SECE, SSHI and MPPT

Intent

Compare what a full-wave bridge, SECE and parallel SSHI take from the same vibrating piezoelectric element.
Know which of them depend on the DC voltage behind them, and where each gives the most.
Track the maximum power point of a source by a fraction of its open-circuit voltage.

Theory

All three interfaces below take their power from the element of source models: a current source $I_p \sin \omega t$ beside its clamped capacitance $C_0$ , vibrating at an amplitude that the harvest does not change. Each half period, between two extrema of the displacement, the source drives the same charge. The two interfaces whose bridge feeds the DC voltage differ in how much of it they spend swinging the element’s voltage before the bridge conducts; SECE lets all of it charge the element and then takes the energy stored there.

The standard interface

A full-wave bridge onto a DC voltage, a storage capacitor or a battery, conducts only once the element’s voltage has swung from one rail to the other. That swing takes part of the charge each half period, and the rest enters the DC side [Guyomar et al. 2005]:

P = \frac{2 V_\mathrm{DC} \left(I_p - \omega C_0 V_\mathrm{DC}\right)}{\pi}

$P$: power
$C_0$: clamped capacitance
$I_p$: piezoelectric current amplitude
$V_\mathrm{DC}$: rectified DC voltage
$\omega$: vibration angular frequency

Power through a full-wave bridge onto a DC voltage (the standard interface)✓ verified by 7 tests

Assumptions, convention and source

Assumptions

piezoelectric element as a current source I_p sin(ωt) in parallel with its clamped capacitance C_0; the vibration's amplitude does not depend on the electrical load (weak coupling)
ideal full-wave bridge rectifier onto a constant DC voltage

Notes

each half period the element's voltage swings from

-V_\mathrm{DC}

V_\mathrm{DC}

before the bridge conducts, which takes

2 C_0 V_\mathrm{DC}

of the

2 I_p/\omega

the source drives; from

V_\mathrm{DC} = V_p

on the bridge never conducts

Source

Guyomar et al. 2005, the standard technique; derived here; Lefeuvre et al. 2006

Derivation

step-by-step sympy derivation

Equation id

piezo.P_std · equations.yaml entry

The power depends on the DC voltage. It is zero at zero volts and at the open-circuit amplitude, where the swing takes all of the charge, and it is largest at half the open-circuit amplitude:

P_\mathrm{std,max} = \frac{I_p^{2}}{2 \pi \omega C_0}

$P_\mathrm{std,max}$: largest power of the standard interface
$C_0$: clamped capacitance
$I_p$: piezoelectric current amplitude
$\omega$: vibration angular frequency

Largest power of the standard interface✓ verified by 7 tests

Assumptions, convention and source

Assumptions

piezoelectric element as a current source I_p sin(ωt) in parallel with its clamped capacitance C_0; the vibration's amplitude does not depend on the electrical load (weak coupling)
ideal full-wave bridge rectifier onto a constant DC voltage

Notes

V_\mathrm{DC} = V_p/2

, half the open-circuit amplitude

Source

Guyomar et al. 2005, the standard technique; derived here; Lefeuvre et al. 2006

Derivation

step-by-step sympy derivation

Equation id

piezo.P_std_max · equations.yaml entry

SECE

Synchronous electric charge extraction leaves the element open between the extrema of its voltage, then takes its whole charge at each extremum through a converter into the storage [Lefeuvre et al. 2005]. In between, the whole charge of the half period raises the element’s voltage, from zero to twice its open-circuit amplitude, and the converter takes the energy then stored in $C_0$ and delivers it at whatever voltage the storage has. The element never connects to the storage, so the power does not depend on the storage’s voltage:

P = \frac{2 I_p^{2}}{\pi \omega C_0}

$P$: power
$C_0$: clamped capacitance
$I_p$: piezoelectric current amplitude
$\omega$: vibration angular frequency

