Mathematical Models for Multi-Source Energy Harvesting

An energy harvester rarely sits in a world with only one source of energy. A sensor on a factory wall feels vibration, a temperature difference, stray light and radio waves at the same time. Each of these inputs is weak on its own, often in the microwatt range, and each changes with time and place. The engineering question is simple to state and hard to answer: how much electrical power can a given piece of material deliver when all of these inputs act on it together? The difficulty lies in the models. Each conversion mechanism has its own physics, its own variables and its own efficiency limits, and the textbook equations treat them one at a time. A designer who adds them up risks counting energy twice or missing the losses that appear when inputs interact inside the same material.

The Established Approaches

Energy harvesting has a set of well-tested models, one for each input channel. Each describes its own channel well, and each stops at the edge of that channel.

Vibration (inertial harvesters). Williams and Yates modelled a proof mass on a spring with electrical and mechanical damping [1]. Roundy, Wright and Rabaey gave the widely used expression for the power at resonance [2]:

P = m · ζ_e · A² / (4 · ω_n · ζ_T²)

where m is the proof mass, A the acceleration amplitude, ω_n the resonant angular frequency, ζ_e the electrical damping ratio and ζ_T the total damping ratio. The model is accurate near resonance and has guided piezoelectric, electromagnetic and electrostatic designs [3, 7]. Its limit is bandwidth: power falls steeply when the vibration frequency drifts away from ω_n, and output scales with the proof mass, which is small in miniature devices.

Thermoelectric conversion. A thermoelectric generator with Seebeck coefficient S and internal resistance R delivers its maximum power into a matched load:

P_max = (S · ΔT)² / (4 · R)

Its efficiency is bounded by the Carnot factor ΔT/T_h, reduced further by the material’s figure of merit ZT [4]. With the few kelvin available in most ambient settings, efficiencies stay well below one per cent.

Photovoltaic conversion. Output is P = η_PV · G · A, with irradiance G and cell area A. For a single junction under sunlight, the detailed-balance limit of Shockley and Queisser caps efficiency at about 33 per cent [5]. Indoors, irradiance drops by two to three orders of magnitude, and output drops with it.

Radio-frequency harvesting. Received power follows the Friis equation, P_r = P_t · G_t · G_r · (λ / 4πd)², and a rectifier converts it to direct current [6]. Rectifier efficiency falls sharply at the microwatt input levels typical of ambient radio fields, so distance to the transmitter dominates the result.

Hybrid harvesters combine two or more of these channels, but usually as separate transducers, each with its own model and its own power electronics, whose outputs are added at the end [8].

The Open Question

The established models describe single channels well. They do not describe several weak ambient inputs coupling into one material at the same time. In a layered nanomaterial, heat, electromagnetic fields, mechanical motion and particle interactions act on the same charge carriers in the same volume. Their contributions are not independent: a temperature change alters how the material responds to an electromagnetic field, and a mechanical strain alters its electronic structure. Adding the single-channel formulas cannot capture this, and it gives no single place to state the energy balance. What is missing is one accounting framework that integrates all coupled inputs over the active volume, applies one overall efficiency, and makes conservation of energy an explicit condition.

A Generalized Approach

A generalized approach to this problem is represented by the Schubart Master Formula, which describes the usable electrical output of a material exposed to several ambient inputs at once:

P(t) = η · ∫_V Φ_eff(r, t) · σ_eff(E) dV

The derivation, notation and open questions are published at pt-schubart.com.

Definitions

• P(t): instantaneous electrical output, the usable electrical power delivered at time t. Unit: watt (W). Energy over an interval is the time integral of P(t), in joules or watt-hours.
• η: total conversion efficiency, dimensionless, between 0 and 1. It combines coupling, resonance, rectification and electrical losses.
• Φ_eff(r, t): effective ambient flux density at position r and time t, the local strength of all ambient inputs acting on the material.
• σ_eff(E): energy-dependent effective interaction cross-section, how strongly the material couples to an input of energy E.
• ∫_V: volume integral over the active material.

