# Mathematical model of solar cell

The basic principle of a solar cell is similar to that of a diode. It can be explained by a simple PN junction. Figure 1 shows the single model and appearance of a solar cell. When sunlight hits the PN junction, the atoms in the semiconductor are released due to light energy. Electrons, at the same time, correspondingly generate electron-hole pairs, so an electromotive force is generated between the PN junction. When the external circuit is connected, there is electrical energy output. The battery cell is the smallest unit of photoelectric conversion, and is generally not used as a separate unit. Power supply, the battery cells are connected in series, parallel and packaged to become solar cells. The power is generally a few watts, tens of watts or even hundreds of watts. Many solar cell modules need to be connected in series and parallel to form a solar cell array, which constitutes a solar cell. “Solar Generator”.

The temperature of photovoltaic cells is affected by various factors, as shown in formula (1)
T=Tair＋kS——(1)
In the formula, T is the temperature of the photovoltaic cell, ℃; Tair is the ambient temperature, C; S is the illuminance.w/㎡; k is the coefficient, ℃·㎡.
Under general test conditions, photovoltaic cells have five parameters: short-circuit current Isc, open-circuit voltage Uoc, maximum power point output power Pm, maximum power point voltage Um, and maximum power point current Im.

The equivalent circuit of a photovoltaic cell is shown in Figure 2, which is equivalent to a current source and a diode in parallel. In the figure, RL is an external load, Rs and Rsh are equivalent to actual internal losses, and the battery output voltage is UL. IL is the output current of the photovoltaic cell. Isc is the current stimulated by the sun’s rays.

In Figure 2. IVD is the diode current, and its expression is shown in (2):
IVD=ID0[e(qE/AKT)﹣1]——(2)
In the formula, q is the charge of the electron, 1.6×10﹣19C; K is Boltzmann’s constant, 1.38×10-23J/K; A is a constant factor, generally 1 or 2: T is the temperature of the panel, ℃; E is the electromotive force of the battery, V.
From Figure 2 we can get formula (3) load current IL:
IL=ISC﹣ID0{e [q(UL＋ILRS)/AKT]﹣1}﹣(UL＋ILRS/RSh)——(3)
In the formula, Rs is the series resistance; Rsh is the bypass resistance. The two resistance values ​​are related to the internal material of the battery panel.
Since Rs in formula (3) is very small. Rsh is very large and can be ignored, we obtain formulas (4) and (5)
IL=ISC﹣ID0[e(qUL/AKT)﹣1]——(4)
UL=(AKT/q)ln{[(ISC﹣IL)/ID0 ]＋1}——(5)
From equations (4) and (5), the output voltage and current of photovoltaic cells are mainly affected by irradiance and temperature. In the short circuit experiment, when RL=0, the output current IL is equal to Isc; in the open circuit experiment, when RL→∞, the voltage across the battery can be measured as UOC.
The open circuit voltage of the photovoltaic cell is shown in equation (6):
UOC=(AKT/q)ln[(ISC/ID0)＋1]≈(AKT/q)ln(ISC/ID0)——(6)
It can be seen from the above formula that Uoc is related to the irradiance and temperature of the photovoltaic cell, and is inversely proportional to the temperature.
According to the engineering calculation method, formula (4) can be converted into formula (7):
IL=ISC{1﹣C1[e(UL/C2UOC)﹣1]}——(7)
In the open circuit state, the output current is 0. The voltage is Uoc: when the photovoltaic cell has the maximum power, the output current is Im and the output voltage is Um. Solutions have to:
{C1=[1﹣(Im/ISC)]e(﹣Um/C2UOC)
{C2=[(Um/Uoc)﹣1]{ln[1﹣(Im/ISC)]}﹣1 ——(8)
Considering the irradiance and temperature changes, the output characteristic formula of the photovoltaic cell is equation (9)
{IL=ISC[1﹣C1(e[(UL﹣DUL)/C2UOC]﹣1]＋DIL
{DI=α·R/Rref·DT＋(R/Rref﹣1)·ISC
{DU=﹣β·DT﹣RS·DI
{DT=T﹣Tref——(9)

In the formula, D is the open-air duty ratio; Rref is the reference value of irradiance, generally 1000w/㎡; Tref is the reference value of photovoltaic cell temperature, 25℃; α is the temperature coefficient of current change, A/℃; β is the voltage Change temperature coefficient, V/°C.
C1 and C2 can be obtained respectively at the maximum power point and the open circuit state. From the above mathematical model, the I-U and P-U characteristic curves of the photovoltaic cell at the reference irradiance Rref=1000W/㎡ and the temperature Tref=25℃ can be determined. The MAT-LAB/SIMULINK simulation model can obtain the characteristic curve at any irradiance S and photovoltaic cell temperature T.