It is well known that nonuniform illumination of solar cells under high concentration may decrease the power output significantly. For the concentrator designer, the key question is what local concentration distribution can be considered good enough for a given cell.
Usually, the only information available for concentrator designers is the cell efficiency under uniform illumination for different concentration factors Cu. With this information, the common rule of thumb is that the cell efficiency under a certain local concentration distribution C(x, y) of average concentration (C) and peak concentration Cmax will be between the cell efficiencies under uniform illuminations of Cu = (C) and Cu = Cmax. This simple rule of thumb does not indicate to which cell efficiency it will be closer, because more parameters appear (how much area of the cell is illuminated with each concentration level, how important is the series resistance power loss at (C) and at Cmax, etc). The concentrator designer usually takes Cmax as the merit function to minimize (without evaluating the probability of any other concentration value), although its correlation with the best performance is uncertain.
In order to improve the merit function, cells under nonuniform illumination need to be modelled. One possible model consists in using equivalent circuits of the discretized cell and solving the resulting system of nonlinear equations using circuit simulators (e. g. PSPICE) [13]. Figure 13.4 shows an example of such modelling for a selected GaAs cell with crossedgrid front metallization.
The numerical resolution with this discretized circuit model provides the concentrator designer with an accurate merit function (the cell efficiency at the maximum power point) with which to optimize the design. However, such simulations are time consuming and do not indicate how to modify the concentration distribution on the cell to produce significant improvements in the merit functions. Therefore, this optimization process, without additional information, is likely to be very inefficient.
The concentrator designer would find faster tools and a deeper knowledge to orientate, speedup and terminate the optimization of the optical design useful.
as shown in (b) to illuminations with which can be used
Assuming that the cell parameters and average concentration (C) on it are fixed and known, the objective is then to have a cell model able to answer the following questions:
(a) How far is a given local concentration distribution C (x, y) from the optimum, measured in terms of loss of cell efficiency?
(b) What modification in C (x, y) will approach the optimum most rapidly?
(c) If a given distribution C (x, y) has a specific feature (for instance, a certain irradiance peak), is it limiting the cell performance?
Question (a) implies the determination of the optimum irradiance
Figure 13.4. (Continued.) 
distribution. Such an optimum should be defined with constraints, taking into account the fact that concentrator designs have limited capabilities in practice. For instance, it is obvious that, for a given cell, the optimum must not illuminate the gridlines in frontcontacted cells. However, if the use of microoptics is excluded, the optimum does not seem to be attainable with lowcost optics and, thus, this optimum will not have any practical interest for the designer. In this case, it would be more practical to define the optimum with the practical constraint that the irradiance will not vary significantly along distances greater than the gridline thickness or even greater than the gridline pitch in high concentration cells (this can formally be stated in terms of the correlation length of the possible functions C(x, y)). Then, with this constraint, the shadow factor of the gridlines will not be avoided and it will be approximately independent of C(x, y).
It is well known that if the gridlines’ contribution to the series resistance is negligible (compared to the summations of the frontcontact, emitter, base and rear contact contributions), this practical optimum is the uniform illumination (see the appendix). Note that if we add the obvious additional constraint to C(x, y) i. e. that it must vanish outside the active cell area, the uniform illumination is not attainable either, because the first constraint does not allow for an abrupt transition near the edge.
It is also well known that in the case of a nonnegligible contribution by the gridlines to the series resistance, uniform illumination is not the optimum: the cell efficiency will improve if nonuniformity reduces the average path of the current along the gridlines. Although the efficiency increase with respect to
uniformity is slight, the optimum distribution C (x, y) can be very different from the uniform one if the gridline series resistance dominates [14]. This indicates to the concentrator designer where and how high local concentrations levels can be allowed for in these cells without degrading their performance.
We will focus the discussion here on the first case (negligible contribution of the gridlines to the series resistance), and we will consider that the average concentration (C) is fixed. This cell can be modelled as a set of parallel connected unit cells of differential area dxdy. We will assume that the single exponential model for the dark current is valid, that the photocurrent density is proportional to the local concentration factor and that the different series resistance contributions can be modelled as a single constant parameter and the specific series resistance rs, is independent of the concentration factor (i. e. high injection and current crowding effects at the emitter are neglected). Then, if J(C, V) is the photocurrent density though a differential unit cell illuminated with a local concentration C and with an applied voltage V, the following equation is fulfilled:
J(C, V) = C/L, isun – /о exp I——— ———– I (13.5)
where JL, 1sun is the photocurrent density at onesun illumination, J0 the saturation current density, VT = kT/e and V the cell voltage (common for all unit cells).
