jefimenko and Walker [27]. From that time, several generators exploiting a mechanical energy of rotation were developed (Jefimenko [27], Tada [28], Genda [29] or Boland [16]). Figure 24 presents an example, developed by Boland in 2003 [16**,** 30] of an electret-based generator able to turn a relative rotation of the upper plate compared to the lower plate into electricity.

**Figure 24.** Boland's electret-based generator prototype [30] (a) perspective view and (b) stator

With the development of energy harvesting and the need to design autonomous sensors for industry, researchers and engineers have decided to exploit electrets in their electrostatic vibration energy harvesters as their everlasting polarization source.

#### *3.2.2. Devices*

Even if the four capacitor shapes presented in subsection 2.3.1 are suitable to develop electret-based vibration energy harvesters, only two architectures have been really exploited: out-of-plane gap closing and patterned in-plane with variable surface structures.

<sup>1</sup> latest results from ESIEE showed that higher output powers are reachable thanks to this device (up to 500nW).

**Figure 25.** Standard architectures for electret-based Vibration Energy Harvesters

This section presents some examples of electret-based vibration energy harvesters from the state-of-the-art. We have decided to gather these prototypes in 2 categories: devices using full-sheet electrets (electret dimensions or patterning higher than 5mm) and devices using patterned electrets (electret dimensions or patterning smaller than 5mm). Indeed, it is noteworthy that texturing an electret is not an easy task as it generally leads to a weak stability (important charge decay) and requires MEMS fabrication facilities.

a. Devices using full-sheet electrets

Full-sheet-electret devices can exploit a surface variation or a gap variation. In 2003, Mizuno [31] developed an out-of-plane gap closing structure using a clamped-free beam moving above an electret. This structure was also studied by Boisseau et al. [15] in 2011. This simple structure is sufficient to rapidly demonstrate the principle of vibration energy harvesting with electrets. Large amount of power can be harvested even with low vibration levels as soon as the resonant frequency of the harvester is tuned to the frequency of ambient vibrations.

**Figure 26.** Cantilever-based electret energy harvesters [15]

The first integrated structure using full-sheet electrets was developed by Sterken et al. from IMEC [32] in 2007. A diagram is presented in Figure 27: a full-sheet is used as the polarization source. The electret layer polarizes the moving electrode of the variable capacitance (Cvar). The main drawback of this prototype is to add a parasitic capacitance in series with the energy harvester, limiting the capacitance's variation and the converter's efficiency.

Today, most of the electret-based vibration energy harvesters use patterned electrets and exploit surface variation.

**Figure 27.** IMEC's first electret-based vibration energy harvester [32]

b. Devices using patterned electrets

112 Small-Scale Energy Harvesting

facilities.

vibrations.

efficiency.

exploit surface variation.

a. Devices using full-sheet electrets

*y*(*t*) *Y*.sin

**Figure 26.** Cantilever-based electret energy harvesters [15]

*t*

beam

electret

(a) (b)

**Figure 25.** Standard architectures for electret-based Vibration Energy Harvesters

This section presents some examples of electret-based vibration energy harvesters from the state-of-the-art. We have decided to gather these prototypes in 2 categories: devices using full-sheet electrets (electret dimensions or patterning higher than 5mm) and devices using patterned electrets (electret dimensions or patterning smaller than 5mm). Indeed, it is noteworthy that texturing an electret is not an easy task as it generally leads to a weak stability (important charge decay) and requires MEMS fabrication

Full-sheet-electret devices can exploit a surface variation or a gap variation. In 2003, Mizuno [31] developed an out-of-plane gap closing structure using a clamped-free beam moving above an electret. This structure was also studied by Boisseau et al. [15] in 2011. This simple structure is sufficient to rapidly demonstrate the principle of vibration energy harvesting with electrets. Large amount of power can be harvested even with low vibration levels as soon as the resonant frequency of the harvester is tuned to the frequency of ambient

counter-electrode

The first integrated structure using full-sheet electrets was developed by Sterken et al. from IMEC [32] in 2007. A diagram is presented in Figure 27: a full-sheet is used as the polarization source. The electret layer polarizes the moving electrode of the variable capacitance (Cvar). The main drawback of this prototype is to add a parasitic capacitance in series with the energy harvester, limiting the capacitance's variation and the converter's

Today, most of the electret-based vibration energy harvesters use patterned electrets and

*R*

*x*(*t*) *X*.sin

*t* 

electrode

**+ + + + + + + + + + + + + + + + + -----------------------**

mass

The first structure using patterned electrets was developed by the university of Tokyo in 2006 [33]. Many other devices followed, each of them, improving the first architecture [10, 34-38]. For example Miki et al. [39] improved these devices by developing a multiphase system and using non-linear effects. Multiphase devices enable to limit the peaks of the electrostatic force and thus to avoid to block the moving mass.

c. Mechanical springs to harvest ambient vibrations

Developing low-resonant frequency energy harvesters is a big challenge for small-scale devices. In most cases, ambient vibrations' frequencies are below 100Hz. This leads to long and thin springs difficult to obtain by using silicon technologies (form factors are large and structures become brittle). Thus, to reduce the resonant frequency of vibration energy harvesters, keeping small dimensions, solutions such as parylene springs [40] were developed. Another way consists in using microballs that act like a slideway. Naruse has already shown that such a system could operate at very low frequencies (<2 Hz) and could produce up to 40 µW [37] (Figure 29).

**Figure 29.** Device on microballs from [37]

Besides, a good review on MEMS electret energy harvesters can be found in [41]. The next subsection presents an overview of some electret-based prototypes from the state of the art.

#### *3.2.3. State of the art – Overview*

An overview of electret-based electrostatic vibration energy harvesters is presented in Table 6.


**Table 6.** Electret-based energy harvesters from the state of the art

Table 6 shows a significant increase of electret-based prototypes since 2003. It is also interesting to note that some companies such as Omron or Sanyo [48] started to study these devices and to manufacture some prototypes.

Thanks to simple cantilever-based devices developed for example by Mizuno [31] and Boisseau [15], the theoretical model of electret-based devices can be accurately validated.

