Source models

Intent

Model a harvester that is linear over its working range as an open-circuit voltage behind a source resistance, and know the most power it can give.
Model a vibrating piezoelectric element as a current source beside its clamped capacitance, and find the resistor that takes the most power from it.
Know what a triboelectric generator’s figure of merit compares.

Theory

A linear source

A source that is linear over the range it works in looks, to its load, like an open-circuit voltage $V_\mathrm{oc}$ behind a source resistance $R_s$ : its Thevenin equivalent [Alexander & Sadiku 2017, Ch. 4]. It delivers the most power into a load equal to its own resistance [Alexander & Sadiku 2017, Sec. 4.8 (Maximum Power Transfer)]:

P_\mathrm{max} = \frac{V_\mathrm{oc}^{2}}{4 R_s}

$P_\mathrm{max}$: largest power available
$R_s$: source resistance
$V_\mathrm{oc}$: source open-circuit voltage

Maximum power from a linear source✓ verified by 7 tests

Assumptions, convention and source

Assumptions

linear (Thevenin) source V_oc behind R_s

Notes

delivered into a matched load

R_L = R_s

Source

Alexander & Sadiku 2017, Sec. 4.8 (Maximum Power Transfer)

Derivation

step-by-step sympy derivation

Equation id

src.Pmax · equations.yaml entry

That is the power available from the source. A converter behind it takes a share of it that depends on what its input looks like; the matching module works that share out.

A vibrating piezoelectric element

A piezoelectric element on a structure that vibrates sinusoidally drives a current proportional to the structure’s velocity. With its clamped capacitance $C_0$ across its terminals, it is a current source $I_p \sin \omega t$ in parallel with $C_0$ [Guyomar et al. 2005], as long as what the electrical side takes does not change the vibration’s amplitude. With nothing connected, the whole current charges and discharges $C_0$ , and the voltage swings each way by

V_p = \frac{I_p}{\omega C_0}

$V_p$: open-circuit voltage amplitude
$C_0$: clamped capacitance
$I_p$: piezoelectric current amplitude
$\omega$: vibration angular frequency

Open-circuit voltage amplitude of a vibrating piezoelectric element✓ 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)

Notes

with nothing connected, the whole current charges and discharges

C_0

Source

Guyomar et al. 2005, the element's electromechanical model; derived here

Derivation

step-by-step sympy derivation

Equation id

piezo.Vp · equations.yaml entry

Its internal impedance is a capacitance, not a resistance. A resistor across the element shares the current with $C_0$ , and takes the most power when it equals the capacitance’s reactance at the vibration’s frequency:

R = \frac{1}{\omega C_0}

$R$: load resistance
$C_0$: clamped capacitance
$\omega$: vibration angular frequency

Resistive load that takes the most power from a vibrating piezoelectric element✓ 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)

Notes

the reactance of the clamped capacitance at the vibration's frequency

Source

Guyomar et al. 2005, the element's electromechanical model; derived here

Derivation

step-by-step sympy derivation

Equation id

piezo.R_opt · equations.yaml entry

P = \frac{I_p^{2}}{4 \omega C_0}

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

Power into the best resistive load✓ 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)

Notes

a resistor

R

across the element takes

I_p^2 R/(2(1 + (\omega R C_0)^2))

, largest at

R = 1/(\omega C_0)

Source

Guyomar et al. 2005, the element's electromechanical model; derived here

Derivation

step-by-step sympy derivation

Equation id

piezo.P_R · equations.yaml entry

The best resistor depends on the frequency: a resistor sized for one vibration is mismatched at another. The energy the load takes comes out of the vibration, so the model holds while the energy the harvest takes each cycle is small against what the structure’s own mechanical losses take, or where the motion is imposed. Otherwise the harvest lowers the vibration’s amplitude, and near a sharp resonance the electrical load can also move the resonance itself; the model leaves both out and overstates the power.

A storage element needs a DC voltage, so the element usually works through a rectifier. A full-wave bridge onto the best DC voltage gives less than the best resistor (largest power of the standard interface); SECE and SSHI, two nonlinear interfaces (SECE, SSHI and MPPT), give more.

A triboelectric generator

Zi et al. proposed to compare triboelectric nanogenerators by the largest energy a cycle of their motion can deliver, their cycles for maximized energy output, per area and per displacement. Their performance figure of merit is a structural figure of merit, set by the design, times the square of the surface charge density, set by the materials [Zi et al. 2015]. It rates the generator by energy per cycle, not by the power a particular load happens to take.

