Practical Nonlinear Energy Harvesting Model

• The Practical Nonlinear Energy Harvesting Model is a framework capturing nonlinear behaviors in RF harvesters, including diode threshold, roll-off, and saturation phenomena.
• It employs an RC charging circuit with exponential dynamics and parameter fitting to accurately model capacitor voltage and energy storage over time.
• The model enhances real-world design in IoT and SWIPT by enabling precise energy management, adaptive operation, and optimal component sizing.

A practical non-linear energy harvesting model provides a physically accurate, implementation-validated framework for predicting, analyzing, and optimizing the behavior of real-world energy harvester circuits—especially those used in sustainable IoT, wireless power transfer, and SWIPT. These models explicitly capture the saturation, threshold, and dynamic effects intrinsic to rectifiers, storage capacitors, and corresponding interface circuits, contrasting sharply with the overly idealized linear-efficiency approximations frequently seen in early literature.

1. Circuit-Level Foundations and Physical Model

A key practical approach models the RF energy harvester—including the rectifier and storage supercapacitor—as an RC charging circuit with a non-linear open-circuit rectifier voltage. Let $P_i$ denote the incident RF power at the antenna, and $V_{\rm OC}(P_i)$ the open-circuit DC voltage output from the rectifier. The voltage across the storage capacitor at time $t$, $V_{\rm cc}(t)$, is governed by the exponential RC law: $$V_{\rm cc}(t) = V_{\rm OC}(P_i)\,\bigl(1 - e^{-t/(RC)}\bigr)$$ with stored energy

$$E_s(t) = \tfrac12 C [V_{\rm cc}(t)]^2 = \tfrac12 C \bigl[V_{\rm OC}(P_i)\bigr]^2 \bigl(1 - e^{-t/(RC)}\bigr)^2$$

where $R$ and $C$ are the effective series resistance and storage capacitance, respectively. Critically, $V_{\rm OC}(P_i)$ is itself a non-linear function of the RF input power due to rectifier diode effects: at low $P_i$, no conduction occurs until the diode threshold is exceeded; at high $P_i$, reverse conduction and circuit losses cause roll-off or saturation (Luo et al., 2019).

This model precisely mirrors experimental observations on, e.g., Powercast P2110 harvester modules, showing that $V_{\rm OC}(P_i)$ exhibits rapid non-linear increase past the diode threshold and saturates at higher input power.

2. Nonlinear Input–Output Characteristics and Parameterization

The non-linear relationship is twofold:

• Temporal (Charging): The exponential rise of $V_{\rm cc}(t)$ captures the fast initial ramp and gradual saturation in time; the stored energy follows a squared exponential.
• RF–DC Conversion: $V_{\rm OC}(P_i)$ exhibits a non-linear (often sigmoidal) profile with respect to $P_i$. Empirical datasets (e.g., for Powercast P2110: at −14 dBm, $V_{\rm OC} = 0.4$ V; at −2 dBm, $V_{\rm OC} = 4$ V) can be fitted by low-order polynomials or lookup tables.

Circuit-level effects encapsulated include: diode forward turn-on, high-input-power reverse-conduction saturation, matching network-induced resistive loss (contained in $R$), and storage capacitor non-idealities.

3. Practical Model Calibration and Validation

Experimental calibration involves fitting measured $\{t_k, V_{\rm cc}(t_k)\}$ points for each device/capacitor, minimizing the mean squared voltage error via least-squares adjustment of $R$ and $V_{\rm OC}$ (Luo et al., 2019). This procedure is robust: for $C = 2.2$ mF, $R \approx 170.6$ Ω, mean voltage error ≈5.1 mV; for $C = 50$ mF, $R \approx 3.7\,\mathrm{k}\Omega$, mean error ≈8.4 mV. Residuals are typically <0.01 V over a 0–4 V range.

Such accuracy enables direct overlay and tracking of measurements and model curves for both initial charge ramp and final saturation.

4. Key Model Features Compared to Linear Approaches

Aspect	RC + Nonlinear $V_{\rm OC}(P_i)$ Model	Linear Model
Diode threshold, sensitivity	Physical threshold modeled	No threshold
Saturation at high input	Yes (via $V_{\rm OC}$ and exponential)	No (unbounded)
Charge/discharge rate	Exponential, $RC$ time constant	Linear
Predictive accuracy (voltage)	<1% error possible	Can be grossly wrong near threshold/saturation

Neglecting the non-linearities leads to significant allocation mismatch in SWIPT and IoT systems especially when some harvesters are near threshold and others are near saturation.

5. System Design Implications and Methodological Guidelines

• Node-Level Calibration: Each energy harvester node should periodically “pulse charge” itself, record $\{t_k, V_k\}$, and fit $R$, $V_{\rm OC}$ in situ to reflect true RF conditions and stochastic environment (Luo et al., 2019).
• Energy Management: For any required delivered energy $\Delta E$, inversion of (2) gives the precise charge time, permitting precise duty-cycle scheduling and ensuring availability for payload transmissions.
• Component Sizing: The capacitance $C$ controls tradeoffs: a smaller capacitor enables faster current ramp-up but limits total stored energy, especially if $R$ is large. Choice of $C$ should match application-specific charge time and energy budget constraints.
• Front-End Optimization: By minimizing $R$ (better matching network, lower-loss designs), both charge rates and peak voltages are improved, directly enhancing total harvested energy.
• Adaptive Operation: As the capacitor approaches saturation ($V_{cc} \sim V_{\rm OC}$), additional harvested energy per unit time diminishes; optimal operation avoids leaving the harvester idle in this regime by scheduling loads or using a bleed resistor to reset the charging cycle.

6. Broader Context and Integration with SWIPT/IoT Paradigms

This model unifies key circuit-level nonlinearities—including diode threshold, roll-off, and capacitor dynamics—into a compact, parametric framework with demonstrated sub-percent voltage prediction error on real commercial RF harvesters (Luo et al., 2019). A similar mathematical structure underpins the widely adopted logistic/sigmoid models in SWIPT resource allocation, where nonlinear RF–DC transfer is paramount for both beamforming and power control under practical constraints (Boshkovska, 2016, Boshkovska et al., 2015).

The practical RC + nonlinear-voltage model forms the foundation for accurate energy supply prediction and real-time system adaptation in sustainable IoT deployments, providing a closed-form, physically justifiable alternative to empirical or linearized approaches and enabling robust operation across a broad spectrum of input powers and circuit architectures.

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References:

(Luo et al., 2019, Boshkovska, 2016, Boshkovska et al., 2015)