### *3.2.4. Validation of theory with experimental data – Cantilever-based electret energy harvesters*

The theoretical model of electret-based energy converters and vibration energy harvesters can be easily validated by experimental data with a simple cantilever-based electret vibration energy harvester [15].

#### a. Device

114 Small-Scale Energy Harvesting

**Figure 29.** Device on microballs from [37]

*3.2.3. State of the art – Overview* 

**Table 6.** Electret-based energy harvesters from the state of the art

devices and to manufacture some prototypes.

Besides, a good review on MEMS electret energy harvesters can be found in [41]. The next subsection presents an overview of some electret-based prototypes from the state of the art.

An overview of electret-based electrostatic vibration energy harvesters is presented in Table 6.

Author Ref Vibrations / Rotations Active Surface Electret Potential Output Power Jefimenko [27] 6000 rpm 730 cm² 500V 25 mW Tada [28] 5000 rpm 90 cm² 363V 1.02 mW Boland [16] 4170 rpm 0.8 cm² 150V 25 µW Genda [29] 1'000'000 rpm 1.13 cm² 200V 30.4 W Boland [42] 7.1G@60Hz 0.12 cm² 850V 6 µW Tsutsumino [33] 1.58G@20Hz 4 cm² 1100V 38 µW Lo [43] 14.2G@60Hz 4.84 cm² 300V 2.26 µW Sterken [32] 1G@500Hz 0.09 cm² 10V 2nW Lo [34] 4.93G@50Hz 6 cm² 1500V 17.98 µW Zhang [35] 0.32G@9Hz 4 cm² 100V 0.13 pW Yang [44] 3G@560Hz 0.3 cm² 400V 46.14 pW Suzuki [40] 5.4G@37Hz 2.33 cm² 450V 0.28 µW Sakane [10] 0.94G@20Hz 4 cm² 640V 0.7 mW Naruse [37] 0.4G@2Hz 9 cm² 40µW Halvorsen [45] 3.92G@596Hz 0.48 cm² 1µW Kloub [46] 0.96G@1740Hz 0.42 cm² 25V 5µW Edamoto [36] 0.87G@21Hz 3 cm² 600 V 12µW Miki [39] 1.57G@63Hz 3 cm² 180V 1µW Honzumi [47] 9.2G@500Hz 0.01 cm² 52V 90 pW Boisseau [15] 0.1G@50Hz 4.16cm² 1400V 50µW

Table 6 shows a significant increase of electret-based prototypes since 2003. It is also interesting to note that some companies such as Omron or Sanyo [48] started to study these The prototype presented in Figure 30 consists in a clamped-free beam moving with regards to an electret due to ambient vibrations. The mechanical-to-mechanical converter is the mass-beam system and the mechanical-to-electrical converter is made of the electrodeelectret-airgap-moving counter-electrode architecture [15].

**Figure 30.** Example of a simple out-of-plane electret-based VEH (cantilever) (a) prototype, (b) diagram, (c) parameters and (d) dimensions

This system can be modeled by equations developed in section 2.

#### b. Model

From equations (1) and (6), one can prove that this device is ruled by the system of differential equations (15).

$$m\ddot{\mathbf{x}} + b\_m \dot{\mathbf{x}} + k\mathbf{x} - \frac{d}{d\mathbf{x}} \left(\frac{Q\_2^2}{2\mathbf{C}(t)}\right) - mg = -m\ddot{\mathbf{y}}$$

$$\left|\frac{d\mathbf{Q}\_2}{dt} = \frac{1}{\left(1 + \frac{\mathbf{C}\_{par}}{\mathbf{C}(t)}\right)} \left(\frac{V}{R} \cdot \mathbf{Q}\_2 \left(\frac{1}{RC(t)} \cdot \frac{\mathbf{C}\_{par}}{C(t)^2} \frac{dC(t)}{dt}\right)\right) \tag{15}$$

Obviously, this system cannot be solved by hand. Yet, by using a numerical solver (e.g. Matlab), this becomes possible. It is also imaginable to use Spice by turning this system of equations in its equivalent electrical circuit (Figure 31).

**Figure 31.** Equivalent electrical model of electret-based vibration energy harvesters

c. Theory vs experimental data

The prototype presented in Figure 30 has been tested on a shaker at 0.1G@50Hz with two different loads (300MΩ and 2.2GΩ) and the corresponding theoretical results have been computed using a numerical solver. Theoretical and experimental output voltages are presented in Figure 32 showing an excellent match.

**Figure 32.** Validation of theory with a cantilever-based electret energy harvester (a) R=300MΩ and (b) R=2.2GΩ

This simple prototype enables to validate the model of electret-based vibration energy harvesters that was presented in section 2. It is also interesting to note that this simple prototype has an excellent output power that reaches 50µW with a low vibration acceleration of 0.1G@50Hz.

Section 3 is concluded by an overview of electret patterning methods. Actually, electret patterning can be a real challenge in electret-based devices because of weak stability problems.

### *3.2.5. Electret patterning*

As presented in section 2, electret patterning is primordial to develop efficient and viable eVEH. Various methods from the state of the art to make stable patterned electrets in polymers and SiO2-based layers are presented hereafter.

#### a. Polymers

116 Small-Scale Energy Harvesting

2

equations in its equivalent electrical circuit (Figure 31).

voltage measurement

m

presented in Figure 32 showing an excellent match.