Power of synchronous electric charge extraction (SECE)✓ verified by 7 tests

Assumptions, convention and source

Assumptions

piezoelectric element as a current source I_p sin(ωt) in parallel with its clamped capacitance C_0; the vibration's amplitude does not depend on the electrical load (weak coupling)
ideal (lossless) switches and passive components

Notes

the element is left open and its whole charge is taken at each extremum of its voltage,

\tfrac12 C_0 (2 V_p)^2

twice a period whatever the load: four times the standard interface's largest power

Source

Lefeuvre et al. 2005, derived here; Lefeuvre et al. 2006

Derivation

step-by-step sympy derivation

Equation id

piezo.P_sece · equations.yaml entry

That is four times the standard interface’s best, or 300% more. Lefeuvre et al. report that, over a standard interface adapted for the most power, the technique “increases the harvested power by 400%” [Lefeuvre et al. 2005]; their percentage is quoted as they state it. The converter must take the element’s voltage at extraction, twice its open-circuit amplitude.

Parallel SSHI

Synchronized switch harvesting on inductor, in its parallel form, keeps the bridge and adds a switch and an inductor across the element. At each extremum of the displacement the switch closes briefly, and the inductor inverts the element’s voltage [Guyomar et al. 2005]. The inversion loses some of it: with the element going from the DC voltage to a fraction $\gamma$ of it of the opposite sign, the swing to the other rail costs only the rest:

P = \frac{2 V_\mathrm{DC} \left(I_p - \frac{\omega C_0 V_\mathrm{DC} \left(1 - \gamma\right)}{2}\right)}{\pi}

$P$: power
$C_0$: clamped capacitance
$I_p$: piezoelectric current amplitude
$V_\mathrm{DC}$: rectified DC voltage
$\gamma$: inversion coefficient
$\omega$: vibration angular frequency

Power of parallel synchronized switch harvesting on inductor (SSHI)✓ verified by 7 tests

Assumptions, convention and source

Assumptions

piezoelectric element as a current source I_p sin(ωt) in parallel with its clamped capacitance C_0; the vibration's amplitude does not depend on the electrical load (weak coupling)
ideal full-wave bridge rectifier onto a constant DC voltage

Notes

at each extremum of the displacement a switch and an inductor invert the element's voltage from

V_\mathrm{DC}

-\gamma V_\mathrm{DC}

, so the swing to the other rail takes only

(1 - \gamma) C_0 V_\mathrm{DC}

Source

Guyomar et al. 2005, derived here; Lefeuvre et al. 2006

Derivation

step-by-step sympy derivation

Equation id

piezo.P_sshi · equations.yaml entry

P_\mathrm{SSHI,max} = \frac{I_p^{2}}{\pi \omega C_0 \left(1 - \gamma\right)}

$P_\mathrm{SSHI,max}$: largest power of parallel SSHI
$C_0$: clamped capacitance
$I_p$: piezoelectric current amplitude
$\gamma$: inversion coefficient
$\omega$: vibration angular frequency

Largest power of parallel SSHI✓ verified by 7 tests

Assumptions, convention and source

Assumptions

piezoelectric element as a current source I_p sin(ωt) in parallel with its clamped capacitance C_0; the vibration's amplitude does not depend on the electrical load (weak coupling)
ideal full-wave bridge rectifier onto a constant DC voltage

Notes

V_\mathrm{DC} = V_p/(1 - \gamma)

2/(1 - \gamma)

times the standard interface's largest power

Source

Guyomar et al. 2005, derived here; Lefeuvre et al. 2006

Derivation

step-by-step sympy derivation

Equation id

piezo.P_sshi_max · equations.yaml entry

The better the inversion, the more power, at a higher DC voltage. Guyomar et al. report that compared with the standard technique the harvested power “may be increased above 900%” [Guyomar et al. 2005]; in this model that is an inversion that keeps about four fifths of the voltage. Even an inversion that keeps nothing, a switch that only empties the element, doubles the standard interface’s best.

