> ## Documentation Index
> Fetch the complete documentation index at: https://docs.plantpredict.com/llms.txt
> Use this file to discover all available pages before exploring further.

# 7-Parameter Model

export const BuiltInVoltage = () => <Tooltip tip="Contact potential across a semiconductor junction at equilibrium, equal to the difference in chemical potential between the two sides in isolation.">
    built-in voltage
  </Tooltip>;

export const Recombination = () => <Tooltip tip="Loss of photo-generated electron-hole pairs before collection as electrical current; reduces cell output.">
    recombination
  </Tooltip>;

export const OpenCircuitVoltage = () => <Tooltip tip="Voltage across a PV cell or module when no current flows (I = 0); decreases with temperature.">
    open-circuit voltage
  </Tooltip>;

export const Photocurrent = () => <Tooltip tip="Light-generated current in a PV cell; slightly exceeds short-circuit current due to internal losses at V = 0.">
    photocurrent
  </Tooltip>;

export const IVCurve = () => <Tooltip tip="Current-voltage characteristic of a PV cell or module; the operating point on this curve determines power output.">
    I-V curve
  </Tooltip>;

## Summary

The 7-parameter model extends the [5-parameter single-diode model](/models/dc-performance/5_parameter_model) with a voltage-dependent <Recombination /> current term, following the equivalent circuit proposed by Merten et al. (1998). The additional term captures carrier recombination at low irradiance. This mechanism is most relevant for thin-film technologies but can improve accuracy for any module where shunt resistance alone underestimates low-light losses. The model adds two parameters—<BuiltInVoltage /> ($V_{bi}$) and recombination parameter ($d_i^2/\mu\tau$)—to the five standard single-diode parameters. All five base parameters are scaled by [Parameter Translation](/models/dc-performance/parameter_translation) in the same way as for the 5-parameter model; $V_{bi}$ and $d_i^2/\mu\tau$ are not scaled.

## Inputs

| Name                          | Symbol          | Units | Description                                                                                                            |
| ----------------------------- | --------------- | ----- | ---------------------------------------------------------------------------------------------------------------------- |
| **Photocurrent**              | $I_{ph}$        | A     | Light-generated current                                                                                                |
| **Saturation Current**        | $I_0$           | A     | Diode reverse saturation current                                                                                       |
| **Series Resistance**         | $R_s$           | Ω     | Series resistance (includes module internal resistance and DC wiring resistance)                                       |
| **Shunt Resistance**          | $R_{sh}$        | Ω     | Shunt resistance of module                                                                                             |
| **Diode Ideality Factor**     | $\gamma$        | —     | Diode ideality factor                                                                                                  |
| **Number of Cells in Series** | $N_c$           | —     | Cells in series within module                                                                                          |
| **Cell Temperature**          | $T_c$           | °C    | Operating cell temperature                                                                                             |
| **Built-in Voltage**          | $V_{bi}$        | V     | Built-in voltage per cell                                                                                              |
| **Recombination Parameter**   | $d_i^2/\mu\tau$ | V     | Lumped recombination parameter combining recombination layer thickness and effective carrier mobility-lifetime product |

***

## Outputs

| Name                     | Symbol   | Units | Description                       |
| ------------------------ | -------- | ----- | --------------------------------- |
| **Max Power Voltage**    | $V_{mp}$ | V     | Voltage at maximum power point    |
| **Max Power Current**    | $I_{mp}$ | A     | Current at maximum power point    |
| **Max Power**            | $P_{mp}$ | W     | $V_{mp} \times I_{mp}$            |
| **Open-Circuit Voltage** | $V_{oc}$ | V     | Voltage at open-circuit ($I = 0$) |

***

## Detailed Description

### Circuit Equation

The 7-parameter <IVCurve /> adds a recombination current to the [5-parameter circuit equation](/models/dc-performance/5_parameter_model#circuit-equation):

$$
I = I_{ph} - I_0 \left(\exp\!\left(\frac{q(V + IR_s)}{N_c k T_c \gamma}\right) - 1\right) - \frac{V + IR_s}{R_{sh}} - \frac{(d_i^2/\mu\tau) \cdot I_{ph}}{N_c V_{bi} - (V + IR_s)}
$$

where $q = 1.602 \times 10^{-19}$ C is the elementary charge and $k = 1.381 \times 10^{-23}$ J/K is the Boltzmann constant. As in the [5-parameter model](/models/dc-performance/5_parameter_model#circuit-equation), $V_{th} = N_c \gamma k T_c / q$ is the modified thermal voltage and $V_{int} = V + IR_s$ is the internal voltage:

$$
I(V_{int}) = I_{ph} - I_0 (e^{V_{int}/V_{th}} - 1) - \frac{V_{int}}{R_{sh}} - \frac{(d_i^2/\mu\tau) \cdot I_{ph}}{N_c V_{bi} - V_{int}}
$$

The last term represents a voltage-dependent current loss that increases as $V_{int}$ approaches $N_c V_{bi}$ and is proportional to <Photocurrent />. It was originally proposed by Merten et al. (1998) to model recombination in amorphous silicon p-i-n junctions, but is used more broadly as an empirical correction that improves low-irradiance accuracy beyond what shunt resistance alone provides.

### Maximum Power Point

PlantPredict uses the same internal-voltage approach as the [5-parameter model](/models/dc-performance/5_parameter_model#maximum-power-point), expressing $I(V_{int})$ and $V(V_{int})$ as explicit functions and solving $dP/dV_{int} = 0$ via [Newton-Raphson iteration](https://en.wikipedia.org/wiki/Newton%27s_method). Once the optimal internal voltage $V_{int,mp}$ has converged:

$$
I_{mp} = I(V_{int,mp})
$$

$$
V_{mp} = V_{int,mp} - I_{mp} R_s
$$

$$
P_{mp} = V_{mp} \times I_{mp}
$$

### $V$ Given $I$

The recombination term's pole at $N_c V_{bi}$ prevents reformulation into the [Lambert W](https://en.wikipedia.org/wiki/Lambert_W_function) canonical form used by the 5-parameter model. PlantPredict instead solves for $V_{int}$ directly via Newton-Raphson iteration on the circuit equation and recovers the terminal voltage as $V = V_{int} - IR_s$. The <OpenCircuitVoltage /> $V_{oc}$ is obtained as the special case with $I = 0$.

### $I$ Given $V$

When the terminal voltage is fixed—for instance, when set by the inverter at an operating point away from MPP (e.g., clipping)—the Lambert W reformulation is again not applicable. PlantPredict solves for $V_{int}$ via Newton-Raphson with $I = (V_{int} - V)/R_s$ substituted into the circuit equation, then computes $I$ from the converged $V_{int}$.

***

## References

* Merten, J., Asensi, J. M., Voz, C., Shah, A. V., Platz, R., & Andreu, J. (1998). *Improved equivalent circuit and analytical model for amorphous silicon solar cells and modules.* IEEE Transactions on Electron Devices, 45(2), 423–429. DOI: [10.1109/16.658676](https://doi.org/10.1109/16.658676)