# A unified N-SECE strategy for highly coupled piezoelectric

## Transcript Of A unified N-SECE strategy for highly coupled piezoelectric

A unified N-SECE strategy for highly coupled piezoelectric energy scavengers

A Morel1,2, A Badel2, Y Wanderoild1 and G Pillonnet1

1 Univ. Grenoble Alpes, CEA, LETI, MINATEC, F-38000 Grenoble, France 2 Univ. Savoie Mont Blanc, SYMME, F-74000 Annecy, France

Email: [email protected]

Abstract. This paper proposes a novel vibration energy harvesting strategy based on an extension of the Synchronous Electric Charge Extraction (SECE) approach, enabling both the maximization of the harvested power and a consequent bandwidth enlargement in the case of highly coupled/lightly damped piezoelectric energy harvesters. The proposed strategy relies on the tuning of the frequency of the energy extraction events, which is either π times greater than the vibration frequency (Multiple SECE case, π > 1) or 1/π times smaller (Regenerative SECE, π < 1). We first prove analytically than increasing or decreasing π both lead to a damping reduction. While π has no impact on the systemβs resonance frequency in the Regenerative case (π < 1), we show that this resonant frequency becomes a function of π in the Multiple SECE case (π > 1). Experimental results on a highly coupled/lowly damped piezoelectric harvester (π( = 0.44, π/ = 20) demonstrates the potential of this strategy, leading to 257% harvested power improvement compared to SECE (π = 1). and the possibility to tune the resonant frequency on a range as large as 35% of the short-circuit resonant frequency of the harvester.

1. Introduction Energy scavenging is a promising alternative to batteries in order to supply sensor nodes.

Vibrations energy harvesting is of particular interest in closed confined environments where there are few solar energies nor thermal gradients. At a cm scale, piezoelectric energy harvesters (PEH) are a good way to convert mechanical strain into useful and storable electrical energy, since they are easier to miniaturize than electromagnetic generators, while exhibiting higher power densities than electrostatic generators [1].

During the last decade, electrical interfaces for this kind of harvester have been studied. Standard interfaces based on a diode bridge rectifying and smothering the piezoelectric voltage have first been proposed in [2,3]. However, these interfaces, in the case of lowly coupled PEH, only allow to extract a small part of the maximal energy that could be harvested [4]. Furthermore, their efficiencies are highly dependent on the voltage of the storage capacitance.

In order to design higher performances interfaces, researchers have developed SECE based strategies [4]. Indeed, they propose simple, efficient, and load independent ways to harvest the energy from lowly coupled and/or highly damped PEH. These strategies have either be implemented using on-the-shelf components [5], or developing dedicated ASIC with external passive components [6].

Thanks to recent progresses in the mechanical design and the use of highly coupled piezoelectric materials, e.g. single crystal, piezoelectric harvesters may now exhibit much higher electromechanical coupling associated with low mechanical damping. For this particular kind of PEH, SECE strategy is not well adapted since it overdamps the mechanical resonator, leading to low displacements and harvested energy [7,8].

Some research has been conducted to investigate new efficient strategies that could be used for highly coupled PEH, in order to optimize the damping induced by the electrical interface [8-10]. Indeed, in [8], we proposed to wait a certain number of semi-period before harvesting the accumulated energy in order to reduce the damping induced by the electrical interface. In [9] and [10], the damping is reduced thanks to the control of the number of electrical charges extracted from the piezoelectric material. Those two approaches, even though they allowed to reduce the damping, did not induce any frequency tuning effect which could although be used to enlarge the harvesting bandwidth.

Electrically-based frequency tuning strategies have also been proposed in the literature [1115]. In [11], a capacitive bank combined with a variable resistive load emulated thanks to a DC/DC converter allows to tune the phase of the piezoelectric voltage while optimizing the damping induced by the interface. However, this approach requires an important number of passive components. In order to overcome this issue, in [12], we proposed to replace the capacitive bank with a short-circuit control which emulates a tunable capacitive behavior. [13] and [14-15] proposed a new frequency tuning approach based on the combination of the tunable SECE explained in [9] and [10] with a phase tuning. This approach is theoretically very efficient, however it introduces two tuning parameters, leading to a complicated control scheme which may take time to converge. Since the harvested energy may be a complex function of these two parameters, it may also converge toward a low local maximum, leading to limited performances.

Recently, researchers have started to propose single tuning harvesting strategies, which could combine damping optimization, some frequency tuning possibilities, with a simpler single dimension control scheme. In [16], Lefeuvre et al proposed a phase-shifted SECE strategy which consists of only controlling the phase between the displacement extrema and the energy extraction event. This strategy allows to reach the maximum harvestable power for two different frequencies with a single tuning parameter. However, this technique is quite sensitive to the phase, requiring a sensitive control loop. Plus, the two frequencies for which the harvested power is maximized are both functions of the coupling coefficient, which tend to complicate the design of applications-dedicated PEHs.