0 0,05 0,1 0,15

**Time [s]** Experiment (300M) Theory with parasitic capacitances (b)

*y . <sup>x</sup> .*

c. Theory vs experimental data

(a)

(b) R=2.2GΩ

**Output voltage [V]**

*bm*

*1/k*

*V*

**Figure 31.** Equivalent electrical model of electret-based vibration energy harvesters


( ) ( ) <sup>1</sup>

Obviously, this system cannot be solved by hand. Yet, by using a numerical solver (e.g. Matlab), this becomes possible. It is also imaginable to use Spice by turning this system of

*1/k <sup>x</sup> C(x)*

*Felec*

Mechanics Electric/Electrostatic

The prototype presented in Figure 30 has been tested on a shaker at 0.1G@50Hz with two different loads (300MΩ and 2.2GΩ) and the corresponding theoretical results have been computed using a numerical solver. Theoretical and experimental output voltages are

**Figure 32.** Validation of theory with a cantilever-based electret energy harvester (a) R=300MΩ and

**Output voltage [V]**

*dQ <sup>V</sup> <sup>C</sup> dC t <sup>Q</sup> dt <sup>C</sup> R RC t dt C t*

*<sup>d</sup> <sup>Q</sup> mx + b x + kx mg = my dx (t)*

1 1 ( )

. 2C

( )

*C t*

*par*

*m*

2 2

*dC/dx Z Vs*

*i*

*V*

voltage measurement

0 0,02 0,04 0,06 0,08

**Time [s]** Experiment (2,2G) Theory with parasitic capacitances

*par*

(15)

2 2

> The problem of polymer electrets patterning has been solved for quite a long time [16, 49]. In fact, it has been proven that it is possible to develop stable patterned electrets in CYTOP by etching the electret layer before charging, as presented in Figure 33 [10]. The patterning size is in the order of 100µm.

**Figure 33.** CYTOP electret patterning [10]

Equivalent results have been observed on Teflon AF [16].

However, making patterned SiO2-based electrets is generally more complicated, leading to a strong charge decay and therefore an extremely weak stability.

#### b. SiO2-based electrets

In fact, an obvious patterning of electret layers would consist in taking full sheet SiO2-based electrets (that have an excellent stability) and by etching them, like it is done on polymer electrets (Figure 34).

Al

**Figure 34.** Obvious patterning that does not work

Unfortunately, this obvious patterning does not work because it makes the electret hard to charge (ions or electrons go directly in the silicon wafer) and the stability of these electrets is not good [37]. This is the reason why new and smart solutions have been developed to pattern SiO2-based electrets.

IMEC

The concept developed by IMEC to make SiO2/Si3N4 patterned electrets is based on the observation that a single SiO2 layer is less stable than a superposition of SiO2 and Si3N4 layers. A drawing of the patterned electrets is provided in Figure 35. This method has been patented by IMEC [50].

These patterned electrets are obtained from a silicon wafer that receives a thermal oxidization to form a SiO2 layer. A Si3N4 layer is deposited and etched with a patterning. This electret is then charged thanks to a corona discharge. Charges that are not on the SiO2/Si3N4 areas are removed thanks to thermal treatments while charges that are on these areas stay trapped inside.

**Figure 35.** Patterned SiO2/Si3N4 electrets from IMEC and used in a structure [45].

The stability of these electrets was proven down to a patterning size of 20µm.

Sanyo and the University of Tokyo

Naruse et al. [37] developed SiO2 patterned electrets thanks to a different concept. The electret manufacturing process starts with a SiO2 layer on a silicon wafer. The SiO2 layer is metalized with aluminum. The aluminum layer is then patterned and the SiO2 layer is etched as presented in Figure 36. The sample is then charged.

**Figure 36.** Patterned SiO2 electrets from Sanyo and the University of Tokyo [37]

The guard electrodes of Aluminum form the low surface voltage and the charged areas of SiO2 form the high surface voltage. This potential difference enables to turn mechanical energy into electricity. The hollow structure of SiO2 prevents charge drifting to the guard electrode.

#### CEA-LETI [51]

118 Small-Scale Energy Harvesting

Al

pattern SiO2-based electrets.

patented by IMEC [50].

areas stay trapped inside.

Sanyo and the University of Tokyo

IMEC

SiO2 Si HMDS 3N4

**Figure 34.** Obvious patterning that does not work

Silicon

SiO2

Unfortunately, this obvious patterning does not work because it makes the electret hard to charge (ions or electrons go directly in the silicon wafer) and the stability of these electrets is not good [37]. This is the reason why new and smart solutions have been developed to

The concept developed by IMEC to make SiO2/Si3N4 patterned electrets is based on the observation that a single SiO2 layer is less stable than a superposition of SiO2 and Si3N4 layers. A drawing of the patterned electrets is provided in Figure 35. This method has been

These patterned electrets are obtained from a silicon wafer that receives a thermal oxidization to form a SiO2 layer. A Si3N4 layer is deposited and etched with a patterning. This electret is then charged thanks to a corona discharge. Charges that are not on the SiO2/Si3N4 areas are removed thanks to thermal treatments while charges that are on these

Silicon substrate

Naruse et al. [37] developed SiO2 patterned electrets thanks to a different concept. The electret manufacturing process starts with a SiO2 layer on a silicon wafer. The SiO2 layer is metalized with aluminum. The aluminum layer is then patterned and the SiO2 layer is

The guard electrodes of Aluminum form the low surface voltage and the charged areas of SiO2 form the high surface voltage. This potential difference enables to turn mechanical energy into

electricity. The hollow structure of SiO2 prevents charge drifting to the guard electrode.

Nitride + + + + + + + + + + + + + + + + + + + + + + + + ------ ------ ------ ------

**Figure 35.** Patterned SiO2/Si3N4 electrets from IMEC and used in a structure [45].

etched as presented in Figure 36. The sample is then charged.

**Figure 36.** Patterned SiO2 electrets from Sanyo and the University of Tokyo [37]

The stability of these electrets was proven down to a patterning size of 20µm.

Silicon

Si HMDS 3N4

Oxide

Al

Contrary to the previous methods, the goal of this electret patterning method is to make continuous electret layers. Actually, instead of patterning the electret, it is the substrate of the electret (the silicon wafer) that is patterned thanks to a Deep Reactive Ion Etching (DRIE). The fabrication process of these patterned electrets is similar to the one of full sheet electrets in order to keep equivalent behaviors and above all equivalent stabilities.

The main difference between the two processes (full-sheet and patterned electrets) is the DRIE step that is used to geometrically pattern the electret. The main manufacturing steps are presented in Figure 37. The process starts with a standard p-doped silicon wafer (a). After a lithography step, the silicon wafer is etched by DRIE (b) and cleaned. Wafers are then oxidized to form a 1µm-thick SiO2 layer (c). SiO2 layer on the rear face is then removed by HF while front face is protected by a resin. A 100nm-thick LPCVD Si3N4 is deposited on the front face (d). Wafers receive a thermal treatment (450°C during 2 hours into N2) and a surface treatment (vapour HMDS) (e). Dielectric layers are then charged by a standard corona discharge to turn them into electrets (f).

