1. Introduction

The angular transmittance function T(θ,φ) of a CPV module is defined as the ratio between the output power P(θ,φ) for a given illumination direction (θ,φ) relative to the maximum output power P(0,0) at the best module alignment with respect to the light source. The angular transmittance function depends not only on the module properties but also on the characteristics of the light source impinging the CPV module, i.e., angular distribution and spectrum. Regarding the angular distribution, the angular transmittance T(θ,φ) can be theoretically expressed as the convolution of the light source angular distribution S(θ,φ) and the impulse-response angular transmittance H(θ,φ) of the module [1]. The impulse-response angular transmittance H(θ,φ) corresponds to the angular transmittance of a module when it is illuminated by a beam of parallel rays with zero angular spread (as given by a Dirac delta (δ) function).

The angular transmittance T(θ,φ) can be obtained by direct methods, (i.e., direct illumination of the module aperture) by measuring the module under the Sun or in a flash solar simulator [2], and recording the maximum power at different angular alignments of the module to the light source. As an alternative to direct methods, one can study the transferred flux when the concentrator is illuminated from the output by indirect methods [3–5]. Specifically, the luminescence inverse (LI) method was proposed [6] to provide the 2-dimensional (2D) impulse-response angular transmittance H(θ,φ) of a CPV module at high resolution, without the need for mechanical module rotation.

In the LI method, the solar cells of the CPV module are forward biased such that their electroluminescence forms a spatially uniform and Lambertian light source located at the exit of the concentrator optics [7]. The direct current values recommended for the luminescence method correspond to 10% of the value of the working photocurrent at which the CPV module is designed to perform best [8]. The light, traveling in the inverse direction (from concentrator exit to entrance) is emitted from the module in a partially collimated beam whose angular distribution will have the form of H(θ,φ). Finally, a large parabolic mirror or other concentrating element can be used to focus the light emission from the module onto a Lambertian target [Fig. 1(a)].The emission reaching the Lambertian surface is equivalent to the impulse-response angular transmittance function at the exit of the module, with the difference that the function is transposed onto spatial axes [Fig. 1(b)]. A CCD camera can be used to record this spatial distribution and thus the impulse-response angular transmittance H(θ,φ) can be obtained. A 2D convolution of H(θ,φ) with the angular distribution of a light source S(θ,φ), for instance, that of the sun, is applied to obtain the 2D angular transmittance T(θ,φ) [Fig. 1(c)]. The 1-dimensional (1D) angular transmittance T(θ) is obtained through the intersection of a given meridian plane of azimuthal angle φ with the 2-dimensional (2D) transmittance T(θ,φ) [Fig. 1(d)] . The meridian plane that defines the 1D function is related to the path the light follows when measuring with direct methods.

 

Fig. 1 (a) Measurement diagram of the luminescence inverse method (b) The irradiance map at the Lambertian target (c) Convolution between the impulse response H(θ,φ) and the light source S(θ,φ) (d) The 1D angular transmittance T(θ) definition.

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The basis of the LI method and its application to concentrators composed of a single optics-cell unit were presented in previous discussion [9]. This article reviews two additional aspects of the LI method: the influence of the quality of the cell emission on the accuracy of the results, and the methodology for composing the global angular transmission function of the module starting from the individual measurements of every single unit (i.e., optical system and solar cell). Additionally, the LI method capabilities for measuring the acceptance angle are shown for 5 different CPV technologies. Direct measurement of the transmission function at a solar simulator is used as reference for comparison.

2. Robustness of the luminescence inverse method against non-idealities of the cell emission

The excess carriers injected into a solar cell by forward biasing are preferentially recombined at the lowest energy gap between the valence band and the conduction band. If this minimum gap is direct, then the photon energies of the emitted radiation are narrowly distributed around the energy of that bandgap. The materials for top and middle junctions of the state of the art three junction (3J) lattice matched solar cells (GaInP/GaInAs/Ge) have direct gaps, and their peak emissions are at 680 nm and 890 nm respectively. The angular transmission measured by the LI method might be incorrect if the cell emission is not Lambertian and uniform in the whole area, thus how the cell emission affects the measurement must be studied.

To evaluate all possible optical paths that rays can follow when coming from the Sun to the module, the cell emission must be Lambertian at least in the angular range needed to cover all those paths. This angular range is given by the rim angle of the light beam impinging on the cell when the optical system is illuminated by the Sun. The radiant intensity curves of common 3J solar cells at different incident angles have been analyzed by using a CCD camera, confirming that the emission is close to Lambertian [Fig. 2].

 

Fig. 2 Lambertian emission vs. GaInP/GaInAs/Ge cell emission.

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To cover all possible optical paths with the same intensity level, the emission of the solar cell must be free of defects (i.e., spatially uniform). Different mechanisms such as impurity density variation, alloy clustering, or interface roughness in heterostructures can vary the electronic properties across the cell, causing non-uniform electroluminescence emission [10], [11].To evaluate how the cell emission is, electroluminescence maps of a number of GaInP/GaInAs/Ge cells (10mmX10mm) have been measured using a CCD camera with a Si sensor, and the addition of high/low pass filters to analyze separately the emission from the top and middle subcells. Two cells were selected from those measured as representing very uniform emission (Cell 1) and very non-uniform emission (Cell 2), and the results measured emission maps of these two cells are presented in Fig. 3(a).

 

Fig. 3 (a) Emission maps for top and middle subcells of Cell 1 and Cell 2 (b) Angular transmittance for the same optical system measured by LI method with Cell 1 vs. Cell 2.