In this paper, we propose a new single tuning harvesting approach. Based on the strategy presented in [8], we generalize the regenerative principal for any case where the energy extraction events happen at a different frequency compared to the vibration frequency. We prove that there is two different ways to reduce the damping, either by harvesting the energy at a lower frequency rate, as detailed in [8], or at a higher frequency rate, which is a new and complementary way to reduce the damping. Interestingly, the energy extraction frequency has indeed a high impact on the dynamics of the electromechanical system, leading to frequency tuning possibilities. This new strategy, while exhibiting equivalent performances as the one presented in [16], is more robust and less sensitive to the variations of the tuning parameter. Indeed, good performance in terms of maximal harvested power and bandwidth can be obtained with only 2 or 3 different values of π. Furthermore, one of the frequency where the harvested power is maximized is insensitive to the electromechanical coupling, which greatly facilitates the PEH design for this kind of approach.

In a first part, we remind why the standard SECE becomes less efficient when the electromechanical coupling of the harvester increases. In a second part, we present the proposed approach called N-SECE strategy. We then define two different cases (π < 1 and π > 1) which are theoretically studied. The power expressions in each case are derived, and lead to a unified analysis of the influences of π on the dynamics and performance of the PEH. Finally, the proposed approach is experimentally validated, and the obtained results are described and discussed.

2. Original background: standard SECE The standard SECE has first been introduced by Lefeuvre et al. in [4]. It is an interesting

alternative to the standard energy harvesting (SEH) interface based on a full bridge rectifier, for two main reasons:

β’ It increases the percentage of mechanical energy of the harvester converted into electrical energy, leading to enhanced performances on lowly coupled and/or highly damped PEH.

β’ Its performances are not dependent on the voltage across the storage element (π23 in Figure 1).

The SECE strategy can be described as follow: during a semi period, the energy is accumulated in the piezoelectric material. When the voltage reaches its maximum value, an inductance

πΏ is connected to the piezoelectric material. The electrical energy stored on the piezoelectric element is then transferred to the inductance. Once the voltage on the piezoelectric element reaches zero, the inductance is disconnected from the piezoelectric element and the energy flows to the storage element

πΆ678. The typical circuit realizing the SECE strategy and its associated waveforms are shown in Figure 1 and Figure 2.

Rectifier

Inductive Switch

Energy storage

Piezo !"

%&'(

#

)*+

$

Max

detector

Figure 1-SECE interface circuit

1

Xm

Energy extraction event

vp

x

0.5

Piezoelectric voltage vp

-Xm

0 0

0.5

1

0 :/2 : 3:/2 2:

3

Figure 2-Typical voltage and displacement waveforms when using the SECE strategy.

In order to theoretically evaluate SECE performance, we will consider a PEH made of a piezoelectric material deposited on a mechanical linear oscillator, as shown in Figure 3.

Piezoelectric material

(

Cantilever beam of stiffness !"# and equivalent mass $

Support

Tip mass

% &'

Neutral axis

Electric interface

)

(

Ambient acceleration

Mechanical system

.

$ 1/!-#

$)

Piezoelectric material

1// *'

Electric interface

Figure 3-Piezoelectric energy harvester made of a cantilever beam, and its equivalent electrical model

This PEH can be seen as a single-degree of freedom mass + damper + stiffness + piezo system, that is classically modelled by the equivalent electrical model shown in figure 3 and whose constitutive equations are given by (1).

πΉ = ππΎ = ππ₯ + π·π₯ + πΎB3π₯ + πΌπ£>

π = πΌπ₯ β πΆ>π£>

(1)

Where πΉ, πΎ, π₯, π and π£> respectively stand for the vibration force, ambient acceleration, the displacement of the inertial mass, the current extracted in the interface circuit, and the voltage across the electrodes of the piezoelectric material. π, π·, πΎB3, πΆ> and πΌ stand for the dynamic mass of the harvester, the mechanical damping, the short-circuit stiffness, the piezoelectric material clamped capacitance, and the piezoelectric coefficient.

This model is accurate as long as the harvester remains linear and that the PEH is driven around one of its resonant frequencies. In order to simplify the calculations, we will consider in the following that π₯ is purely sinusoidal and of constant amplitude, as expressed by (2). This physically means that the harvester is considered in steady state operation and that any potential mechanical nonlinearity is neglected.

π₯ π‘ = π/ πππ (π) = π/πππ (ππ‘)

(2)

As explained extensively in [17], the studies of the electrical interfaces for piezoelectric harvesters can be unified and simplified thanks to normalized variables:

π(

πΌ(

π/( = 1 β π( = πΎ πΆ

B3 >

πΎB3 π

π/ = π·

(3)

πΎB3 πT = π

πΊ/ = π/πT

π63 = πT 1 + π/(

π( β [0,1] is the square coupling coefficient, while π/( β βS is defined as the square modified electromechanical coupling. π/, πT and πΊ/ are the quality factor of the mechanical resonator, its short circuit angular resonant frequency, and the normalized vibration frequency respectively. The two normalized variables π/( and π/are sufficient to characterize and predict the performances of a linear PEH, and their product π/( π/ is commonly used to evaluate the potential of a piezoelectric harvester [8,17]. That is why it has been defined as a figure of merit in previous works

[17]. π63 is the open circuit angular resonant frequency of the harvester. If the PEH is highly coupled, this frequency is slightly different than the short circuit angular resonant frequency, πT.