Tracking the maximum power point

For a linear source, the power into a constant voltage is largest at half the open-circuit voltage (constant-voltage extraction), and the standard interface above is also best at half the element’s open-circuit amplitude. A converter that holds its input at a fixed fraction of the source’s open-circuit voltage therefore stays at the maximum power point as the source’s amplitude changes: the fractional open-circuit voltage method [Esram & Chapman 2007]. To learn the open-circuit voltage, such a converter stops drawing power now and then and samples it: one harvesting charger periodically disables itself and samples a fraction of the open-circuit voltage, typically 50 % for thermoelectric generators [Texas Instruments, bq25570 Nano Power Boost Charger and Buck Converter for Energy Harvester Powered Applications]. A source that is not linear, such as a solar cell, has its best point at another fraction [Esram & Chapman 2007].

Worked example

One element behind the three interfaces:

Piezoelectric interfaces example

Symbol	Value	How it is obtained
$I_p$ piezoelectric current amplitude	100 µA	given
$\omega$ vibration angular frequency	1 krad/s	given
$C_0$ clamped capacitance	50 nF	given
$V_\mathrm{DC}$ rectified DC voltage	1.5 V	given
$\gamma$ inversion coefficient	0.8	given
$V_p$ open-circuit voltage amplitude	2 V	Open-circuit voltage amplitude of a vibrating piezoelectric element
${P}_{\mathrm{std}}$ power	23.9 µW	Power through a full-wave bridge onto a DC voltage (the standard interface)
$P_\mathrm{std,max}$ largest power of the standard interface	31.8 µW	Largest power of the standard interface
${P}_{\mathrm{sece}}$ power	127 µW	Power of synchronous electric charge extraction (SECE)
$P_\mathrm{SSHI,max}$ largest power of parallel SSHI	318 µW	Largest power of parallel SSHI

Operating mode: the bridge conducts ( $V_\mathrm{DC} < V_p$ : holds)

Behind a bridge at 1.5 V, the element gives 23.9 µW. At the best DC voltage, half its open-circuit amplitude of 2 V, it would give 31.8 µW. SECE takes 127 µW at any storage voltage, and parallel SSHI with an inversion of 0.8 gives up to 318 µW.

Try it

▶ Open in the equation explorer: Power through a full-wave bridge onto a DC voltage (the standard interface)
Preset from the example piezo-interfaces

Sweep the DC voltage behind the bridge: the power rises to its peak at half the open-circuit amplitude and falls to zero at the amplitude.

▶ Open in the equation explorer: Largest power of parallel SSHI
Preset from the example piezo-interfaces

Sweep the inversion coefficient: the best power grows without bound as the inversion approaches a perfect one, which the inductor’s and switch’s losses never allow.

Bench exercise

On a shaker at a steady amplitude, connect the element through a full-wave bridge to a storage capacitor held at the DC voltage by a source-measure unit, which sinks the bridge’s current, microamperes, and measures it. Step the DC voltage from zero to the open-circuit amplitude, recording the current the unit sinks, and plot the power; compare the best DC voltage with half the open-circuit amplitude. For parallel SSHI, add the switch and inductor, trigger the switch at each voltage extremum, and read the element’s voltage just before and just after the inversion on the oscilloscope: their ratio is the inversion coefficient to put in the formula.

Gotchas

Above the open-circuit amplitude a bridge harvests nothing. A storage charged beyond it stops charging (power of the standard interface).
The equations take an ideal bridge. In a real one two diodes conduct in series [Mohan et al. 2003, Ch. 5 (Line-Frequency Diode Rectifiers)], so the element must swing to the DC voltage plus two forward drops before the bridge conducts: for an element whose open-circuit amplitude is a few volts, a large part of the swing.
SECE’s converter sees twice the open-circuit amplitude, since the element’s voltage climbs from zero across a whole half period (power of SECE).
SSHI’s gain is the inversion’s quality. Losses in the switch and the inductor lower the coefficient, and the power with it (largest power of parallel SSHI).
Sampling the open-circuit voltage costs harvest time: the converter takes no power while it samples.
These results keep the vibration’s amplitude fixed. A harvest that takes energy not small against what the structure’s own losses take lowers the amplitude, and the model then overstates every interface.