**Figure 37.** Fabrication process of CEA-LETI's DRIE-patterned electrets

Manufacturing results are presented in Figure 38 for a (e,b,h)=(100µm, 100µm, 100µm) electret (Figure 38(a)). SEM images in Figure 38(b, c, d) show the patterning of the samples and the different constitutive layers. It is interesting to note the continuity of the electret layer even on the right angle in Figure 38(d). The long-term stability of these patterned electrets has been proven thanks to various surface potential decays measurements.

**Figure 38.** Patterned SiO2/Si3N4 electrets from CEA-LETI [51]

We have presented in this section several prototypes of electrostatic VEH and their output powers that may reach some tens or even hundreds of microwatts. This is in agreement with WSN' power needs. Yet, the output voltages are not appropriate for supplying electronic devices as is. This is the reason why a power converter is required.

That power management unit is essential for Wireless Sensor Nodes; this is the topic of the next section.

## **4. Power Management Control Circuits (PMCC) dedicated to electrostatic VEH (eVEH)**

The next section is aimed at presenting some examples of PMCC for electrostatic VEH.

### **4.1. Need for Power Management Control Circuit (PMCC)**

As presented in section 3, electrostatic vibration energy harvesters are characterized by a high output voltage that may reach some hundreds of volts and a low output current (some 100nA). Obviously, it is impossible to power any application, any electronic device with such a supply source. This is the reason why a power converter and an energetic buffer are needed to develop autonomous sensors. Figure 39 presents the conversion chain.

Power Management Control Circuits (PMCC) can have many functions: changing eVEH resonant frequency, controlling measurement cycles... here, we focus on the power converter and on its control circuit.

**Figure 39.** Power Management Control Circuit to develop viable VEH

As eVEH output powers are low (generally <100µW), Power Management Control Circuit must be simple and above all low power. For example, it is difficult to supply a MMPT (Maximum Power Point Tracker) circuits and the number of transistors and operations must be highly limited. We present in the next subsections some examples of Power Management Control Circuit for electret-free and electret-based eVEH.

### **4.2. PMCC for electret-free electrostatic VEH**

As the mechanical-to-electrical conversion is not direct, electret-free eVEH need a PMCC able to charge and to discharge the capacitor at the right time. Once more, we will focus on voltage-constrained and charge-constrained cycles.

#### *4.2.1. Voltage-constrained cycles*

120 Small-Scale Energy Harvesting

next section.

**VEH (eVEH)** 

conversion chain.

converter and on its control circuit.

energy

**Figure 39.** Power Management Control Circuit to develop viable VEH

Control Circuit for electret-free and electret-based eVEH.

**4.2. PMCC for electret-free electrostatic VEH** 

voltage-constrained and charge-constrained cycles.

We have presented in this section several prototypes of electrostatic VEH and their output powers that may reach some tens or even hundreds of microwatts. This is in agreement with WSN' power needs. Yet, the output voltages are not appropriate for supplying electronic

That power management unit is essential for Wireless Sensor Nodes; this is the topic of the

**4. Power Management Control Circuits (PMCC) dedicated to electrostatic** 

As presented in section 3, electrostatic vibration energy harvesters are characterized by a high output voltage that may reach some hundreds of volts and a low output current (some 100nA). Obviously, it is impossible to power any application, any electronic device with such a supply source. This is the reason why a power converter and an energetic buffer are needed to develop autonomous sensors. Figure 39 presents the

Power Management Control Circuits (PMCC) can have many functions: changing eVEH resonant frequency, controlling measurement cycles... here, we focus on the power

Application Ambient

As eVEH output powers are low (generally <100µW), Power Management Control Circuit must be simple and above all low power. For example, it is difficult to supply a MMPT (Maximum Power Point Tracker) circuits and the number of transistors and operations must be highly limited. We present in the next subsections some examples of Power Management

As the mechanical-to-electrical conversion is not direct, electret-free eVEH need a PMCC able to charge and to discharge the capacitor at the right time. Once more, we will focus on

VEH-based supply source

Power Management Control (PMC)

VEH Buffer power converter

The next section is aimed at presenting some examples of PMCC for electrostatic VEH.

devices as is. This is the reason why a power converter is required.

**4.1. Need for Power Management Control Circuit (PMCC)** 

The voltage-constrained cycle is not often used, and no specific example is available. Yet, Figure 40 presents an example of a PMCC to implement voltage-constrained cycles on electret-free electrostatic converters.

**Figure 40.** Example of a PMCC to implement voltage-constrained cycles

When the electrostatic converter's capacitance reaches its maximum, a quantity of energy is transferred from the electrical energetic buffer E and stored in the magnetic core M1 by closing K1 during few µs, negligible compared to the mechanical period. This energy is then transferred to the variable capacitor Cvar by closing K2 during few µs. The voltage UCvar through Cvar reaches UCV, the constant voltage. The mechanical movement induces a decrease of the electrostatic structure's capacitance Cvar and a charge *Q UCV C*var is transferred to the constant voltage storage CCV. To maintain UCV approximately constant, a second electrical converter is used. When UCV becomes higher than a threshold voltage, a quantity of energy is transferred from the constant voltage capacitor CCV to the energetic buffer E by closing K4 and then K3. Finally, when the electrostatic structure's capacitance reaches its minimum, the remaining energy stored in the electrostatic structure Cvar is sent to the energetic buffer by closing K2 and then K1.

Even though this PMCC works, it nevertheless requires two electrical converters that cost in price, space, losses and complexity. In order to have only one electrical converter, the MIT proposed in 2005 the following structure that applies a partial constant voltage cycle [26]:

**Figure 41.** Example of a PMCC from MIT for voltage-constrained cycles [26]

This electronic circuit keeps the electrostatic structure's voltage between two values (Vres and Vstore). When the electrostatic structure's capacitance Cvar increases, its voltage decreases and finishes to reach the low voltage storage Ures. Then, diode D1 becomes conductive and a current is transferred from Cres (storage capacitor) to the electrostatic device. When Cvar's capacitance decreases, its voltage increases and finally reaches the high voltage storage Ustore. Then diode D2 becomes conductive and a current is transferred from the electrostatic structure to Cstore. This structure works as a charge pump from Cres to Cstore. And, in order to close the cycle, one part of the energy transferred to Cstore is transferred to Cres by using an inductive electrical converter. Although this structure uses only one inductive component, it requires a complex electronic circuit to drive the floating transistor connected to the high voltage.