From (1) and (2) and considering the diode threshold voltages negligible compared to the

piezoelectric voltage, it can be proven that (for a SECE interface) the displacement amplitude π/ can be given by the following expression when the vibration frequency matches the resonance frequency

of the PEH:

πΉ/

π/ =

4πΌ(

(4)

π·π + ππΆ>

Where πΉ/ stands for the amplitude of the vibration force πΉ. The harvested power thanks to the SECE interface is given by (5).

2ππΌ(

πΉ/(

16π/( π/

π/WX

π= ππΆ>

4πΌ( ( β π

4π( π (

(5)

π·π + ππΆ>

1+

//

π

Where π/WX is the maximum power which can be extracted from a linear PEH under a constant vibration of amplitude πΉ/. The expression of this maximum power is given by (6).

πΉ/( ππΎ/( π/WX = 8π· = 8πT π/ (6)

Where πΎ/ is the amplitude of the driving acceleration. Figure 4 represents the evolution of the harvested power at resonance with both the SECE and the standard interface as a function of the product π/( π/ . We can clearly see that for π/( π/ = π/4, the harvested power is maximized. When the product π/( π/ gets too important, the SECE overdamps the mechanical system, leading to a reduction of the vibration amplitude and lower performances than what can be obtained with a simple rectifier. This can also be seen in Figure 5, which shows the displacement amplitude at the resonant

frequency as a function of π/( π/. When the PEH is too highly coupled and/or too lightly damped, the displacement amplitude goes below its optimal value, leading to a lower extracted energy [7,8].

1

Area where the standard

0.8

SECE is more efficient

than the standard energy

harvesting (SEH) circuit.

0.6

SECE Standard interface (SEH)

Overdamping

Normalized power P/Pmax

0.4

0.2

k2 Q =:/4

mm

0

10-2

10-1

100

101

102

k2 Q

mm

Figure 4- Normalized harvested power using SECE (black) and standard interface (blue) as a

function of π/( π/.

1

Area where the standard 0.8 SECE is more efficient

than the standard energy harvesting (SEH) circuit. 0.6

SECE Standard interface (SEH)

Displacement amplitude Xm/(Xm )sc

0.4

Optimized displacement amplitude

Overdamping

0.2

k2 Q =:/4

mm

0

10-2

10-1

100

101

102

k2 Q

mm

Figure 5-Normalized displacement amplitude using SECE (black) and standard interface (blue) at

resonance as a function of π/( π/.

(π/)B3 given by πΉ//(π·πT) is the short circuit amplitude displacement of the PEH at resonance. This is the maximum value that can take the displacement, as there is no damping induced by the electrical interface when the piezoelectric material electrodes are short circuited.

Because of this overdamping phenomenon, we should find ways to reduce the damping induced by the SECE interface in order to be able to use it on high coupling interfaces. This could lead to an interface:

β’ Which is effective on any PEH, whether it is highly coupled/damped or not. β’ Whose effectiveness is not related to the voltage across the storage element.

The SECE strategy increasing the forces induced by the electrical interface on the mechanical system, we could use this characteristic to electrically change the resonant frequency of the electromechanical system, and enlarge the harvesting energy bandwidth.

3. Proposed concept: N-SECE In the standard SECE, there are two voltage extrema every vibrationβs period, thus the SECE

harvesting frequency is two times higher than the vibration frequency (π^W_`abcb = 2π`de). In the proposed N-SECE, instead of harvesting the energy at a frequency fixed by the vibrations, similarly as the standard SECE strategy, we propose to harvest the energy at a different frequency, controlled by the parameter π, where π=π^W_`/π^W_`fghg = π^W_`/(2π`de) as depicted in Figure 6.

Vibration frequency = ,034 Rectifier

Inductive Switch

Energy storage

Piezo !"

%&'(

#

)*+

$

Oscillator , = 2, 5

5

-./0

034

Figure 6-Proposed N-SECE system based on a tunable parameter π

By doing so, we obtain the piezoelectric voltage waveforms shown in Figure 7 when the harvesting frequency π^W_` is higher than 2π`de (π > 1). We also obtain the voltage waveforms shown in Figure 8 when π^W_` is lower than 2π`de (π < 1). Obviously, when π = 1, π^W_` = π^W_`abcb, and the N-SECE interface behaves like a standard SECE interface. Increasing or decreasing π leads in any case to a reduction of the damping induced by the electrical interface, and hence to improved performance, as demonstrated in the next sections.