The cell current I under an illumination C (x, y) is obtained by integrating the photocurrent density J in the cell area AC. From equation (13.5):
With this function, equation (13.6) can be written as
Equation (13.8) is the IV characteristic of the nonuniformly illuminated cell. Assuming that the cell parameters and C (x, y) are known, the maximum power point (Vm, Im) can be calculated from equation (13.8). This calculation can be performed numerically with a simple fast program (or even with a spreadsheet) in a computer.
Note that this calculation answers question (a), giving the concentrator designer a quick tool with which to evaluate the cell efficiency, which is the best merit function for the optimization.
The effect of langleEC(V)) in equation (13.8) can be interpreted as an apparent (voltagedependent) increase in I0. If an apparent voltage drop is preferred for the analysis, the function
Vs(V) = Vrln« EC(V))) (13.9)
should be defined and used.
The minimum (EC(V)) for each V is obtained at the optimum case, i. e. under uniform illumination (see appendix). Consider now a nonuniform local concentration distribution C(x, y) with the same average concentration (C). Recalling question (b): What modification in C(x, y) will approach the optimum most rapidly?
Let us imagine that the designer can freely select two equal areas (dxdy)A and (dxdy )B of the cell surface illuminated with local concentrations CA and CB, respectively, and that the designer can modify the concentrator to decrease the local concentration on (dxdy)A to CA – dC and correspondingly increase it on (dxdy)B to CB + dC (note that the average concentration (C) is maintained). Then, for a given voltage V, the variation in (EC(V)) with the concentrator modification can be immediately calculated from equation (13.6):
d((EC(V))) = dC[EC(Cb, V) – EC(Ca, V)]. (13.10)
The fastest decrease in (EC(V)) towards the minimum is obtained when the right – hand side in equation (13.10) is the most negative. Due to the monotonically increasing dependence of EC on C (which can be easily deduced from equations (13.5) and (13.7)), the modification to C(x, y) which approaches the optimum most rapidly will be produced if the concentrator designer selects (dxdy)A and (dxdy)B such that CA = CMAX and CB = CMIN.
This result was clearly expected: we must reduce the local concentration where it is at a maximum to increase where it is at a minimum. However, equation (13.10) gives more information than that. To show this, let us point out that in practice, the concentrator designer does not have such wide degrees of freedom in the design and the fastest modification rate of the concentration distribution is, in general, not attainable. However, it is not unusual for the designer to be able to select between two alternatives. Let us simplify them as follows: the designer can move the concentration dC either from (dx dy)Ai to (dx dy)B1 or from (dx dy)A2 to (dx dy)B2. Which to choose? According to equation (13.10), the best option is not that for which the concentration difference CA, k – Cb* is the greatest but that for which the exponential concentration difference ECa* — ECB, k (at the maximum power point before modification) is the greatest.
Therefore, for the cells considered in this model, it can be concluded that it is not the concentration distribution but the exponential concentration distribution that plays the key role. Then, the concentrator designer when raytracing should display the function EC(x, y, Vm) over the cell area rather than C(x, y). Note that
the exponential will cause a rather abrupt dependence of EC on C. This means that if there is even a small area of high local concentration (containing even a small power!), it may dominate the series resistance effect, because it dominates the value of (EC(Vm)) over the rest of the lower local concentration levels. Note that if such a high concentration peak on a small area is not dominant, the cell efficiency gain by eliminating that peak will be unimportant (note that this does not contradict the fact that the fastest optimization will need to reduce that peak).
Determining which concentrator levels are dominantcan be deduced by integrating EC(Vm) within the area between isotopes of the function C(x, y). However, this is much easier to do if the exponential concentration EC(x, y, Vm) is expressed as a function of the local concentration C, instead of the spatial variables (x, y), as shown next. This will make the information about the distributions more precise, due to the reduction to a single dimension.
