**3. Solar cell equivalent circuit**

The mathematical model of a photovoltaic cell has to take into account the following factors capable to influence the solar cell behaviour.


The solar cell model, derived from the Mottet-Sombrin's one, is basically a current generator driven by the value of the voltage applied at its terminal according to the equivalent circuit reported below. Generally speaking a solar cell is a particular p-n junction where the diffusion process (diode D1) co-exists with the generation and recombination effect of the charge carrier (diode D2) induced by the presence of crystalline defects. This model was tested using data relevant to the AZUR SPACE 28% solar cell, as reported in the datasheet provided by the Manufacturer, and available on company web-site.

Fig. 4. Equivalent Circuit of solar cell

The relevant Kirchhoff equations are:

$$\dot{i}\_o = \dot{i}\_L - \dot{i}\_D \cdot \left[ \exp\left(\frac{q \cdot V\_D}{k \cdot T}\right) - 1 \right] - \dot{i}\_R \cdot \left[ \exp\left(\frac{q \cdot V\_D}{2k \cdot T}\right) - 1 \right] - \frac{V\_D}{R\_p} \tag{1a}$$

Architectural Design Criteria for Spacecraft Solar Arrays 165

Where Jtot is light intensity (W/ m2), η(T) is the efficiency of the cell, K(T) is a coefficient to

At this point all the terms of the equations (1) can be defined at any temperature and by setting as input the operating voltage Vo and solving the system by the Newton-Raphson

Figure 5 shows the V-I curves relevant to Triple Junction AZUR SPACE solar cell starting from the datasheet available on the web site, as function of temperature at Begin Of Life (BOL); the black asterisks are the maximum power points calculated according to the

**AZUR Space 3G 28% BOL, 1367 W/m2**

Fig. 5. Computed V-I curves as function of temperature using AZUR SPACE 3G 28% data

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> <sup>0</sup>

voltage [V]

**Azur Space 3G 28%, BOL @300K**

Fig. 6. Computed V-I curves as function of illumination using AZUR SPACE 3G 28% data

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> <sup>0</sup>

Voltage [V]

(mA/cm2) (4)

**440 K 400 K 360 K 320 K 280 K 240 K 200 K 160 K 120 K**

**100 W/m2**

**250 W/m2**

**500 W/m2**

**1000 W/m2**

**1367 W/m2**

*<sup>L</sup> tot i T KT T J* 

n1 e n2 are two coefficients depending on the adopted solar cell technology:

datasheet. In figure 6 V-I curves for different illumination levels are reported.

numerical scheme is possible to calculate the output current io.

With Eg0 = 1.41 eV, αe=-6.6×10-4 eV/°K, and βe=552 °K. The current **iL** due to illumination is given instead by

be determined as function of the temperature.

0.1

0.1

0.2

0.3

Current [A ] 0.4

0.5

0.6

0.7

0.2

0.3

Current [A]

0.4

0.5

0.6

0.7

*V V Ri o D So* (1b)

Where:

K=1.381×10-23 (J/°K) is the Boltzmann constant;

q=1.602×10-19 (C) is the electron charge;

iL, iD e iR are respectively the current due to illumination, and the reverse currents of the diodes D1 e D2; they are function of the temperature.

The equations (1) give the output voltage Vo, and current Io as function of the voltage drop Vd over the diodes D1 and D2. The second and third term of (1a) represent the typical voltage-current laws of the diodes, and the currents iD and iR are the reverse currents of the diodes dependent from the physics of the solar cell.

In general, the solar cell is characterised by the following data provided in the manufacturer's data sheet, the table below gives the values relevant to the one used for testing the model:


Such data are given in AM0 (1367.0 W/m2) conditions at Tref=28 °C (301.15 °K) reference temperature.

Usually the series resistance is around 300mΩ for a triple junction cell, while for the shunt one 500Ω maybe assumed. Such resistances may be considered in a first approximation as constant in the operating temperature range of the cell.

The values of iL, iD and iR at the reference temperature can be calculated with the (1) in the three main point of the V-I curve; short circuit, maximum power and open circuit, by the least square method.

The next step is to define how these currents change with temperature.

Concerning **iD** e **iR** it is possible to write:

$$\dot{\mathbf{u}}\_D = \mathbf{C}\_D \cdot T^{5/2} \cdot \exp\left(\frac{E\_g}{n\_1 \cdot T}\right) \tag{2a}$$

$$\dot{n}\_R = \mathbf{C}\_R \cdot \exp\left(\frac{E\_\mathbf{g}}{n\_2 \cdot k \cdot T}\right) \tag{2b}$$

Where **CD** and **CR** are constants independent from temperature, and Eg is the Energy of the prohibited band gap:

$$E\_g = E\_{\mathcal{g}^0} - \frac{\left(\alpha\_\varepsilon \cdot T^2\right)}{\left(T + \beta\_\varepsilon\right)} \quad \text{(mA/cm2)}\tag{3}$$

iL, iD e iR are respectively the current due to illumination, and the reverse currents of the

The equations (1) give the output voltage Vo, and current Io as function of the voltage drop Vd over the diodes D1 and D2. The second and third term of (1a) represent the typical voltage-current laws of the diodes, and the currents iD and iR are the reverse currents of the