Figure 7-Piezoelectric voltage waveforms for various π when π > 1

Piezoelectric voltage vp

N = 1

N = 1/3

N = 1/5

N = 1/7

Time

Figure 8-Piezoelectric voltage waveforms for various π when π < 1 4. Theoretical analysis: Multiple SECE (N>1 case)

In this section, the case where the harvesting frequency is higher than the vibration frequency (π > 1) is considered. In this case, π β ββ and π β [1, π] β© ββ respectively represent the number of harvesting events in a vibration semi-period, and an index of the harvesting event in the semi-period. First, the PEH works in open circuit condition: the voltage across the piezoelectric material electrodes starts to increase as charges accumulate in the dielectric capacitance. During every harvesting event, this piezoelectric voltage drops almost instantly to zero as the electrical charges are extracted from the material. Then the voltage starts to grow again, until the next harvesting event. Consequently, the piezoelectric voltage during a semi-period can be divided in π subset voltages, each related to a particular harvesting event. The expression of the π8^ subset voltage is given by (7)

Vibration semi-period

Piezoelectric voltage vp

vp

vp

vp

1

2

3

vp 1

vp

vp

2

3

vp=vp +vp +vp 123

Energy extraction events

0

:

2:

3

Figure 9-Theoretical voltage waveforms obtained for π = 3, revealing 3 different subset voltages

πΌr

π β 1 ππ

π₯ π ππ , π£ ΞΈ = πΆ> stn v

βπ β ]

π, [

π

π

(7)

>o

u

πβ1

ππ

0,

βπ β [0,

π] βͺ [ , π]

π

π

Consequently, the piezoelectric voltage expression during a semi-period is the sum of π periodic subset voltages, and can be expressed as a Fourier series since it is a periodical signal.

u

uβ

π£> π = π£>s(π) =

[π7o cos π’π + π7osin (π’π)]

(8)

syn

syn 7yn

Figure 7 shows the piezoelectric voltage waveforms simulated for various π while Figure 9 illustrates the case π = 3. In order to linearize the piezoelectric voltage expression given by (8), we find the first harmonic coefficients of the Fourier series associated with π£>. These coefficients are given by (9).

u

π/πΌ

2ππ

2 π β 1 π 2π

π β 1 π ππ

πn = 2πΆ π

sin

+ sin

π

π

+ β 2cos(

)sin( )

π

π

π

>

(9)

syn

u

π/πΌ

2ππ

2 πβ1 π

πβ1 π

ππ

πβ1 π

πn = 2πΆ π

βcos

+ cos

π

π

+ 4cos(

)(cos( ) β cos(

))

π

π

π

> syn

Considering the filtering effect of the mechanical resonator, it is reasonable to consider that

only the first voltage harmonic π£>n will have an influence on the dynamics of the system. Its expression in the Fourier domain π£>n is given by (10).

πn

πn

(10)

π£>n π = (π/ β π π/)π₯(π)

Where π₯ stands for the mass displacement written in the Fourier domain. Substituting (10) in

(1) expressed in the Fourier domain leads to the expression of the displacement magnitude π/, given by (11).

πΉ

π/ = π΄( + π΅(

π΄ = πΎ β ππ( + πΌ πn

(11)

B3

π/

π΅ = ππ· β πΌ πn π/

From the displacement amplitude obtained in (11), we can eventually determine the harvested

power, given by (12).

πΌ(

u

π

ππ

πβ1 π (

(12)

π = π/(

cos

β cos

πΆ> 2π

π

π

syn

Thus, from (9), (11) and (12), the harvested power can be determined for any value of π as a

function of the PEHβs parameters and of the ambient acceleration frequency and magnitude.

5. Theoretical analysis: Regenerative SECE (N<1 case)

When the harvesting frequency π^W_` is lower than the vibration frequency π`de, the voltage waveforms is different, leading to a completely different analytical formulation, fully detailed in [8]. In this case, π can be defined as the inverse of π β ββ, where π represents the number of semi-period of vibrations between each harvesting events. The piezoelectric voltage is then expressed by (13).

1 r/u πΌπ₯ π ππ ,

πΆ> T

βπ β ]0, π/π[

π£> π =

0, ππ π = π/π

(13)

1 r/u

πΌπ₯ π ππ , βπ β ]π/π, 2π/π[

πΆ> β

0, ππ π = 2π/π

This piezoelectric voltage has been drawn numerically for various values of π, as depicted in Figure 8. The new voltage expression leads to another Fourier expression. The expressions of the first Fourier coefficients are given by (14).

πΌπ/

πn = πΆ

>

πΌπ/π

n/u

(14)

πn = 2 πΆ π β1 β 1

>

Thus, in the regenerative case (N<1), as extensively explained in [8], the displacement amplitude (15) as well as the harvested power expressions (16) can be derived. Note that when 1/π is even, the additional damping due to energy harvesting is null, and thus the harvesting power is zero. We will only consider odd values of 1/π in the following.