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To validate the LI method with non-optimum cell emission, these two cells shown in Fig. 3(a) were both placed in the focal plane of the same optical system and the angular transmission of the concentrator thus formed was measured using the LI method. In Fig. 3(b) it may be seen that the differences between measured angular transmission functions corresponding to uniform vs. non-uniform light emission from the cell are very small (error RMSE = 0.03). These differences may be negligible when measuring the angular transmission function of a module composed by several optics-cell units as will be shown in next section, because maximum differences between the angular transmission functions of the units [Fig. 5(a)] are usually higher than the measured differences regarding the cell emission.

3. Optical characterization of CPV modules using the LI method

To obtain the angular transmission function of a concentrator, the monochromatic flux transferred by the optics or the concentrator electrical performance (i.e., P_mp or I_sc) are studied in the the LI and direct method respectively. In both methods, the angular transmission of the CPV module depends on the performance of every unit in the module according to their electrical connections. A case study based on a module composed by 6 optical system-3J cell units connected in series is analyzed in this section. Each of the 6 optical units is composed by a Fresnel lens as primary optical system (POE) and a glass homogenizer [12] as secondary optical system (SOE), and each unit has a bypass diode for reverse current protection.

The I-V curves of the module case study measured at different light source angular misalignments can be seen in Fig. 4 (a). When the module is biased at a point (i,v), only those units whose photocurrent I_L is greater than the current i will be forward biased. The units generating less current than I_L will be in reverse biased (i.e., the bypass diode is forward biased) and therefore will take away power from the module. The characteristic curve of the module satisfies the following equation:

 

Fig. 4 (a) I-V curves of the CPV module deviated a given direction related to the light source. (b) Angular transmittance measured at the solar simulator recording Pmp and Isc.

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where _k is the cell index, V_c is the contribution of each forward biased cell, and V_d is the bypass diode contribution. The number of forward biased units in the module varies with the bias point of the module, increasing with voltage. At the module maximum power point, the module current Imp is related to the worst illuminated unit in the module. At module short circuit point, the module current Isc value corresponds to the current generated by the best illuminated cell, if the sum of the voltage of the forward biased bypass diodes of the rest of units is lower than the voltage at maximum power point Vmp of the best illuminated unit [13].

The 1D angular transmittance curves with I_sc and P_mp as output signal (i.e., T_Pmp(θ) and T_Isc(θ)) are presented in Fig. 4(b). The most significant function for a CPV module is that for the P_mp, which provides data of module power as a function of misalignment in real operation.

In the LI method, the 2D function T(θ,φ) of the module can be reconstructed as a composition of the 2D angular transmission function of every single unit. Electroluminescent emission varies from one solar cell to the next; Although a perfect solar cell would be a perfect emitter via the principle of detailed balance, it is not true that all good absorbers are good emitters (i.e., it would depend on non-radiative and radiative recombination rates) [14]. Thus, the curve measured for each unit is normalized to its integrated or maximum value so that only relative differences on the emissions as function of the angles are considered. In performing this operation, we must also assume that the following parameters are constant for all units across the module:

To calculate the angular transmission of the CPV module, the electrical connection scheme between optics-cell units has to be considered. For the case study module with 6 units series connected, the unit that emits with lower intensity at a given angular position can be assumed to be the one with lower photo-generated current at that angular position and therefore the one defining the Imp. Therefore, the equivalent power transmission of the module T_Pmp(θ,φ) calculated by the inverse method is the lower envelope of the transmission curves T(θ,φ) of every unit in the module. At the same time, the equivalent short circuit current transmission of the module is calculated as the upper envelope of all curves T(θ,φ), which corresponds to the best illuminated unit an each particular angular position. In Fig. 5(a) the 1D T(θ) curves corresponding to all single units in the module, and the equivalent P_mp (i.e., minimum curves envelope) and I_sc (i.e., maximum curves envelope) transmission curves measured by the LI method are presented. The curves obtained by the LI method are compared with the ones from solar simulator with a good agreement as shows Fig. 5(b). Table 1 summarizes the acceptance angles measured for 5 different CPV technologies (included the module case study of Fig. 5). The maximum relative error between results from direct (i.e., solar simulator) and indirect measurements (i.e., SLI) is 2.44% (related to the module case study).

 

Fig. 5 (a). Angular transmittance curves for every single unit in the module from LI method (b). Angular transmittance curves from LI and solar simulator.

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Table 1. Acceptance angles measured by direct method (AA_D) and indirect (AA_I) methods for five different CPV technologies, and the relative error (AA_D-AA_I)/AA_D

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It must be pointed out that the angular transmission curves calculated by the lower and upper envelopes in the LI method do not corresponds always to the transmission curves calculated by direct method when recording Pmp or Isc values. For example, the power output could not be given by the worst illuminated cell if its photocurrent is much lower than the others so the transmission function given by the LI method would be slightly narrower compared to the module performance; also the short circuit current of the module does not correspond exactly to the photo-generated current of the best single unit when many units are connected in series because one single unit cannot compensate the rest of the forward biased bypass diodes which leads to a transmission function slightly wider.

3. Conclusions

We have evaluated the requirements of the cell emission (i.e., Lambertian and spatially uniform) to validate the viability of the LI method. The solar cell emission has been measured to be Lambertian, and it has been shown that the angular transmission measured using the LI method is almost un-affected even when solar cells have non-uniform light emission.

The LI method provides useful information not only for single optics-cell units but also for CPV modules. We have presented a method for calculating CPV module performance from the individual optics-cell unit measurements, and discussed the assumptions implicit in this method. CPV module-level results using this method have been compared with results from conventional methods (e.g., solar simulator) for 5 different CPV technologies, and they have shown good agreement. Therefor the LI method is a good candidate for production line testing of CPV module acceptance angle characteristics.