Let us consider the distribution of values of the local concentration factor C, which is studied in terms of the probability density function f (C). The differential of area of the cell that is illuminated with local concentration in the interval (C, C + dC) is given by f (C) dC. The function f (C) can be calculated by standard statistical methods considering C (x, y) as a function of the twodimensional uniform random variable (X, Y) on the cell area. In practice, during the optical design, it can be calculated directly from the ray trace results.
Note that the nomenclature in equation (13.8) is general, independent of the change invariables. However, equation 6 must be rewritten in terms of f (C) as
I(V) = (C)/L, iSUn – k exp (Jr’j /o°°exp(J(C^) /(C) dC. (13.11) Recalling the integrand in equation (13.11) as
w(C, v) = exp (ЛСутУ)Г§) /(О = EC(C, V)f(C). (13.12)
The function w(C, V) gives us the measure at a given voltage V of the relative importance of the different local concentration intervals (C, C + dC) on the cell performance degradation due to the series resistance losses. And obviously (EC(Vm)) is given by the area enclosed by w(C, Vm).
Consider, as an example, the cell parameters rs = 2.1 m^ cm2, VT = 25 mV, JL, isun = 25 mA cm2. These parameters are close to those of high – concentration GaAs cells. Assume also that a given concentrator produces an irradiance distribution on the cell with the probability density function f (C) shown in figure 13.5, whose average is (C) = 1490 suns. This function f (C) will illustrate the concepts presented, although it does not fulfil the smoothness constraint mentioned for the definition of the optimum distribution C(x, y).
w(C, Vm)
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Figure 13.5. Example of probability density function f ( C) of the local concentration level and the contribution of each concentration interval (C, C + dC) at low voltages given by the function w(C, Vm).
From equation (13.13), which is the IV characteristic of the whole cell, the maximum power point has been calculated, leading to Vm = 1.02 V and a cell efficiency nC = 24.8% (to be compared with the uniform illumination of (C) = 1490 suns, nC = 25.4%, the maximum achievable).
In figure 13.5 the function w(C, Vm) is also shown and the amplification at high concentrations caused by the exponential function is noticeable. The maximum concentration is Cmax = 4500 suns. However, the maximum contribution per concentration interval (C, C + dC) occurs at 4000 suns. Let us compare this contribution to that around other local maximum, at 2200 suns: Since f (4000)/f (2200) = 0.162 and w(4000, Vm)/w(2200, Vm) = 3.42, the portion of cell area that is illuminated with concentration levels in (4000, 4000 + dC) is 16.2% of the corresponding illuminated area for (2200,2200 + dC) but the contribution to (EC(Vm)) is 3.42 times greater around 4000.
Talking about the contribution of wider concentration intervals, there are two lobes in the function w(C, Vm): one above and the other below 3500 suns. The lobe above 3500 only corresponds to 2.1% of the incident light power (calculated by integrating Cf(C)). However, its relative contribution to (EC(Vm)), which is calculated by integrating w(C, Vm) = EC(Vm) f (C), is much higher: 23.6%. Therefore, the relative contributions of the two lobes are of the same order of magnitude, and neither of them dominate (EC(Vm)) (this comparison in terms of orders of magnitude results from the fact that cell performance is affected only by the changes of order of magnitude of (EC(Vm)), as occurs with I0).
For a deeper understanding of the analysis, it is interesting to make a
Figure 13.6. Graphical analysis of the relative contributions of the local concentration factors to the degradation by nonuniform illumination. The tilted nearlystraight lines are isotopes with an equal contribution to {EC(Vm)). The ratio of the contribution values of adjacent isotopes is exp(6) = 0.25%. 
graphical representation of the oneparametric family of functions f (C) defined as
u>(C v) = exp (/(CVtV),S) f{C) = В (13.14)
where B is a positive parameter. For a given value of B, this function f (C) is the
loci of the points of the plane f (C)—— C that contribute equally to {EC(Vm))
per concentration interval (C, C + dC). For V = Vm, the oneparametric family of functions is:
( J(C, Vm)rs
f(C) = В exp (—– m 4 ^ in /(C) = in В – J(C, Vm) (13.15)
Figure 13.6 shows some curves of this family in the plane ln f (C)C. The selected values for parameter B fulfil Bk+1 = exp(6)Bk, where exp(6) & 0.25%.