A Morel1,2, A Badel2, Y Wanderoild1 and G Pillonnet1

1 Univ. Grenoble Alpes, CEA, LETI, MINATEC, F-38000 Grenoble, France 2 Univ. Savoie Mont Blanc, SYMME, F-74000 Annecy, France

Email: [email protected]

Abstract. This paper proposes a novel vibration energy harvesting strategy based on an extension of the Synchronous Electric Charge Extraction (SECE) approach, enabling both the maximization of the harvested power and a consequent bandwidth enlargement in the case of highly coupled/lightly damped piezoelectric energy harvesters. The proposed strategy relies on the tuning of the frequency of the energy extraction events, which is either π times greater than the vibration frequency (Multiple SECE case, π > 1) or 1/π times smaller (Regenerative SECE, π < 1). We first prove analytically than increasing or decreasing π both lead to a damping reduction. While π has no impact on the systemβs resonance frequency in the Regenerative case (π < 1), we show that this resonant frequency becomes a function of π in the Multiple SECE case (π > 1). Experimental results on a highly coupled/lowly damped piezoelectric harvester (π( = 0.44, π/ = 20) demonstrates the potential of this strategy, leading to 257% harvested power improvement compared to SECE (π = 1). and the possibility to tune the resonant frequency on a range as large as 35% of the short-circuit resonant frequency of the harvester.

1. Introduction Energy scavenging is a promising alternative to batteries in order to supply sensor nodes.

Vibrations energy harvesting is of particular interest in closed confined environments where there are few solar energies nor thermal gradients. At a cm scale, piezoelectric energy harvesters (PEH) are a good way to convert mechanical strain into useful and storable electrical energy, since they are easier to miniaturize than electromagnetic generators, while exhibiting higher power densities than electrostatic generators [1].

During the last decade, electrical interfaces for this kind of harvester have been studied. Standard interfaces based on a diode bridge rectifying and smothering the piezoelectric voltage have first been proposed in [2,3]. However, these interfaces, in the case of lowly coupled PEH, only allow to extract a small part of the maximal energy that could be harvested [4]. Furthermore, their efficiencies are highly dependent on the voltage of the storage capacitance.

In order to design higher performances interfaces, researchers have developed SECE based strategies [4]. Indeed, they propose simple, efficient, and load independent ways to harvest the energy from lowly coupled and/or highly damped PEH. These strategies have either be implemented using on-the-shelf components [5], or developing dedicated ASIC with external passive components [6].

Thanks to recent progresses in the mechanical design and the use of highly coupled piezoelectric materials, e.g. single crystal, piezoelectric harvesters may now exhibit much higher electromechanical coupling associated with low mechanical damping. For this particular kind of PEH, SECE strategy is not well adapted since it overdamps the mechanical resonator, leading to low displacements and harvested energy [7,8].

Some research has been conducted to investigate new efficient strategies that could be used for highly coupled PEH, in order to optimize the damping induced by the electrical interface [8-10]. Indeed, in [8], we proposed to wait a certain number of semi-period before harvesting the accumulated energy in order to reduce the damping induced by the electrical interface. In [9] and [10], the damping is reduced thanks to the control of the number of electrical charges extracted from the piezoelectric material. Those two approaches, even though they allowed to reduce the damping, did not induce any frequency tuning effect which could although be used to enlarge the harvesting bandwidth.

Electrically-based frequency tuning strategies have also been proposed in the literature [1115]. In [11], a capacitive bank combined with a variable resistive load emulated thanks to a DC/DC converter allows to tune the phase of the piezoelectric voltage while optimizing the damping induced by the interface. However, this approach requires an important number of passive components. In order to overcome this issue, in [12], we proposed to replace the capacitive bank with a short-circuit control which emulates a tunable capacitive behavior. [13] and [14-15] proposed a new frequency tuning approach based on the combination of the tunable SECE explained in [9] and [10] with a phase tuning. This approach is theoretically very efficient, however it introduces two tuning parameters, leading to a complicated control scheme which may take time to converge. Since the harvested energy may be a complex function of these two parameters, it may also converge toward a low local maximum, leading to limited performances.

Recently, researchers have started to propose single tuning harvesting strategies, which could combine damping optimization, some frequency tuning possibilities, with a simpler single dimension control scheme. In [16], Lefeuvre et al proposed a phase-shifted SECE strategy which consists of only controlling the phase between the displacement extrema and the energy extraction event. This strategy allows to reach the maximum harvestable power for two different frequencies with a single tuning parameter. However, this technique is quite sensitive to the phase, requiring a sensitive control loop. Plus, the two frequencies for which the harvested power is maximized are both functions of the coupling coefficient, which tend to complicate the design of applications-dedicated PEHs.

