# Low-Cost Three-Quadrant Single Solar Cell _I-V_ Tracer
## 2 Methods and Materials
### Principles of Operation  
As shown in Figure 3, the proposed \(I\)-\(V\) tracer is just based on the current transfer between two capacitors of equal capacitances across a solar cell. Figure 3: Schematic topology for the proposed \(I\)-\(V\) tracer. DUT: device under test. Figure 2: \(I\)-\(V\) curve of a photovoltaic cell. (**a**) Maintaining the current and voltage criteria from the diode. (**b**) Inverting the current criterion from the diode. \(V_{\rm OC}\): open circuit voltage; \(I_{\rm SC}\): short circuit current. Our design for the three-quadrant \(I\)-\(V\) tracing takes advantage of the fact that most of the curve (from the second quadrant at the left to the zero current point in the first quadrant) corresponds to positive (forward) values of the current and, for a wide range of voltage values, is approximately constant in the current. If we connect our solar cell under test to the positive plates of these capacitors, a forward current gives rise to a charge transfer from one capacitor to another and, consequently, the voltages change (one rises and the other diminishes). The changing rate of one capacitor voltage is expressed as shown (1):  
\[\frac{dV}{dt}=\frac{I}{C} \tag{1}\]  
So, as the capacitances of both capacitors are the same, their voltage changing rates are the same, with one rising and the other reducing, proportional to the current. Considering that the voltages across the solar cells are the differences of the voltages across each capacitor, these will vary at rates proportional to the currents. This means that, if we start from a difference between the capacitor voltages representing an initial reverse voltage for the cell, we will stand on the flat region of the curve; the solar cell voltage evolution will be (more or less) a linear sweep (as shown in Figure 23 below). As the cell voltage reaches values next to \(V_{\text{OC}}\), the cell current diminishes, and the voltage sweep speed slows down (as shown in Figure 22 below). This way, a voltage sweep will be performed naturally, with the curve traveling from the beginning negative voltage in the second quadrant to near the zero current (positive voltage \(V_{\text{OC}}\)) point, where the charge transfer will stop, and the voltage sweep finishes. To trace the fourth quadrant section of the curve, another voltage sweep needs to be performed (in this case, starting from an initial cell voltage over \(V_{\text{OC}}\) and finishing again on \(V_{\text{OC}}\)). After these two voltage sweeps, we will have traced a complete three-quadrant solar cell curve. The measurement strategy could also be carried out on small associations of solar cells in series (small modules) and even regular photovoltaic modules (PV modules). For this reason, we designate it as the device under test (DUT). **Measurement process. **  
The process, controlled by opening or closing the needed switches at each step, is conducted as follows:  
1. Firstly, both capacitors (\(C_{1}\) and \(C_{2}\)) are discharged--\(V(C_{1})=V(C_{2})=0\)--and all switches (\(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\)) are opened. 2. Then, by closing \(S_{1}\), \(C_{1}\) is charged to the source voltage (\(V_{S}\)). This voltage (\(V_{S}\)), as will be explained later, will be the starting reverse voltage of the curve and could be conveniently adjusted. 3. \(S_{1}\) is then opened again. At this point, \(C_{1}\) stays charged at vs. and \(C_{2}\) is still discharged. So, \(V(C_{1})=\) vs. and \(V(C_{2})=0\) V.
4. \(S_{2}\) is then closed; at this point, the voltage at the DUT terminal is the differential voltage between the positive plate of \(C_{2}\) and the positive plate of \(C_{1}\)--that is, \(V(\text{DUT})=V(C_{2})-V(C_{1})\)--. That means a reverse polarisation of the solar cell, which, considering its expected behaviour (described in Section 2.1), will supply a current proportional to its irradiation flowing from \(C_{1}\) toward \(C_{2}\) and, consequently, increasing the \(C_{2}\) voltage and lowering the \(C_{1}\) voltage by means of the charge transfer.