The representation has been done in the plane ln f (C)C because the curves of this family are close to straight lines. In fact, since J(C, Vm)rs ^ VT and J & CJL,1sun for the lower concentration values, the function in equation (13.15) coincides, for those values, with the straight line:
C VT
/(С) = В exp(C/C0) ^ in /(C) = ln В—————— where C0 =—————— —.
Co Jl, lsunr s
(13.16)
For the cell parameters in this example, C0 = 475.
Equation (13.16) suggests using the approximation J & CJL,1sun for evaluating the exponential concentration EC(C, V) for any concentration C. This
Illumination 
Vm (mV) 
VC (%) 

Uniform C = 1490 
1043 
25.4 

Uniform C = 2000 
1017 
24.9 

Uniform C = 4000 
945 
22.9 

Uniform C = 4500 
926 
22.3 

f (C) in figure 13.5, 
exact EC 
1014 
24.8 
f (C) in figure 13.5, 
approx EC 
1001 
24.5 
Table 13.1. Comparison for the selected example of the simulation at the maximum power point for calculations done with the models presented and uniform illuminations 
leads to very simple expressions. For instance:
which is independent of V. This approximation is accurate for low C and low V, and its effect in the IV characteristic of the cell (equation (13.8)) is that it is accurate close to shortcircuit operation and very inaccurate close to opencircuit operation.
Near the maximum power point, since the function J (C, V) will be intralinear with respect to C, this approximation is pessimistic in the sense J & C JL, isun will overestimate EC. This overestimation can be also seen in figure 13.6, where the isotopes of this approximation would be the straight lines given by equation (13.16), which are the tangents lines, at low concentration, to the isotopes shown in that figure.
As (EC(C, V)) will also be independent of V, using the apparent voltage drop defined in equation (13.9), the series resistance effect with this approximation is equivalent to a constant voltage drop of
For the example under analysis, Vs = 114 mV.
The accuracy of this approximation for calculating the maximum efficiency has not been studied in general but only for the selected example. Table 13.1 shows the comparison between the model with and without the approximation of EC, along with the calculation for several uniform illuminations with remarkable values in the local concentration distribution ((C) and the two peaks in f (C) in figure 13.5).
From the comparison in table 13.1, it can be deduced that the illumination in figure 13.5 is similar to a uniform illumination of 2000 suns. This implies that a concentration over 3500 suns, which has only 2.1% of the incident power, a
dominant concentration at 4000 and a maximum of 4500 suns does not degrade the cell performance. The question is: how much area or concentration in a certain area is needed to dominate the series resistance degradation?
Let us finalize this section answering this question with a simple analysis, using the approximated model of equation (13.17). Consider a concentration distribution which only takes two values, CMAX and CMiN, each one occupying the areas Amax and AmIN, respectively. Then, the highest concentration level will dominate when ECMAXAmax > ECMINAmIN. For instance, the highest concentration will produce 90% of the value of (EC) when
EC(Cmax)Amax ^ 90 ^
EC(Cmin)Amin ~ І0 ( j
which leads to
( 9 Amin Vt
Cmax > Cmin + О, ІП —— where С0 = 1 . (13.20)
V Amax / JL, 1sunA
For C0 = 475, Cmin = 1490, Amax = 0.1 Ac and Amin = 0.9Ac, we get CMAX > 3580. Again, note that the light power impinging on the area Amax is only 0.1*3580/(0.1*3580 + 0.9*1490) = 21.1% of the total incident power on the cell. As the approximation is pessimistic, the exact calculation with equation (13.10) would show that the threshold is greater than 3580 suns.
Note that all the results in this section are independent of the function J(x, y), depending only on f (C). This means that the performance of the cell will depend on the local concentration values but not on the spatial position where each value is produced. The reason for this property is clear: this cell model is completely symmetric with respect to the differential unit cells. As an example, if the irradiance distribution is translated inside the cell active area, the cell’s IV characteristic will remain unchanged. Of course, this does not happen in cells with nonnegligible gridline series resistance, which have been excluded from this model.