In this paper, we propose a new single tuning harvesting approach. Based on the strategy presented in [8], we generalize the regenerative principal for any case where the energy extraction events happen at a different frequency compared to the vibration frequency. We prove that there is two different ways to reduce the damping, either by harvesting the energy at a lower frequency rate, as detailed in [8], or at a higher frequency rate, which is a new and complementary way to reduce the damping. Interestingly, the energy extraction frequency has indeed a high impact on the dynamics of the electromechanical system, leading to frequency tuning possibilities. This new strategy, while exhibiting equivalent performances as the one presented in [16], is more robust and less sensitive to the variations of the tuning parameter. Indeed, good performance in terms of maximal harvested power and bandwidth can be obtained with only 2 or 3 different values of π. Furthermore, one of the frequency where the harvested power is maximized is insensitive to the electromechanical coupling, which greatly facilitates the PEH design for this kind of approach.

In a first part, we remind why the standard SECE becomes less efficient when the electromechanical coupling of the harvester increases. In a second part, we present the proposed approach called N-SECE strategy. We then define two different cases (π < 1 and π > 1) which are theoretically studied. The power expressions in each case are derived, and lead to a unified analysis of the influences of π on the dynamics and performance of the PEH. Finally, the proposed approach is experimentally validated, and the obtained results are described and discussed.

2. Original background: standard SECE The standard SECE has first been introduced by Lefeuvre et al. in [4]. It is an interesting

alternative to the standard energy harvesting (SEH) interface based on a full bridge rectifier, for two main reasons:

β’ It increases the percentage of mechanical energy of the harvester converted into electrical energy, leading to enhanced performances on lowly coupled and/or highly damped PEH.

β’ Its performances are not dependent on the voltage across the storage element (π23 in Figure 1).

The SECE strategy can be described as follow: during a semi period, the energy is accumulated in the piezoelectric material. When the voltage reaches its maximum value, an inductance

πΏ is connected to the piezoelectric material. The electrical energy stored on the piezoelectric element is then transferred to the inductance. Once the voltage on the piezoelectric element reaches zero, the inductance is disconnected from the piezoelectric element and the energy flows to the storage element

πΆ678. The typical circuit realizing the SECE strategy and its associated waveforms are shown in Figure 1 and Figure 2.

Rectifier

Inductive Switch

Energy storage

Piezo !"

%&'(

#

)*+

$

Max

detector

Figure 1-SECE interface circuit

1

Xm

Energy extraction event

vp

x

0.5

Piezoelectric voltage vp

-Xm

0 0

0.5

1

0 :/2 : 3:/2 2:

3

Figure 2-Typical voltage and displacement waveforms when using the SECE strategy.

In order to theoretically evaluate SECE performance, we will consider a PEH made of a piezoelectric material deposited on a mechanical linear oscillator, as shown in Figure 3.

Piezoelectric material

(

Cantilever beam of stiffness !"# and equivalent mass $

Support

Tip mass

% &'

Neutral axis

Electric interface

)

(

Ambient acceleration

Mechanical system

.

$ 1/!-#

$)

Piezoelectric material

1// *'

Electric interface

Figure 3-Piezoelectric energy harvester made of a cantilever beam, and its equivalent electrical model

This PEH can be seen as a single-degree of freedom mass + damper + stiffness + piezo system, that is classically modelled by the equivalent electrical model shown in figure 3 and whose constitutive equations are given by (1).

πΉ = ππΎ = ππ₯ + π·π₯ + πΎB3π₯ + πΌπ£>

π = πΌπ₯ β πΆ>π£>

(1)

Where πΉ, πΎ, π₯, π and π£> respectively stand for the vibration force, ambient acceleration, the displacement of the inertial mass, the current extracted in the interface circuit, and the voltage across the electrodes of the piezoelectric material. π, π·, πΎB3, πΆ> and πΌ stand for the dynamic mass of the harvester, the mechanical damping, the short-circuit stiffness, the piezoelectric material clamped capacitance, and the piezoelectric coefficient.

This model is accurate as long as the harvester remains linear and that the PEH is driven around one of its resonant frequencies. In order to simplify the calculations, we will consider in the following that π₯ is purely sinusoidal and of constant amplitude, as expressed by (2). This physically means that the harvester is considered in steady state operation and that any potential mechanical nonlinearity is neglected.

π₯ π‘ = π/ πππ (π) = π/πππ (ππ‘)

(2)

As explained extensively in [17], the studies of the electrical interfaces for piezoelectric harvesters can be unified and simplified thanks to normalized variables:

π(

πΌ(

π/( = 1 β π( = πΎ πΆ

B3 >

πΎB3 π

π/ = π·

(3)

πΎB3 πT = π

πΊ/ = π/πT

π63 = πT 1 + π/(

π( β [0,1] is the square coupling coefficient, while π/( β βS is defined as the square modified electromechanical coupling. π/, πT and πΊ/ are the quality factor of the mechanical resonator, its short circuit angular resonant frequency, and the normalized vibration frequency respectively. The two normalized variables π/( and π/are sufficient to characterize and predict the performances of a linear PEH, and their product π/( π/ is commonly used to evaluate the potential of a piezoelectric harvester [8,17]. That is why it has been defined as a figure of merit in previous works

[17]. π63 is the open circuit angular resonant frequency of the harvester. If the PEH is highly coupled, this frequency is slightly different than the short circuit angular resonant frequency, πT.

