How to Calculate Ambient Energy Harvesting

Before a harvester is built, someone has to put a number on it. Will it keep a sensor running? How large must it be? How much storage is needed for the night, the weekend or the quiet hours of a machine? The answer depends on three things that are hard to pin down: how much ambient energy is present at the installation site, how much of it the device can couple in, and how much is lost on the way to the load. Ambient sources are weak and variable. Light switches off, temperature differences shrink, machines stop vibrating. A calculation that uses peak values will promise power that never arrives, and one that ignores conversion losses will overstate the result by a factor of ten or more. A reliable estimate needs a repeatable method.

The Established Approaches

The field has settled on a power-budget method built from per-channel models and measured site data.

Start from the load. The average power a device needs is set by its duty cycle: sleep current, active current and how often it wakes. Ultra-low-power designs bring averages into the tens of microwatts [1, 2].

Measure the site. Ambient inputs are characterised on location: vibration spectra on the actual machine [3], temperature differences across the actual surface, light levels at the actual position, radio field strength at the actual distance. Published surveys give typical ranges, but only site data gives a usable number [1, 4].

Apply the channel model. Each input has a standard formula:

photovoltaic: P = η_PV · G · A, with irradiance G and area A; indoor irradiance is two to three orders of magnitude below sunlight [5];
thermoelectric: P_max = (S · ΔT)² / (4 · R) at a matched load, with Seebeck coefficient S and internal resistance R [6];
vibration at resonance: P = m · ζ_e · A² / (4 · ω_n · ζ_T²), with proof mass m and acceleration amplitude A [3];
radio frequency: P_r = P_t · G_t · G_r · (λ / 4πd)², multiplied by the rectifier efficiency, which falls at low input power [7].

Close the energy balance over time. A system is energy-neutral when harvested energy covers consumed energy over a full cycle, with storage bridging the gaps [8]. The calculation therefore uses averages and minimum periods, not peaks.

This method works well when one source dominates. Its weakness is that each channel is estimated with a different model, and the results are simply added.

The Open Question

Adding per-channel estimates assumes the channels do not influence each other and are captured by separate transducers. That holds for a solar cell next to a thermoelectric module. It does not hold when one layered material absorbs several weak inputs at once, because the inputs share the same charge carriers and the same losses. In that case there is no clean way to split the budget, and the risk of counting energy twice is real. The engineer needs one expression that sums all coupled inputs over the material, applies a single efficiency and enforces conservation of energy, so the estimate can be checked against a measurement.

A Generalized Approach

A generalized approach to this problem is represented by the Schubart Master Formula, which turns the budget into one integral over the active material:

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

The derivation, notation and open questions are published at pt-schubart.com. The underlying model is described in Mathematical Models for Multi-Source Energy Harvesting.

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.
Coupled power densities and η must come from measurements on the specific material, not from assumptions.
Inputs vary in time. A budget uses the average over a full cycle and checks the weakest period separately.
Contributions of different inputs can interact; each must be isolated experimentally before its share can be stated.
Storage, power management and the load’s own electronics add losses that are outside the integral and must be budgeted separately.

Worked Example

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

Step 1: the load. A wireless sensor needs an average of 80 µW, which is 1.92 mWh per day.

Step 2: the active volume. A layered film of 5 cm × 5 cm with a thickness of 0.2 mm has V = 5 × 10⁻⁷ m³.

Step 3: coupled input per channel. With uniform input, each channel contributes (Φ_eff · σ_eff) × V:

Input	Φ_eff · σ_eff	Coupled power
Thermal fluctuations	400 W/m³	0.20 mW
Electromagnetic radiation	200 W/m³	0.10 mW
Mechanical vibration	100 W/m³	0.05 mW
Total ΣP_in		0.35 mW

Step 4: output. With η = 0.15, P = 0.15 × 0.35 mW = 52.5 µW.

Step 5: check the boundary. 52.5 µW ≤ 350 µW. The result is consistent with conservation of energy.

Step 6: daily balance. The harvester delivers 52.5 µW × 24 h = 1.26 mWh per day against a demand of 1.92 mWh. It covers about two thirds of the load; the remaining 0.66 mWh per day comes from storage.

Step 7: what it means. A coin cell storing about 0.675 Wh would run the load alone for about 350 days. With the harvester covering part of the load continuously, the same cell lasts about 1,020 days, roughly three times as long, depending on consumption and operating profile.

Where It Leads

The coupling values in Step 3 are set by the material, which is why the choice of layered graphene and semiconductor structures matters. The same budget applied to a real sensor node is worked through in Powering Autonomous Sensors Without Batteries. How the method extends from one sensor to devices, buildings and microgrids is described in From Sensor to Grid.

References

J. A. Paradiso and T. Starner, “Energy scavenging for mobile and wireless electronics,” IEEE Pervasive Computing 4(1), 18–27 (2005).
R. J. M. Vullers, R. van Schaijk, I. Doms, C. Van Hoof and R. Mertens, “Micropower energy harvesting,” Solid-State Electronics 53, 684–693 (2009).
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).
S. Priya and D. J. Inman (eds.), Energy Harvesting Technologies, Springer (2009).
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).
G. J. Snyder and E. S. Toberer, “Complex thermoelectric materials,” Nature Materials 7, 105–114 (2008).
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).
A. Kansal, J. Hsu, S. Zahedi and M. B. Srivastava, “Power management in energy harvesting sensor networks,” ACM Transactions on Embedded Computing Systems 6(4), article 32 (2007).
Schubart Master Formula: derivation, notation and open questions, pt-schubart.com.

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