Go deeper

Paper Piezoelectric Energy Harvesting Device Optimization by Synchronous Electric Charge Extraction — E. Lefeuvre, A. Badel, C. Richard, D. Guyomar · retrieved 2026-09-24
The paper that introduced synchronous electric charge extraction, whose harvested power does not depend on the load.
Paper Toward energy harvesting using active materials and conversion improvement by nonlinear processing — D. Guyomar, A. Badel, E. Lefeuvre, C. Richard · retrieved 2026-09-24
The paper that introduced synchronized switch harvesting on inductor; the PubMed record carries its abstract.
Paper Comparison of Photovoltaic Array Maximum Power Point Tracking Techniques — T. Esram, P. L. Chapman · retrieved 2026-09-24
A survey of maximum power point tracking methods, the fractional open-circuit voltage method among them.
Data sheet bq25570 Nano Power Boost Charger and Buck Converter for Energy Harvester Powered Applications — Texas Instruments · retrieved 2026-09-24
A harvesting charger that tracks the maximum power point by sampling a fraction of the source's open-circuit voltage.

Quiz

A piezoelectric element has a 2 V open-circuit amplitude. At what DC voltage does a full-wave bridge take the most power from it?
1. 2 V
2. 4 V
3. 1 V
4. 0.5 V
Answer and explanation
Correct answer: C The standard interface is best at half the open-circuit amplitude.
The storage behind that bridge has charged to 2 V. What power does the bridge deliver now?
1. None: swinging between the rails takes the whole charge
2. Its largest power
3. Half its largest power
4. Twice its largest power
Answer and explanation
Correct answer: A At the open-circuit amplitude the element's voltage only just reaches the other rail each half period, and no charge is left for the bridge.
How does SECE's power compare with the standard interface's largest?
1. The same
2. Twice
3. Half
4. Four times
Answer and explanation
Correct answer: D SECE takes the element's whole energy at each extremum: two times I_p squared over pi omega C_0, four times the standard interface's best.
The storage behind a SECE interface charges from 2 V to 4 V. What happens to the harvested power?
1. It stays the same
2. It halves
3. It doubles
4. It falls to zero
Answer and explanation
Correct answer: A The element never connects to the storage: the converter takes the energy the element holds at each extremum and delivers it at whatever voltage the storage has, so the power does not depend on that voltage.
A parallel SSHI inversion keeps half the voltage. How does its largest power compare with the standard interface's largest?
1. Twice
2. Four times
3. Ten times
4. The same
Answer and explanation
Correct answer: B Two over one minus the inversion coefficient: 2 over 0.5.
At what DC voltage is that SSHI interface best, for the 2 V element?
1. 1 V
2. 2 V
3. 8 V
4. 4 V
Answer and explanation
Correct answer: D At the open-circuit amplitude over one minus the inversion coefficient: 2 V over 0.5.
A converter tracks a linear source's maximum power point by holding its input at a fixed fraction of the open-circuit voltage. Which fraction?
1. 80 %
2. 50 %
3. 100 %
4. 25 %
Answer and explanation
Correct answer: B A linear source gives the most into a constant voltage at half its open-circuit voltage.
What voltage must a SECE converter take from a 2 V element?
1. 2 V
2. 1 V
3. 4 V
4. 8 V
Answer and explanation
Correct answer: C Left open from zero, the element's voltage climbs to twice its open-circuit amplitude before each extraction.

The numbers in the examples, presets and exercises on this site are synthetic: round teaching values, not data from any real design.