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Helios Core — Physics Reference

This document outlines the mathematical models, physical constants, and conversion logic used by the Helios Core solver for parameter extraction and data analysis.

1. Physical Constants

The solver uses standard CODATA fundamental constants for all thermal voltage calculations.

Constant Symbol Value Unit
Boltzmann Constant $k_B$ $1.380649 \times 10^{-23}$ J/K
Elementary Charge $q$ $1.602176 \times 10^{-19}$ C
Standard Temperature $T_{STC}$ $298.15$ K (25°C)
Thermal Voltage (STC) $V_t$ $\approx 0.0257$ V

Thermal Voltage Equation: $$V_t = \frac{k_B T}{q}$$

2. Diode Models

Helios Core utilizes global optimization (Differential Evolution) followed by local refinement (Levenberg-Marquardt) to fit raw $I-V$ data to these models.

2.1 One-Diode Model

The standard model for most solar cell technologies (Si, Perovskite, Organic).

$$I = I_{ph} - I_0 \left[ \exp\left(\frac{V + I R_s}{n V_t}\right) - 1 \right] - \frac{V + I R_s}{R_{sh}}$$

  • $I_{ph}$: Photocurrent (A)
  • $I_0$: Dark saturation current (A)
  • $n$: Ideality factor (typically 1.0–2.0)
  • $R_s$: Series resistance ($\Omega$)
  • $R_{sh}$: Shunt resistance ($\Omega$)

2.2 Two-Diode Model

Used for cells with significant recombination at the space-charge region or grain boundaries.

$$I = I_{ph} - I_{01}\left(\exp\left(\frac{V_j}{n_1 V_t}\right) - 1\right) - I_{02}\left(\exp\left(\frac{V_j}{n_2 V_t}\right) - 1\right) - \frac{V_j}{R_{sh}}$$ Where $V_j = V + I R_s$

3. Performance Metrics

These values are extracted either directly from the data (using interpolation) or derived from the fitted model parameters.

3.1 Short-Circuit Current ($J_{sc}$)

The current density when $V = 0$. $$J_{sc} \text{ [mA/cm}^2\text{]} = \frac{|I(V=0)|}{\text{Area}} \times 1000$$

3.2 Open-Circuit Voltage ($V_{oc}$)

The voltage when $I = 0$. Found using linear interpolation between the two points surrounding the zero-crossing. $$V_{oc} = |V(I=0)|$$

3.3 Fill Factor ($FF$)

The ratio of maximum power to the theoretical maximum. $$FF = \frac{P_{max}}{V_{oc} \cdot I_{sc}}$$ Note: Helios Core ensures $P{max}$ is calculated within the power-producing quadrant._

3.4 Power Conversion Efficiency ($\eta$)

$$P_{in} = 100 \text{ mW/cm}^2 \text{ (1 Sun)}$$ $$\eta \text{ [%]} = \frac{P_{max}}{\text{Area} \cdot P_{in}} \times 100$$

4. Normalization and Scaling

Helios Core converts all absolute values ($\Omega$, A) to normalized values ($\Omega \cdot cm^2$, $mA/cm^2$) for standardized reporting.

4.1 Resistance Normalization

To compare cells of different sizes, resistances must be normalized by area: $$R_{\text{normalized}} \text{ [}\Omega \cdot cm^2\text{]} = R_{\text{absolute}} \text{ [}\Omega\text{]} \times \text{Area [}cm^2\text{]}$$

4.2 Current Normalization

$$J \text{ [mA/cm}^2\text{]} = \frac{I \text{ [A]} \times 1000}{\text{Area [}cm^2\text{]}}$$

5. Automated Data Handling

5.1 Sign Convention

Helios Core automatically detects the data's sign convention. If current at $V > 0$ is negative (sink convention), the engine automatically flips it for fitting against the generator-based diode equations.

5.2 Unit Detection

The engine automatically scales current based on detected column headers:

  • mA/cm² or mA: Scale by $10^{-3}$
  • µA/cm² or µA: Scale by $10^{-6}$
  • A: No scaling