Worked example

A piezoelectric element, open and then across its best resistor:

Piezoelectric source 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
$V_p$ open-circuit voltage amplitude	2 V	Open-circuit voltage amplitude of a vibrating piezoelectric element
$R$ load resistance	20 kΩ	Resistive load that takes the most power from a vibrating piezoelectric element
${P}_{\mathrm{R}}$ power	50 µW	Power into the best resistive load

The element drives 100 µA at 1 krad/s beside 50 nF. Open, its voltage swings 2 V each way. The best resistor is 20 kΩ, and it takes 50 µW.

Try it

▶ Open in the equation explorer: Resistive load that takes the most power from a vibrating piezoelectric element
Preset from the example piezo-source

Sweep the vibration’s frequency: the best resistor falls as the frequency rises, so a fixed resistor matches only one vibration.

▶ Open in the equation explorer: Maximum power from a linear source
Preset from the example match-basic

Sweep the source resistance of a linear source: halving it doubles the available power.

Bench exercise

Mount the element on a shaker driven at a steady sinusoidal amplitude. Measure its open-circuit voltage amplitude with a probe whose input resistance is far above the best resistor, and note the probe’s input capacitance, which adds to $C_0$ (probes). Then load the element with a decade resistance box, record the voltage amplitude at each value and compute the power each resistor takes. Find the resistor that takes the most, and compare it with the reactance of the element’s capacitance at the vibration’s frequency, measured with an LCR meter at a frequency away from the structure’s resonance: near it the motion adds to the measured capacitance, and away from it, for a weakly coupled element, adds little. Repeat at a second frequency and see the best resistor move.

Gotchas

A probe loads a high-impedance source. Its input resistance and capacitance sit in parallel with the source; compare them with the best resistor (best resistor) and with $C_0$ before trusting a reading.
The best resistor holds at one frequency. It is the reactance of $C_0$ at the vibration’s frequency, and moves with it.
A bridge onto a DC voltage takes less than the best resistor (largest power of the standard interface).
The model does not damp the vibration. A load that takes energy not small against what the structure’s own losses take lowers the vibration’s amplitude, and the model then overstates the power.

Go deeper

Paper Standards and figure-of-merits for quantifying the performance of triboelectric nanogenerators — Y. Zi, S. Niu, J. Wang, Z. Wen, W. Tang, Z. L. Wang · retrieved 2026-09-24
Open access: the cycles for maximized energy output, and the figure of merit that compares triboelectric nanogenerators by the energy a cycle delivers per area and per displacement.
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.

Quiz

A linear source has a 20 V open-circuit voltage behind 50 ohm. What is the most power it can give?
1. 8 W
2. 2 W
3. 4 W
4. 0.5 W
Answer and explanation
Correct answer: B The square of the open-circuit voltage over four times the source resistance: 400 over 200 is 2 W, into a 50 ohm load.
Into which load does that source give its most power?
1. A short circuit, which draws the most current
2. An open circuit, which has the most voltage
3. A resistance equal to the source resistance, 50 ohm
4. A resistance twice the source resistance
Answer and explanation
Correct answer: C At the matched load half the open-circuit voltage falls across the load and half across the source resistance; a short or an open takes no power at all.
A piezoelectric element drives a 50 uA current amplitude at 2000 rad/s beside a 25 nF clamped capacitance. What is its open-circuit voltage amplitude?
1. 1 V
2. 0.5 V
3. 2 V
4. 4 V
Answer and explanation
Correct answer: A With nothing connected, the whole current charges and discharges the capacitance: 50e-6 over (2000 times 25e-9), which is 1 V.
What power does the best resistor take from that element?
1. 25 uW
2. 50 uW
3. 6.25 uW
4. 12.5 uW
Answer and explanation
Correct answer: D The best resistor is the capacitance's reactance, 1/(2000 times 25e-9) = 20 kohm, and it takes the current amplitude squared over four times omega C_0: 2.5e-9 over 2e-4, which is 12.5 uW.
The vibration's frequency doubles and the current amplitude stays the same. What happens to the best resistor?
1. It doubles
2. It halves
3. It stays the same
4. It falls to a quarter
Answer and explanation
Correct answer: B It is the reactance of the clamped capacitance, one over omega C_0, which halves when omega doubles.
Why can an oscilloscope probe change the reading of a high-impedance source?
1. Its input resistance and capacitance sit in parallel with the source and take part of its current
2. It adds a DC offset to every reading
3. It changes the vibration's frequency
4. It cannot: a probe draws no current
Answer and explanation
Correct answer: A The probe is a load. Against a source whose best resistor is large, a probe's input resistance and capacitance are not negligible.
What does the performance figure of merit of Zi et al. multiply to rate a triboelectric nanogenerator?
1. The open-circuit voltage times the short-circuit current
2. The power into a fixed resistor times the area
3. The capacitance times the frequency
4. A structural figure of merit set by the design, times the square of the surface charge density
Answer and explanation
Correct answer: D It rates the energy a cycle can deliver, per area and per displacement: a design part and a material part, the square of the surface charge density.

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