# Appendix: Uniform distribution as the optimum illumination

We will formally prove with the model presented in section 13.1.5 that for cells whose grid-line series resistance is negligible, when the local concentration distribution is assumed to vary slowly between the grid-lines, the distribution providing maximum cell efficiency is the uniform one.

Let us consider the cell model presented in section 13.1.5. In the trivial case in which the parameter rs is negligible (i. e. J(C, V)rs ^ VT for any C < CMAX and any V < Voc), any local concentration distribution f (C) produces the same efficiency (equation (13.11) does not depend on f (C)).

Assuming now that rs is not negligible, consider two concentration distributions with the same average concentration (C): one uniform distribution, i. e. fU(C) = S(C – (C)), where 8 denotes the Dirac-delta, and another non­uniform one, with probability density function fNU(C). From equations (13.5) and (13.11) we can write:

( V + JU(V)rs

Iu(V) = (C)IL, lsun-I0exPl——————– with /и = Ac/и (13.21)

, ,„T, {V f°° {Jwj(C, V)rs f

/nu (V) = (C) /L, lsun – /о exp I — 1 J exp I ————- ——— 1 /nu (C) dC

(13.22)

and:

/nu(V) ^ Jnu(C, V)fNU(C)dC = Ac < Jnu(V)). (13.23)

J0

We shall prove that for every value of V, ZU(V) > ZNU( V) and, thus, PU(V) = VIu(V ))VInu (V) = Pnu(V ).

From the properties of the exponential function, we can write

exp(x) > exp(x0)(1 + (x – X0)) (13.24) for any value of x, and the equality is only fulfilled at x = x0. Let us apply this property to x = JNUrs/ VT and x0 = Jurs/ VT, we can obtain, first, from equation (13.22):

As the second factor is positive, we get Iu(V) > Inu(V).