Finally, the constant voltage cycle is not frequently used due to the complex electronic circuits associated.

### *4.2.2. Charge-constrained cycles*

Charge-constrained cycles are easier to implement than voltage-constrained cycles as the conversion consists in charging the capacitor when the capacitance is maximal and to let it in open-circuit till it reaches its minimum. On the minimal capacitance, corresponding to the maximal voltage, charges are collected from the converter.

Usually, to reach a high conversion power density, the capacitor must be polarized at a high voltage (V1>100V). Yet, in autonomous devices, only 3V supply sources are available: a first DC-to-DC converter (step-up) is therefore needed to polarize the capacitor at a high voltage (step 1). In the same way, the output voltage on the capacitor after the mechanical-toelectrical conversion (step 2) may reach several hundreds of volts (V2>200-300V) and is therefore not directly usable to power an application: a second converter (step-down) is then necessary (step 3). Obviously, to limit the number of sources, it is interesting to use the same 3V-supply source to charge the electrostatic structure and to collect the charges at the end of the mechanical-to-electrical conversion. Figure 42 sums up the 3-steps conversion process with the two DC-to-DC conversions (DC-to-DC converters) and the mechanical-to-electrical conversion (energy harvester).

**Figure 42.** DC-to-DC conversions needed to develop an operational electret-free electrostatic converter and conversion steps

Furthermore, in order to limit the size and the cost of the power converters and the power management control circuit, it is worth combining the step-up and the step-down converters into a single DC-to-DC converter: a bidirectional converter is then used. The two most wellknown bidirectional converters are the buck-boost and the flyback converters.

**Figure 43.** Bidirectional DC-to-DC converters (a) buck-boost and (b) flyback

c. Bidirectional buck-boost converter

The operating principle of the bidirectional buck-boost converter (Figure 43(a)) is summed up below:

#### **Step 1.** Capacitor charging

122 Small-Scale Energy Harvesting

circuits associated.

*4.2.2. Charge-constrained cycles* 

conversion (energy harvester).

and conversion steps

maximal voltage, charges are collected from the converter.

This electronic circuit keeps the electrostatic structure's voltage between two values (Vres and Vstore). When the electrostatic structure's capacitance Cvar increases, its voltage decreases and finishes to reach the low voltage storage Ures. Then, diode D1 becomes conductive and a current is transferred from Cres (storage capacitor) to the electrostatic device. When Cvar's capacitance decreases, its voltage increases and finally reaches the high voltage storage Ustore. Then diode D2 becomes conductive and a current is transferred from the electrostatic structure to Cstore. This structure works as a charge pump from Cres to Cstore. And, in order to close the cycle, one part of the energy transferred to Cstore is transferred to Cres by using an inductive electrical converter. Although this structure uses only one inductive component, it requires a

complex electronic circuit to drive the floating transistor connected to the high voltage.

Finally, the constant voltage cycle is not frequently used due to the complex electronic

Charge-constrained cycles are easier to implement than voltage-constrained cycles as the conversion consists in charging the capacitor when the capacitance is maximal and to let it in open-circuit till it reaches its minimum. On the minimal capacitance, corresponding to the

Usually, to reach a high conversion power density, the capacitor must be polarized at a high voltage (V1>100V). Yet, in autonomous devices, only 3V supply sources are available: a first DC-to-DC converter (step-up) is therefore needed to polarize the capacitor at a high voltage (step 1). In the same way, the output voltage on the capacitor after the mechanical-toelectrical conversion (step 2) may reach several hundreds of volts (V2>200-300V) and is therefore not directly usable to power an application: a second converter (step-down) is then necessary (step 3). Obviously, to limit the number of sources, it is interesting to use the same 3V-supply source to charge the electrostatic structure and to collect the charges at the end of the mechanical-to-electrical conversion. Figure 42 sums up the 3-steps conversion process with the two DC-to-DC conversions (DC-to-DC converters) and the mechanical-to-electrical

> mechanical-to-electrical conversion V1>100V V2>200V

②

①

3V supply

**Figure 42.** DC-to-DC conversions needed to develop an operational electret-free electrostatic converter

③

Kp is closed for a time t1. The energy Ec, that has to be sent to the energy harvester to polarize it, is transferred from the supply source E to the inductance L.

Kp is open, and Ks is closed till current is becomes equal to 0, corresponding to the time needed to transfer the energy stored in inductance L to the capacitor of the energy harvester C.

**Step 2.** Mechanical-to-electrical conversion step

Kp and Ks are open to let the electrostatic converter in open circuit so that the voltage across C may vary freely.

**Step 3.** Capacitor discharging

Ks is closed for a time t2, to transfer the energy stored in the capacitor C to inductance L and the storage element E.

Ks is open and Kp is closed till ip becomes equal to 0 corresponding to the time needed to transfer the energy stored in L to the storage element E.

The waveforms of currents in buck–boost converters are presented in Figure 44.

This converter has a good conversion efficiency that can reach up to 80-90%. Yet, Flyback converters are generally more suitable for electrostatic energy harvesters where conversion ratios are higher than 30.

**Figure 44.** Waveforms of currents in buck–boost converters

#### d. Bidirectional flyback converter

The operating principle of the bidirectional flyback converter (Figure 43(b)) is summed up below:

#### **Step 1.** Capacitor charging

Kp is closed for a time t1. The energy Ec, that has to be sent to the energy harvester to polarize it, is transferred from the supply source E to the inductance Lp that charges the magnetic core M.

Kp is open, and Ks is closed till current is becomes equal to 0, corresponding to the time needed to transfer the energy stored in the magnetic core M to the capacitor of the energy harvester C.

