# $I/V$ Characteristics of a Solar Panel

A solar panel is an assembly of solar cells that uses light energy from the sun to produce electricity. A solar cell usually consists of a silicon based large area pn-junction with a thin and heavily doped n-type layer above a more lightly doped p-layer. When the cell is exposed to light and a photon with sufficient energy enters the depletion region through the n-typed layer, electron-hole pairs are created. Due to the electric field present in the depletion region electrons will be pushed to the n-side while holes are pushed to the p-side, forward biasing the diode. By connecting a load with a smaller impedance than the forward biased diode the generated photo current can be extracted. In a solar panel, the individual solar cells are connected electrically in series to achieve the desired voltage and to then in parallel to increase the current.

The goal of this exercise was to measure the $I/V$ characteristics of a solar panel for different light levels and temperatures. The measurements were carried out on a Phaesun Sun Plus 5 solar module using the Keithley 2636A System SourceMeter as a voltage supply and current measurement device. To operate the solar panel in full darkness and to perform measurements in the temperature range of 10 to 62 °C, a Vötsch VT4002 climate chamber was used. The following measurements were performed:

• Measurement of $I/V$ characteristic at room temperature for different light levels
• Calculation of the saturation current $I_S$ via log-linear fit
• Calculation of maximum power output point at ambient light exposure
• Measurement of temperature dependence of the $I/V$ characteristic

## Measurement of $I/V$ characteristic at room temperature for different light levels

A voltage sweep from -20 to 20 V with the solar panel at ambient light exposure (in the lab near the window on a slightly cloudy day, no direct sunlight exposure), no light exposure (inside the climate chamber) and with its face down was performed at ambient temperature in order to measure the $I/V$ characteristic of the module (see Fig. 2).

## Calculation of the saturation current $I_S$ via log-linear fit

The $I/V$ characteristic of an ideal diode at a given Temperature $T$ can be approximated by the Shockley equation $$I(V) = I_S \cdot (\exp{(\frac{e \cdot V}{n \cdot k_B \cdot T})}-1) \stackrel{\frac{e \cdot V}{n \cdot k_B \cdot T} \gg 1}{\approx} I_S \cdot \exp{(\frac{e \cdot V}{n \cdot k_B \cdot T})}$$ where $I_S$ is the reverse bias saturation current and $n$ the ideality factor. For high enough voltages (about a few hundred mV at room temperature) the Shockley equation can be approximated with an exponential function and the saturation current $I_S$ can be calculated via a log-linear fit of the current. Applying this method to the $I/V$ data measured on the solar panel with no light exposure in the log-linear part between 3 and 18 V results in the following value: \begin{equation*} I_S = (13.1 \pm 0.4) ~ \mu \text{A} \end{equation*} Comparing this with the measurement data in reverse bias (see Fig. 3) shows that the log-linear fit only provides an accurate estimate of $I_S$ at a voltage of about -4 V. Using a static value as approximation for the saturation current over a bigger voltage range in reverse bias does not make much sense as $I_S$ rises linearly with higher applied negative voltage. The calculated value for $I_S$ deviates by a factor of approximately $\frac{1}{6}$ from the measured value at a voltage of -20 V.

## Calculation of maximum power output point at ambient light exposure

The characteristics of the solar panel as stated by the manufacturer at standard test conditions STC (1000 W/m$^2$ irradiance, 25 °C, 1.5 air mass) are \begin{align*} \text{Open-circuit voltage:} &~V_{OC} = 21.0~\text{V} \\ \text{Short-circuit current:} &~I_{SC} = 0.34~\text{A} \\ \text{Peak power:} &~P_{max} = 5~\text{W} \\ \text{Voltage at peak power:} &~V_{max,P} = 16.8~\text{V} \\ \text{Current at peak power:} &~P_{max,P} = 0.30~\text{A} \\ \end{align*} Assuming that all 36 solar cells in the solar panel are connected in series to increase the output voltage this leads to per-cell values \begin{align*} \text{Open-circuit voltage per cell:} &~V_{OC,pc} = 0.583~\text{V} \\ \text{Peak power per cell:} &~P_{max,pc} = 0.14~\text{W} \\ \text{Voltage at peak power per cell:} &~V_{max,P,pc} = 0.467~\text{V} \\ \end{align*} which are approximately in line with expected values for commercial solar cell devices [1].