In general, the solar cell is characterised by the following data provided in the manufacturer's data sheet, the table below gives the values relevant to the one used for

Such data are given in AM0 (1367.0 W/m2) conditions at Tref=28 °C (301.15 °K) reference

Usually the series resistance is around 300mΩ for a triple junction cell, while for the shunt one 500Ω maybe assumed. Such resistances may be considered in a first approximation as

The values of iL, iD and iR at the reference temperature can be calculated with the (1) in the three main point of the V-I curve; short circuit, maximum power and open circuit, by the

5/2

Where **CD** and **CR** are constants independent from temperature, and Eg is the Energy of the

 

*T*

*e*

*T*  2

*e*

2 exp *<sup>g</sup>*

*E*

*n kT*

1 exp *<sup>g</sup>*

*n T*

(2a)

(2b)

(mA/cm2) (3)

*E*

**dIsc/dT** 0.32 mA/°K Short circuit current temperature coefficient **dImp/dT** 0.28 mA/°K Max. power current temperature coefficient **dVmp/dT** -6.1 mV/°K Max. power voltage temperature coefficient; **dVoc/dT** -6.0 mV/°K Open circuit voltage temperature coefficient.

Where:

testing the model:

temperature.

least square method.

prohibited band gap:

K=1.381×10-23 (J/°K) is the Boltzmann constant;

diodes D1 e D2; they are function of the temperature.

diodes dependent from the physics of the solar cell.

**Isc** 506.0 mA Short circuit current; **Imp** 487.0 mA Maximum power current; **Vmp** 2371.0 mV Maximum power voltage; **Voc** 2667.0 mV Open circuit voltage;

constant in the operating temperature range of the cell.

Concerning **iD** e **iR** it is possible to write:

The next step is to define how these currents change with temperature.

*D D*

*i CT*

*R R*

*i C*

0

*g g*

*E E*

q=1.602×10-19 (C) is the electron charge;

*V V Ri o D So* (1b)

With Eg0 = 1.41 eV, αe=-6.6×10-4 eV/°K, and βe=552 °K. The current **iL** due to illumination is given instead by

$$\operatorname{Li}\_L\left(T\right) = \operatorname{K}\left(T\right) \cdot \eta\left(T\right) \cdot f\_{tot} \quad \text{(mA/cm2)}\tag{4}$$

Where Jtot is light intensity (W/ m2), η(T) is the efficiency of the cell, K(T) is a coefficient to be determined as function of the temperature.

n1 e n2 are two coefficients depending on the adopted solar cell technology:

At this point all the terms of the equations (1) can be defined at any temperature and by setting as input the operating voltage Vo and solving the system by the Newton-Raphson numerical scheme is possible to calculate the output current io.

Figure 5 shows the V-I curves relevant to Triple Junction AZUR SPACE solar cell starting from the datasheet available on the web site, as function of temperature at Begin Of Life (BOL); the black asterisks are the maximum power points calculated according to the datasheet. In figure 6 V-I curves for different illumination levels are reported.

Fig. 5. Computed V-I curves as function of temperature using AZUR SPACE 3G 28% data

Fig. 6. Computed V-I curves as function of illumination using AZUR SPACE 3G 28% data

Architectural Design Criteria for Spacecraft Solar Arrays 167

It is the contribution of the direct sun radiation J that is not converted into electrical power

1 2 1 *Qalb*

<sup>12</sup> ( ) *Q FTT Earth* 

<sup>12</sup> (1 ) *QSpace* 

*front rear*

*T T Q k x*

Is the heat transmitted by conduction between the front and rear faces of the panel at Tfront

C = thermal capacitance of the panel per unit area, main contribution is provided the

The radiating view factor of a flat surface with respect to the Earth surface is function of the altitude h, earth radium R and the angle λ between the nadir and the normal to the panel.

> *Falb* 1 cos cos

> > *app* arcsin *<sup>R</sup>*

The integration in the time domain of the equation (5) gives the actual temperature of the panel along the propagation of the orbit in eclipse and sunlight, taking into account the orientation of the panel itself with respect to the earth and the sun. The thermal model

*app*

*h R*

(11)

(12)

*F Al F J* (7)

*<sup>E</sup>* (8)

*F T* (9)

(10)

4 4

4

 

*cond*

The parameters appearing in these equations have the following meanings:

The albedo view factor is computed according to the following formulas:

Where β is the angle between the nadir and the earth-sun direction:

by the photovoltaic cell;

It is the contribution of the albedo radiation;

It is the heat exchanged with the Earth surface.

It is the heat released to the deep space.

and Trear temperature respectively.

Al = albedo coefficient, about 0.3 for earth;

k = panel transverse thermal conductivity;

F12 = View factor between radiating surface and planet = Stephan-Boltzmann, constant: 5.67210-8 W/(m2°K4); TE = Black body equivalent temperature of the earth; = incidence angle of the sunlight on the panel;

honeycomb structure; = solar cell absorptivity; = solar cell emissivity;

Δx =panel thickness;

F = albedo visibility factor;