**Step 2.** Mechanical-to-electrical conversion step

Kp and Ks are open to let the energy harvester in open circuit so that the voltage across C may vary freely.

**Step 3.** Capacitor discharging

Ks is closed for a time t2, to transfer the energy stored in the capacitor C to the magnetic core M through Ls.

Ks is open and Kp is closed till ip becomes equal to 0 corresponding to the time needed to transfer the energy stored in the magnetic core M to the storage element E.

The waveforms of currents in flyback converters are presented in Figure 45.

Contrary to buck-boost converters, flyback converters do not need bidirectional transistors (Ks must be bidirectional in buck-boost converters) that complicate the power management circuit and increase losses. Moreover, flyback converters enable to optimize both the windings for the high voltages and the low voltages (while buck-boost converters have only one winding).

These two DC-to-DC conversions (step-up and step-down) can be simplified by using electret-based devices. The next sub-section is focused on the power converters and the power management control circuits for these energy harvesters.

**Figure 45.** Waveforms of currents in flyback converters

#### **4.3. PMCC for Electret-Based Electrostatic VEH**

Electret-based eVEH enable to have a direct mechanical-to-electrical conversion without needing any cycles of charges and discharges. As a consequence, it is possible to imagine two kinds of power converters.

#### *4.3.1. Passive power converters*

t

t1 (Kp) t3 (Ks t ) 2 (Ks) t4 (Kp)

charge (step 1) discharge (step 3)

The operating principle of the bidirectional flyback converter (Figure 43(b)) is summed up

Kp is closed for a time t1. The energy Ec, that has to be sent to the energy harvester to polarize it, is transferred from the supply source E to the inductance Lp that charges the

Kp is open, and Ks is closed till current is becomes equal to 0, corresponding to the time needed to transfer the energy stored in the magnetic core M to the capacitor of the energy

Kp and Ks are open to let the energy harvester in open circuit so that the voltage across C

Ks is closed for a time t2, to transfer the energy stored in the capacitor C to the magnetic core

Ks is open and Kp is closed till ip becomes equal to 0 corresponding to the time needed to

Contrary to buck-boost converters, flyback converters do not need bidirectional transistors (Ks must be bidirectional in buck-boost converters) that complicate the power management circuit and increase losses. Moreover, flyback converters enable to optimize both the windings for the high voltages and the low voltages (while buck-boost converters have only

transfer the energy stored in the magnetic core M to the storage element E. The waveforms of currents in flyback converters are presented in Figure 45.

**Figure 44.** Waveforms of currents in buck–boost converters

**Step 2.** Mechanical-to-electrical conversion step

d. Bidirectional flyback converter

**Step 1.** Capacitor charging

magnetic core M.

harvester C.

may vary freely.

M through Ls.

one winding).

**Step 3.** Capacitor discharging

124 Small-Scale Energy Harvesting

i p(t) i s(t)

Ipcmax=Iscmax

below:

Ipdmax=Isdmax

Passive power converters are the easiest way to turn the AC high-voltage low-current eVEH output into a 3V DC supply source for WSN. An example of these circuits is presented in Figure 46(a). It consists in a diode bridge and a capacitor that stores the energy from the eVEH.

**Figure 46.** (a) Simple passive power converter – diode bridge-capacitor and (b) optimal output voltage on Ucb

Such a power converter does not need any PMCC as the energy from the energy harvester is directly transferred to the capacitor. This power conversion is quite simple, but the drawback is the poor efficiency.

Actually, to maximize power extraction from an electret-based electrostatic converter, the voltage across Cb must be close to the half of the eVEH's output voltage in open circuit. This optimal value (Ucb,opt) is generally equal to some tens or hundreds of volts. To power an electronic device, a 3V source is required: this voltage cannot be maintained directly on the capacitor as it greatly reduces the conversion efficiency of the energy harvester (Figure 46(b)).

The solution to increase the efficiency of the energy harvester consists in using active power converters.

#### *4.3.2. Active power converters*

As eVEH' optimal output voltages are 10 to 100 times higher than 3V, a step-down converter is needed to fill the buffer. The most common step-down converters are the buck, the buckboost and the flyback converters. We focus here on the flyback converter that gives more design flexibilities (Figure 47).

Many operation modes can be developed to turn the eVEH high output voltages into a 3V supply source. Here, we focus on two examples: (i) energy transfer on maximum voltage detection and (ii) energy transfer with a pre-storage to keep an optimal voltage across the electrostatic converter.

a. Energy transfer on a maximum voltage detection

The concept of this power conversion is to send the energy from the energy harvester to the 3V energy buffer when the eVEH output voltage reaches its maximum.

The power management control circuit is aimed at finding the maximum voltage across the energy harvester and to close Kp (Figure 48) to send the energy from the eVEH to the magnetic circuit. Then Ks is closed to send the energy from the magnetic circuit to the buffer Cb. The winding ratio m is determined from the voltage ratio between the primary and the secondary.

**Figure 47.** Energy transfer on maximum voltage detection

Figure 48 presents the voltages and the currents on the primary and on the secondary during the power transfer.

**Figure 48.** Voltages and currents during power conversion

46(b)).

converters.

*4.3.2. Active power converters* 

design flexibilities (Figure 47).

+++++++

C

during the power transfer.

EH

i

**Figure 47.** Energy transfer on maximum voltage detection

a. Energy transfer on a maximum voltage detection

3V energy buffer when the eVEH output voltage reaches its maximum.

electrostatic converter.

secondary.

