Photovoltaic Solar Cell Dynamic Impedance (Electrical & Energy Engineering)

An electrical energy systems engineer is designing a Maximum Power Point Tracking (MPPT) inverter for a utility-scale photovoltaic solar array. The non-linear current-voltage ($I\text{-}V$) relationship of the solar cell is defined implicitly by the diode equation:

where:

• $I$ is output current in Amperes ($\text{A}$),
• $V$ is terminal voltage in Volts ($\text{V}$),
• $I_{sc} = 8.0\text{ A}$ is the short-circuit illumination current,
• $I_0 = 1.0 \times 10^{-6}\text{ A}$ is the reverse saturation dark current,
• $\beta = \frac{q}{n k T} = 25.0\text{ V}^{-1}$ is the thermal voltage scaling coefficient.

At the target operating point, the cell operates at voltage $V = 0.60\text{ V}$, which yields a current of $I \approx 4.73\text{ A}$.

1. State the multivariable implicit differentiation formula for the incremental voltage-current derivative $\frac{dV}{dI}$.
2. Compute the partial derivatives $F_I = \frac{\partial F}{\partial I}$ and $F_V = \frac{\partial F}{\partial V}$.
3. Calculate the small-signal dynamic output resistance $R_d = -\frac{dV}{dI}$ of the solar cell at $V = 0.60\text{ V}$.
4. Explain how MPPT control loops use the ratio $\frac{dV}{dI}$ to verify that the array is operating at peak power ($P_{\max}$).