The open-circuit voltage $V_{OC}$ is the maximum voltage that can be drawn from a solar cell module, and it occurs when the net current through the device is zero. It can also be thought of as the amount of forward bias that is generated by the incident light in the depletion region. The short-circuit current $I_{SC}$ is the current when the solar cell module is operated at short circuit, $V$ = 0. For a high-quality solar cell with a low series resistance the short-circuit current is approximately equal to the light-generated photo current $I_L$. For the examined solar panel $V_{OC}$ and $I_{SC}$ at ambient light exposure are measured to be (see Fig. 4): \begin{align*} V_{OC} &= 11.700~\text{V} \\ I_{SC} &= 1.139~\text{mA} \end{align*} It is not possible to extract any power from the solar cell module when operating at either open-circuit or short-circuit condition. Starting from 0 V, the generated power $P = V \cdot I$ that can be drawn from the solar panel increases approximately linearly with the applied voltage until it reaches a maximum and then sharply drops to zero towards the open-circuit voltage. The maximum power output $P_{max}$ point for the solar panel is calculated according to Fig. 4: \begin{align*} P_{max} &= 6.813~\text{mW at} \\ V_{max,P} &= 8.000~\text{V} \\ I_{max,P} &= 0.852~\text{mA} \end{align*} The measured values lie far below the characteristics given by the manufacturer. This is due to the fact that our measurement has been performed inside the lab room with indirect sunlight exposure instead of the standard test conditions used by the manufacturer (comparable to direct light exposure on a sunny day at an average european latitude [2]).

## Temperature dependence of $I/V$ characteristic

A temperature sweep from 10 °C to 62 °C in steps of 4 °C in combination with a voltage sweep from -20 V to 20 V at every temperature step is performed inside the climate chamber (no light exposure). The measured data is shown in Fig. 6 and the measurement code provided below:

Given the Shockley equation (1) above as an approximation for the diode $I/V$ characteristic, one may expect the forward current to decrease with an increase in Temperature at a fixed voltage. However, exactly the opposite is the case. Reason for this is the temperature dependence of the reverse saturation current term $I_S$. The saturation current is a combination of the generation current caused by thermal generation of electron-hole pairs in the depletion region and the diffusion current of the minority carriers in the n and p regions diffusing across the depletion region. Detailed theoretical calculations on the $I/V$ characteristics of a pn-diode and the current-temperature characteristics of a pn-diode can be found on the Physics of Semiconductor Devices Website. For a pn-diode the saturation current is proportional to the squared intrinsic carrier density $n_i$ $$I_S \propto n_i^2$$ The intrinsic carrier density has a strong exponential temperature dependence which allows the saturation current to be approximated as $$I_S = A \exp \left( -\frac{E_g}{k_B T} \right)$$ where $E_g$ is the band gap energy and A is a fit parameter that is to be determined. Inserting this into the Shockley-equation (1) leads to $$I = A \exp \left(-\frac{E_g}{k_B T}\right) \left[ \exp \left( \frac{eV}{k_BT} \right) - 1 \right]$$ For sufficiently large voltages $V$ in forward bias the -1 term can be neglected. By then taking the logarithm and rewriting the equation as a function of $V$ one obtains a linear dependence of $V$ in $T$ $$V = \frac{E_g}{e} + \frac{k_B T}{e} \ln \left( \frac{I}{A} \right)$$ In reverse bias the exponential term containing $V$ can be neglected and only the exponential dependence containing $E_g$ remains. By taking the logarithm of the absolute value this can be written as a linear dependence of $I$ over the inverse temperature $T^{-1}$ $$\ln |I| = \ln A - \frac{E_g}{k_B T}$$ With equation (5) and (6) we now have a way to calculate the band gap energy $E_g$ in forward bias at constant current and in reverse bias at constant voltage for different temperatures via a linear or log-linear fit over $T$ or $T^{-1}$ respectively.

Fig. 6 shows the measured voltage values taken at the fixed current $I$ = 10 mA in forward bias over the temperature together with a linear fit. From the first fit parameter the bang gap energy is calculated via equation (5) to \begin{align*} E_g = (40.8 \pm 0.5)~\text{eV}~~~\Rightarrow~~~ (1.132 \pm 0.014)~\text{eV}~~~~\text{(per cell)} \\ \end{align*} which is in accordance with the typical band gap energy of a single silicon solar cell.

Fig. 7 shows the measured current values taken at a fixed voltage $V$ = -15 V in reverse bias together with a log-linear fit over the inverse temperature. Using equation (6) the band gap energy calculates from the slope of the semilogarithmic line to \begin{align*} E_g = (0.28 \pm 0.04)~\text{eV} \end{align*} This value is far off the expected band gap energy of about 1.12 eV for our silicon solar cells. In Fig. 7 can be seen that the measured data does not match the expected exponentially decreasing behaviour and that a much steeper slope in the semilogarithmic plot would be needed to obtain the expected band gap energy. This suggests that the derivations above for the saturation current $I_S$ do not hold for the measured temperature range and that other (probably thermal) effects influence the saturation current. Additional considerations in the theoretical derivation or measurements (on single solar cells) need to be taken into account to correctly describe the temperature dependence of the saturation current in reverse bias.