Actually, to maximize power extraction from an electret-based electrostatic converter, the voltage across Cb must be close to the half of the eVEH's output voltage in open circuit. This optimal value (Ucb,opt) is generally equal to some tens or hundreds of volts. To power an electronic device, a 3V source is required: this voltage cannot be maintained directly on the capacitor as it greatly reduces the conversion efficiency of the energy harvester (Figure

The solution to increase the efficiency of the energy harvester consists in using active power

As eVEH' optimal output voltages are 10 to 100 times higher than 3V, a step-down converter is needed to fill the buffer. The most common step-down converters are the buck, the buckboost and the flyback converters. We focus here on the flyback converter that gives more

Many operation modes can be developed to turn the eVEH high output voltages into a 3V supply source. Here, we focus on two examples: (i) energy transfer on maximum voltage detection and (ii) energy transfer with a pre-storage to keep an optimal voltage across the

The concept of this power conversion is to send the energy from the energy harvester to the

The power management control circuit is aimed at finding the maximum voltage across the energy harvester and to close Kp (Figure 48) to send the energy from the eVEH to the magnetic circuit. Then Ks is closed to send the energy from the magnetic circuit to the buffer Cb. The winding ratio m is determined from the voltage ratio between the primary and the

i

Figure 48 presents the voltages and the currents on the primary and on the secondary

M

Controlled switches

DC-to-DC converter

Kp Ks

<sup>p</sup> i

s

Uc Ucb

+ -

Cb

As eVEH capacitances are quite small, parasitic capacitances of the primary winding may have a strong negative impact on the output powers, increasing conversion losses. An alternative consists in using a pre-storage capacitor.

b. Energy transfer with a pre-storage capacitor

In this operation mode, a pre-storage capacitor Cp is used to store the energy from the eVEH and to maintain an optimal voltage across the diode bridge in order to optimize the energy extraction from the eVEH.

**Figure 49.** (a) eVEH output power vs imposed output voltage and (b) Ucp(t)

The goal of the PMCC is to maintain the voltage quite constant across the diode bridge (+/- 10% Ucp,opt). Then, when Ucp reaches Ucp,opt+10%, one part of the energy stored in Cp is sent to Cb through the flyback converter.

**Figure 50.** Energy transfer with pre-storage

Voltages and currents during the electrical power transfer are presented in Figure 51.

**Figure 51.** Voltages and currents during power conversion

As Cp can be in the order of some tens to hundreds of nanofarads, transformer's parasitic capacitances have smaller impacts on eVEH's output power.

This power conversion principle also enables to use multiple energy harvesters in parallel with only one transformer and above all only one PMCC (which is not the case with the maximum voltage detection).

We have presented some examples of power converters able to turn the raw output powers of the energy harvesters into supply sources able to power electronic devices. Thanks to this, and low power consumptions of WSN' nodes, it is possible to develop autonomous wireless sensors using the energy from vibrations from now on. The last section gives an assessment of this study.

### **5. Assessments and perspectives**

In this last section, we present our vision of eVEH and their perspectives for the future.

### **5.1. Assessments**

Electrostatic VEH are doubtless the less known vibration energy harvesters, and especially compared to piezoelectric devices. Yet, these devices have undeniable advantages: the possibility to develop structures with high mechanical-to-electrical couplings, to decouple the mechanical-to-mechanical converter and the mechanical-to-electrical converter, to develop low-cost devices able to withstand high temperatures…

Moreover, even if these devices have incontestable drawbacks as well, such as low capacitances, high output voltages and low output currents, it has been proven that they can be compatible with WSN needs as soon as a power converter is inserted between the VEH and the device to supply.

### **5.2. Limits**

128 Small-Scale Energy Harvesting

p(t) i

s(t) Ucb U (t) cp(t)

Ucp+

Ucb-

maximum voltage detection).

**5. Assessments and perspectives** 

of this study.

**5.1. Assessments** 

and the device to supply.

Voltages and currents during the electrical power transfer are presented in Figure 51.

Ismax

As Cp can be in the order of some tens to hundreds of nanofarads, transformer's parasitic

This power conversion principle also enables to use multiple energy harvesters in parallel with only one transformer and above all only one PMCC (which is not the case with the

We have presented some examples of power converters able to turn the raw output powers of the energy harvesters into supply sources able to power electronic devices. Thanks to this, and low power consumptions of WSN' nodes, it is possible to develop autonomous wireless sensors using the energy from vibrations from now on. The last section gives an assessment

In this last section, we present our vision of eVEH and their perspectives for the future.

Electrostatic VEH are doubtless the less known vibration energy harvesters, and especially compared to piezoelectric devices. Yet, these devices have undeniable advantages: the possibility to develop structures with high mechanical-to-electrical couplings, to decouple the mechanical-to-mechanical converter and the mechanical-to-electrical converter, to

Moreover, even if these devices have incontestable drawbacks as well, such as low capacitances, high output voltages and low output currents, it has been proven that they can be compatible with WSN needs as soon as a power converter is inserted between the VEH

Ipmax

T1 T3 <sup>i</sup>

**Figure 51.** Voltages and currents during power conversion

capacitances have smaller impacts on eVEH's output power.

develop low-cost devices able to withstand high temperatures…

t

Ucp-

Ucb+

Obviously, eVEH have drawbacks and limitations. We present in this subsection the four most important limits of these devices.


### **5.3. Perspectives**

Like all VEH (piezoelectric, electromagnetic or electrostatic), the most critical point to improve is the frequency bandwidth that must be largely increased to develop viable and adaptable devices.

Indeed, a wide frequency bandwidth is firstly necessary to develop robust devices. VEH are submitted to a large amount of cycles (16 billion cycles for a device that works at 50Hz during 10 years), that may change the resonant frequency of the energy harvester due to fatigue. Then, the energy harvester's resonant frequency is not tuned to the ambient vibrations' frequency anymore. Therefore, it is absolutely primordial to develop devices able to maintain their resonant frequency equal to the vibration frequency.

Wideband energy harvesters are also interesting to develop adaptable devices, able to work in many environments and simple to set up and to use. There is a real need for Plug and Play devices.

Figure 52 presents our vision of VEH today: VEH market as a function of the time and the two technological bottlenecks linked to working frequency bandwidths. In our opinion, today's VEH are yet suitable for industry; increasing working frequency bandwidths and developing plug and play devices are the only way to conquer new markets.

**Figure 52.** Vibration Energy Harvesters – Perspectives [52]

## **6. Conclusions**

We have presented in this chapter the basic concepts and theories of electrostatic converters and electrostatic vibration energy harvesters together with some prototypes from the state of the art, adopting a "global system" vision.

Electrostatic VEH are increasingly studied from the early 2000s. Unfortunately, no commercial solution is on the market today, dedicating these devices to research.

