Illuminated Solar Cell Physical Parameters Extraction Using Mathematica

S. Yadir, S. Aazou, N. Maouhoub, K. Rais, M. Benhmida and E. Assaid

Département de Physique, Equipe d’Optique et Electronique du Solide, Faculté des Sciences El Jadida, Maroc eassaid@yahoo.fr

Abstract— In this work, we present three new Mathematica [1] codes written to extract physical parameters of illuminated solar cell from its current-voltage characteristic. The solar cell is modelled by a circuit containing a p n− junction photocurrent

phI , a reverse saturation current sI , an ideality factor n , a

shunt resistance SHR and a series resistance SR (Fig. 1). The three codes are based on the following methods: the five points method [2], the Ortiz-conde method [3] and a new method developed in our laboratory, where we determine experimentally the value of series resistance SR , define a new variable

Sx V R I= − and fit numerically the expression of the current I

as a function of x to the simulated data.

For a given illumination, the solar cell current I flowing through the series resistance SR (Fig 1) is written as:

( )( ) ( )exp 1S S th S SH phI I V R I nV V R I R I= − − + − − (1)

where th BV K T q= .

In the five points method [2], we express the current I and its first derivative with respect to the voltage dI dV at the short circuit point ( 0)V = and at the open circuit point ( 0)I = . We then obtain four equations with five unknowns. The fifth equation is the expression of the current at the maximum power point. In our Mathematica code we use the Newton method to determine numerically the values of the five physical parameters. Fig.2 presents the curves of simulated data and optimized characteristics in photovoltaic mode where a charge resistance is connected at the solar cell terminals and in photodiode mode where the solar cell is biased by a varying voltage source.

In the case of the Ortiz-conde method, the analytical solution of the transcendental equation (1) is given in term of the LambertW function [3]:

( )exp

(1 ) (1 )

( )1

S S phT S S

S T S P T S P

P S ph

S P

V R I IAV R II LambertWR AV R G AV R G

VG I IR G

⎧ ⎫+ +⎛ ⎞⎪ ⎪= ⎨ ⎬⎜ ⎟+ +⎪ ⎪⎝ ⎠⎩ ⎭− +

++

(2)

The authors define the co-content function by:

0( , ) ( )

V

scCC I V I I dV= −∫ (3)

By replacing equation (2) into equation (3), they find a long expression which contains the LambertW function and the variables I and V . After some algebraic manipulations the authors lead to the following equation:

1 1 1 12 2

2 2

( , ) ( ) ( )

( )V I sc I V sc

V I sc

CC I V C V C I I C V I I

C V C I I

= + − + −

+ + − (4)

In our Mathematica code, we calculate numerically ( ),CC I V from the simulated data. Then we perform a two-dimensional fitting of equation (4) to the numerical function ( ),CC I V . The solar cell physical parameters , , ,S P SR G I n and phI are determined from the coefficients 1 1 2, ,I V IC C C and 2VC . Fig.3 gives the curves of simulated data and numerical characteristics obtained by the second method in photovoltaic and photodiode modes.

The new method starts by replacing the series resistance SR in equation (1) by its experimental value. This value is obtained from the solar cell dark characteristic [4]:

( )( )S SC OC SCR V I V I= − . By putting Sx V R I= − , equation (1) becomes:

( )( )exp . 1S th SH phI I x nV x R I= − + − (5) In our Mathematica code, equation (5) is fitted to the simulated data ( )I f x= in order to determine the remaining four physical parameters , ,SH SR I n and phI . Fig.4 gives the curves of simulated data and numerical characteristics of the solar cell obtained by the third method in photovoltaic and photodiode modes.

Tables I and II present the numerical values of solar cell physical parameters obtained by the three methods for simulated characteristics given by PSpice [5] respectively in photovoltaic and photodiode modes.

CONCLUSION A comparative study of the current deviation between the optimized models and the simulated data shows that in photovoltaic mode the Ortiz-Conde method, gives the best results. However in photodiode mode the best results are obtained by our method.

FIGURES AND TABLES

978-1-4244-3806-8/09/$25.00 © 2009 IEEE 63

TABLE I: SUMMARY OF RESULTS IN PHOTOVOLTAIC MODE

TABLE II : SUMMARY OF RESULTS IN PHOTODIODE MODE

References [1] Mathematica 4.0, Copyright 1988-1999, Wolfram Research Inc. [2] D. S. H. Chan, J. R. Phillips and J. C. H. Phang, Solid-State Electronics,

29, No. 3. pp. 329-337, 1986. [3] A. Ortiz-conde, F. .J. Garcia Sanchez, J. Muci, Solar Energy Materials &

Solar Cells, 90, 352–361, 2006. [4] K. Rajkanan and J. Shewchun, Solid-State Electronics, Vol. 22, pp. 193-

197, 1979. [5] PSpice 9.1. Copyright 1998, OrCAD Inc.

Parameters Methods used

PSpice Five points method RS method Ortiz-Conde method

Iph (A) 2 1.999 1.999 1.966

Is (A) 1.4 10-8 1.71 10-8 1.43 10-8 1. 36 10-8

n 1.984 2.018 1.998 1.981

Rs (Ω) 20 19.51 19.82 20.11

RSH (Ω) 3000 3001.33 3030.44 2999.08

Parameters Methods used

PSpice Five points method RS method Ortiz-Conde method

Iph (A) 2 2.01 1.999 2

Is (A) 1.4 10-8 1.41 10-8 1.43 10-8 9.83 10-8

n 1.984 1.984 1.986 1.926

Rs (Ω) 20 20 20 21.19

RSH (Ω) 3000 3000 3005.33 2771.08

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