From (1) and (2) and considering the diode threshold voltages negligible compared to the

piezoelectric voltage, it can be proven that (for a SECE interface) the displacement amplitude π/ can be given by the following expression when the vibration frequency matches the resonance frequency

of the PEH:

πΉ/

π/ =

4πΌ(

(4)

π·π + ππΆ>

Where πΉ/ stands for the amplitude of the vibration force πΉ. The harvested power thanks to the SECE interface is given by (5).

2ππΌ(

πΉ/(

16π/( π/

π/WX

π= ππΆ>

4πΌ( ( β π

4π( π (

(5)

π·π + ππΆ>

1+

//

π

Where π/WX is the maximum power which can be extracted from a linear PEH under a constant vibration of amplitude πΉ/. The expression of this maximum power is given by (6).

πΉ/( ππΎ/( π/WX = 8π· = 8πT π/ (6)

Where πΎ/ is the amplitude of the driving acceleration. Figure 4 represents the evolution of the harvested power at resonance with both the SECE and the standard interface as a function of the product π/( π/ . We can clearly see that for π/( π/ = π/4, the harvested power is maximized. When the product π/( π/ gets too important, the SECE overdamps the mechanical system, leading to a reduction of the vibration amplitude and lower performances than what can be obtained with a simple rectifier. This can also be seen in Figure 5, which shows the displacement amplitude at the resonant

frequency as a function of π/( π/. When the PEH is too highly coupled and/or too lightly damped, the displacement amplitude goes below its optimal value, leading to a lower extracted energy [7,8].

1

Area where the standard

0.8

SECE is more efficient

than the standard energy

harvesting (SEH) circuit.

0.6

SECE Standard interface (SEH)

Overdamping

Normalized power P/Pmax

0.4

0.2

k2 Q =:/4

mm

0

10-2

10-1

100

101

102

k2 Q

mm

Figure 4- Normalized harvested power using SECE (black) and standard interface (blue) as a

function of π/( π/.

1

Area where the standard 0.8 SECE is more efficient

than the standard energy harvesting (SEH) circuit. 0.6

SECE Standard interface (SEH)

Displacement amplitude Xm/(Xm )sc

0.4

Optimized displacement amplitude

Overdamping

0.2

k2 Q =:/4

mm

0

10-2

10-1

100

101

102

k2 Q

mm

Figure 5-Normalized displacement amplitude using SECE (black) and standard interface (blue) at

resonance as a function of π/( π/.

(π/)B3 given by πΉ//(π·πT) is the short circuit amplitude displacement of the PEH at resonance. This is the maximum value that can take the displacement, as there is no damping induced by the electrical interface when the piezoelectric material electrodes are short circuited.

Because of this overdamping phenomenon, we should find ways to reduce the damping induced by the SECE interface in order to be able to use it on high coupling interfaces. This could lead to an interface:

β’ Which is effective on any PEH, whether it is highly coupled/damped or not. β’ Whose effectiveness is not related to the voltage across the storage element.

The SECE strategy increasing the forces induced by the electrical interface on the mechanical system, we could use this characteristic to electrically change the resonant frequency of the electromechanical system, and enlarge the harvesting energy bandwidth.

3. Proposed concept: N-SECE In the standard SECE, there are two voltage extrema every vibrationβs period, thus the SECE

harvesting frequency is two times higher than the vibration frequency (π^W_`abcb = 2π`de). In the proposed N-SECE, instead of harvesting the energy at a frequency fixed by the vibrations, similarly as the standard SECE strategy, we propose to harvest the energy at a different frequency, controlled by the parameter π, where π=π^W_`/π^W_`fghg = π^W_`/(2π`de) as depicted in Figure 6.

Vibration frequency = ,034 Rectifier

Inductive Switch

Energy storage

Piezo !"

%&'(

#

)*+

$

Oscillator , = 2, 5

5

-./0

034

Figure 6-Proposed N-SECE system based on a tunable parameter π

By doing so, we obtain the piezoelectric voltage waveforms shown in Figure 7 when the harvesting frequency π^W_` is higher than 2π`de (π > 1). We also obtain the voltage waveforms shown in Figure 8 when π^W_` is lower than 2π`de (π < 1). Obviously, when π = 1, π^W_` = π^W_`abcb, and the N-SECE interface behaves like a standard SECE interface. Increasing or decreasing π leads in any case to a reduction of the damping induced by the electrical interface, and hence to improved performance, as demonstrated in the next sections.