We believe that this is a pity because they have undeniable advantages compared to piezoelectric or electromagnetic devices. The first in importance is probably the possibility to manufacture low cost devices (low cost and standard materials). Obviously, the limited frequency bandwidth of vibration energy harvesters does not help the deployment of these devices, even if some solutions are currently under investigation. Yet, with this increasing need to get more information from our surroundings, we can expect that these systems will match industrial needs and find industrial applications.

Anyway, electrostatic converters and electrostatic vibration energy harvesters remain an interesting research topic that gathers material research (electrets), power conversion, low consumption electronics, mechanics and so on…

### **Author details**

S. Boisseau, G. Despesse and B. Ahmed Seddik *LETI, CEA, Minatec Campus, Grenoble, France* 

### **Acknowledgement**

The authors would like to thank their VEH coworkers: J.J. Chaillout, A.B. Duret, P. Gasnier, J.M. Léger, S. Soubeyrat, S. Riché and S. Dauvé for their contributions to this chapter.

### **7. References**

130 Small-Scale Energy Harvesting

Market

**6. Conclusions** 

**Author details** 

Today **Figure 52.** Vibration Energy Harvesters – Perspectives [52]

match industrial needs and find industrial applications.

consumption electronics, mechanics and so on…

S. Boisseau, G. Despesse and B. Ahmed Seddik *LETI, CEA, Minatec Campus, Grenoble, France* 

of the art, adopting a "global system" vision.

today's VEH are yet suitable for industry; increasing working frequency bandwidths and

Industry Transport Defense Infrastructures

We have presented in this chapter the basic concepts and theories of electrostatic converters and electrostatic vibration energy harvesters together with some prototypes from the state

Electrostatic VEH are increasingly studied from the early 2000s. Unfortunately, no

We believe that this is a pity because they have undeniable advantages compared to piezoelectric or electromagnetic devices. The first in importance is probably the possibility to manufacture low cost devices (low cost and standard materials). Obviously, the limited frequency bandwidth of vibration energy harvesters does not help the deployment of these devices, even if some solutions are currently under investigation. Yet, with this increasing need to get more information from our surroundings, we can expect that these systems will

Anyway, electrostatic converters and electrostatic vibration energy harvesters remain an interesting research topic that gathers material research (electrets), power conversion, low

commercial solution is on the market today, dedicating these devices to research.

Environment Industry

Tomorrow After Tomorrow

Industry Transport Defense Infrastructures Healthcare Public at large

developing plug and play devices are the only way to conquer new markets.


[33] Tsutsumino T, Suzuki Y, Kasagi N, Sakane Y. Seismic Power Generator Using High-Performance Polymer Electret. Proc. MEMS 2006: 98-101.

132 Small-Scale Energy Harvesting

2003:538-41.

2011;9(1): 64-75.

1317/19/11/115025

Technology. 2005.

1317/13/2/307

10.1088/0964-1726/19/7/075015

[15] Boisseau S, Despesse G, Ricart T, Defay E, Sylvestre A. Cantilever-based electret energy harvesters. IOP Smart Materials and Structures 2011; 20(105013).

[16] Boland J, Chao Y, Suzuki Y, Tai Y. Micro electret power generator. Proc. MEMS

[17] Boisseau S, Despesse G, Sylvestre A. Optimization of an electret-based energy harvester. Smart Materials and Structures 2010;19(075015). http://dx.doi.org/

[18] Meninger S, Mur-Miranda J O, Amirtharajah R, Chandrakasan A, Lang J. Vibration-toelectric energy conversion. IEEE transactions on very large scale integration (VLSI)

[19] Tashiro R, Kabei N, Katayama K, Tsuboi E, Tsuchiya K. Development of an electrostatic generator for a cardiac pacemaker that harnesses the ventricular wall

[20] Roundy S. Energy Scavenging for Wireless Sensor Nodes with a Focus on Vibration to

[21] Despesse G, Chaillout J J, Jager T, Léger J M, Vassilev A, Basrour S, Charlot B. High damping electrostatic system for vibration energy scavenging. Proc. sOc-EUSAI 2005:283-6. [22] Basset P, Galayko D, Paracha A, Marty F, Dudka A, Bourouina T. A batch-fabricated and electret-free silicon electrostatic vibration energy harvester. IOP Journal of Micromechanics and Microengineering 2009;19(115025). http://dx.doi.org/10.1088/0960-

[23] Hoffmann D, Folkmer B, Manoli Y. Fabrication and characterization of electrostatic

[24] Roundy S. Energy Scavenging for Wireless Sensor Networks with Special Focus on

[25] Mitcheson P, Green T C, Yeatmann E M, Holmes A S. Architectures for vibration-

[26] Chih-Hsun Yen B, Lang J. A variable capacitance vibration-to-electric energy

[27] Jefimenko O, Walker D K. Electrostatic Current Generator Having a Disk Electret as an

[28] Tada Y. Experimental Characteristics of Electret Generator, Using Polymer Film

[29] Genda T, Tanaka S, Esashi M. High power electrostatic motor and generator using

[30] Boland J. Micro electret power generators. PhD thesis. California Institute of

[31] Mizuno M, Chetwynd D. Investigation of a resonance microgenerator. IOP Journal of micromechanics and Microengineering 2003;13: 209-16. http://dx.doi.org/10.1088/0960-

[32] Sterken T, Fiorini P, Altena G, Van Hoof C, Puers R. Harvesting Energy from Vibrations by a Micromachined Electret Generator. Proc. Transducers 2007: 129-32.

driven micropower generators. J. of Microelect. Systems 2004;13: 429-40.

Active Element. Transactions on Industry Applications 1978;IA-14: 537-40.

Electrets. Japanese Journal of Applied Physics 1992;31: 846-51.

Electricity Conversion. PhD Thesis. University of California, Berkeley, 2003.

http://dx.doi.org/ 10.1088/0964-1726/20/10/105013

motion. Journal of Artificial Organs 2002;5:239-45.

micro-generators. Proc. PowerMEMS 2008: 15.

harvester. IEEE Trans. Circuits Syst. 2006;53: 288-95.

Vibrations. Hardcover, Springer, 2003.

electrets. Proc. Transducers 2003;1: 492-5.