Together, Φ_eff · σ_eff is the coupled power per unit volume (W/m³), so the integral over V yields watts. How that product divides between the two factors depends on the input channel, so no separate units are assigned to them here.

Assumptions and Boundaries

The Schubart Master Formula is a proposed generalized framework, not an established equation of physics. Its assumptions and limits:

• The material is an open, non-equilibrium system that exchanges energy with its surroundings.
• Output is bounded by coupled input power: P_out ≤ ΣP_in. The framework creates no energy; it accounts for energy that is coupled in.
• The single-channel models above appear as special cases when only one input is present.
• Contributions of different inputs can interact. Each must be isolated experimentally before its share can be stated.
• η and σ_eff are properties of a specific material and design. The framework does not predict them from first principles; they have to be measured.
• If the input is uniform across the material, the integral reduces to a product: P = η · Φ_eff · σ_eff · V.
• The formula describes power. Storage, load matching and power management are separate design steps.

Worked Example

All values below are illustrative values, chosen to show the method. They are not measured results.

An active film of 10 cm × 10 cm with a thickness of 0.1 mm has a volume V = 1 × 10⁻⁶ m³. Assume three inputs act on it uniformly, with coupled power densities Φ_eff · σ_eff of:

• thermal fluctuations: 2,000 W/m³
• electromagnetic radiation: 500 W/m³
• mechanical vibration: 300 W/m³

The coupled input is ΣP_in = (2,000 + 500 + 300) W/m³ × 10⁻⁶ m³ = 2.8 mW. With a total conversion efficiency η = 0.2, the usable output is P = 0.2 × 2.8 mW = 0.56 mW. The boundary holds: 0.56 mW ≤ 2.8 mW. Over 24 hours, this corresponds to 0.56 mW × 24 h ≈ 13.4 mWh of electrical energy.

The same arithmetic shows where the design levers are: efficiency, input strength, coupling and active volume. Three belong to physics and materials; the fourth belongs to manufacturing.

Where It Leads

The coupling term σ_eff is decided by the material, which is why layered graphene and semiconductor structures are studied for this purpose. The practical side of the calculation is set out in How to Calculate Ambient Energy Harvesting, and the smallest application in Powering Autonomous Sensors Without Batteries. How the same accounting scales from sensors to buildings and grids is described in From Sensor to Grid.

References

1. C. B. Williams and R. B. Yates, “Analysis of a micro-electric generator for microsystems,” Sensors and Actuators A 52, 8–11 (1996).
2. S. Roundy, P. K. Wright and J. Rabaey, “A study of low level vibrations as a power source for wireless sensor nodes,” Computer Communications 26, 1131–1144 (2003).
3. A. Erturk and D. J. Inman, Piezoelectric Energy Harvesting, Wiley (2011).
4. G. J. Snyder and E. S. Toberer, “Complex thermoelectric materials,” Nature Materials 7, 105–114 (2008).
5. W. Shockley and H. J. Queisser, “Detailed balance limit of efficiency of p-n junction solar cells,” Journal of Applied Physics 32, 510–519 (1961).
6. H. J. Visser and R. J. M. Vullers, “RF energy harvesting and transport for wireless sensor network applications: principles and requirements,” Proceedings of the IEEE 101, 1410–1423 (2013).
7. P. D. Mitcheson, E. M. Yeatman, G. K. Rao, A. S. Holmes and T. C. Green, “Energy harvesting from human and machine motion for wireless electronic devices,” Proceedings of the IEEE 96, 1457–1486 (2008).
8. S. Priya and D. J. Inman (eds.), Energy Harvesting Technologies, Springer (2009).
9. P. M. Thibado, P. Kumar, S. Singh, M. Ruiz-Garcia, A. Lasanta and L. L. Bonilla, “Fluctuation-induced current from freestanding graphene,” Physical Review E 102, 042101 (2020).
10. Schubart Master Formula: derivation, notation and open questions, pt-schubart.com.

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