Figure 7-Piezoelectric voltage waveforms for various π when π > 1

Piezoelectric voltage vp

N = 1

N = 1/3

N = 1/5

N = 1/7

Time

Figure 8-Piezoelectric voltage waveforms for various π when π < 1 4. Theoretical analysis: Multiple SECE (N>1 case)

In this section, the case where the harvesting frequency is higher than the vibration frequency (π > 1) is considered. In this case, π β ββ and π β [1, π] β© ββ respectively represent the number of harvesting events in a vibration semi-period, and an index of the harvesting event in the semi-period. First, the PEH works in open circuit condition: the voltage across the piezoelectric material electrodes starts to increase as charges accumulate in the dielectric capacitance. During every harvesting event, this piezoelectric voltage drops almost instantly to zero as the electrical charges are extracted from the material. Then the voltage starts to grow again, until the next harvesting event. Consequently, the piezoelectric voltage during a semi-period can be divided in π subset voltages, each related to a particular harvesting event. The expression of the π8^ subset voltage is given by (7)

Vibration semi-period

Piezoelectric voltage vp

vp

vp

vp

1

2

3

vp 1

vp

vp

2

3

vp=vp +vp +vp 123

Energy extraction events

0

:

2:

3

Figure 9-Theoretical voltage waveforms obtained for π = 3, revealing 3 different subset voltages

πΌr

π β 1 ππ

π₯ π ππ , π£ ΞΈ = πΆ> stn v

βπ β ]

π, [

π

π

(7)

>o

u

πβ1

ππ

0,

βπ β [0,

π] βͺ [ , π]

π

π

Consequently, the piezoelectric voltage expression during a semi-period is the sum of π periodic subset voltages, and can be expressed as a Fourier series since it is a periodical signal.

u

uβ

π£> π = π£>s(π) =

[π7o cos π’π + π7osin (π’π)]

(8)

syn

syn 7yn

Figure 7 shows the piezoelectric voltage waveforms simulated for various π while Figure 9 illustrates the case π = 3. In order to linearize the piezoelectric voltage expression given by (8), we find the first harmonic coefficients of the Fourier series associated with π£>. These coefficients are given by (9).

u

π/πΌ

2ππ

2 π β 1 π 2π

π β 1 π ππ

πn = 2πΆ π

sin

+ sin

π

π

+ β 2cos(

)sin( )

π

π

π

>

(9)

syn

u

π/πΌ

2ππ

2 πβ1 π

πβ1 π

ππ

πβ1 π

πn = 2πΆ π

βcos

+ cos

π

π

+ 4cos(

)(cos( ) β cos(

))

π

π

π

> syn

Considering the filtering effect of the mechanical resonator, it is reasonable to consider that

only the first voltage harmonic π£>n will have an influence on the dynamics of the system. Its expression in the Fourier domain π£>n is given by (10).

πn

πn

(10)

π£>n π = (π/ β π π/)π₯(π)

Where π₯ stands for the mass displacement written in the Fourier domain. Substituting (10) in

(1) expressed in the Fourier domain leads to the expression of the displacement magnitude π/, given by (11).

πΉ

π/ = π΄( + π΅(

π΄ = πΎ β ππ( + πΌ πn

(11)

B3

π/

π΅ = ππ· β πΌ πn π/

From the displacement amplitude obtained in (11), we can eventually determine the harvested

power, given by (12).

πΌ(

u

π

ππ

πβ1 π (

(12)

π = π/(

cos

β cos

πΆ> 2π

π

π

syn

Thus, from (9), (11) and (12), the harvested power can be determined for any value of π as a

function of the PEHβs parameters and of the ambient acceleration frequency and magnitude.

5. Theoretical analysis: Regenerative SECE (N<1 case)

When the harvesting frequency π^W_` is lower than the vibration frequency π`de, the voltage waveforms is different, leading to a completely different analytical formulation, fully detailed in [8]. In this case, π can be defined as the inverse of π β ββ, where π represents the number of semi-period of vibrations between each harvesting events. The piezoelectric voltage is then expressed by (13).

1 r/u πΌπ₯ π ππ ,

πΆ> T

βπ β ]0, π/π[

π£> π =

0, ππ π = π/π

(13)

1 r/u

πΌπ₯ π ππ , βπ β ]π/π, 2π/π[

πΆ> β

0, ππ π = 2π/π

This piezoelectric voltage has been drawn numerically for various values of π, as depicted in Figure 8. The new voltage expression leads to another Fourier expression. The expressions of the first Fourier coefficients are given by (14).

πΌπ/

πn = πΆ

>

πΌπ/π

n/u

(14)

πn = 2 πΆ π β1 β 1

>

Thus, in the regenerative case (N<1), as extensively explained in [8], the displacement amplitude (15) as well as the harvested power expressions (16) can be derived. Note that when 1/π is even, the additional damping due to energy harvesting is null, and thus the harvesting power is zero. We will only consider odd values of 1/π in the following.