Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Westgate, Timothy M.
Master of Applied Science - MASc
University of British Columbia
Governments around the world are striving to develop clean energy solutions to lessen their impact on the environment while addressing the growing need for energy generation. Solar technology has emerged as a major contender and will undoubtedly play a role in this endeavour. However, solar technology has been hampered by its fundamental challenge of cost versus performance. The proposed work recognizes this challenge and introduces a corner-cube-cell capable of being uniformly arrayed for scalability. It is shown that the corner-cube-cell facilitates low-cost implementations, due to its compatibility with standard silicon solar cells, and enables improved performance. Performance improvements shown in this work exhibit an increase in generated electrical power density of up to 3.80% at noon for an 8.29% increase in generated electrical energy density by the end of the day. This improvement comes about due to the geometry of the corner-cube-cell, which captures internal reflections and can exploit multi-channel maximum power point tracking. The corner-cube-cell also shows substantial improvements in generated electrical energy density during sun-rise and sun-set due to its protrusion out of the two-dimensional (horizontal) plane. These performance improvements result in a 6.10% increase in semi-annual generated electrical energy density over a conventional flat-lying solar cell. The introduced technology has strong prospects for realizing low-cost and high-performance solar power system implementations.
A Novel Photovoltaic Device for Improvements in Solar Energy Collection by Timothy M. Westgate B.A.Sc., The University of British Columbia, Canada, 2014 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Electrical Engineering) The University of British Columbia (Okanagan) June 2019 ©Timothy Westgate, 2019 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis entitled: A Novel Photovoltaic Device for Improvements in Solar Energy Collection submitted by Timothy M. Westgate in partial fulfillment of the requirements of the degree of Master of Applied Science . Dr. Jonathan F. Holzman Supervisor Dr. Wilson Eberle Supervisory Committee Member Dr. Liwei Wang Supervisory Committee Member Dr. Jake Bobowski University Examiner iii Abstract Governments around the world are striving to develop clean energy solutions to lessen their impact on the environment while addressing the growing need for energy generation. Solar technology has emerged as a major contender and will undoubtedly play a role in this endeavour. However, solar technology has been hampered by its fundamental challenge of cost versus performance. The proposed work recognizes this challenge and introduces a corner-cube-cell capable of being uniformly arrayed for scalability. It is shown that the corner-cube-cell facilitates low-cost implementations, due to its compatibility with standard silicon solar cells, and enables improved performance. Performance improvements shown in this work exhibit an increase in generated electrical power density of up to 3.80% at noon for an 8.29% increase in generated electrical energy density by the end of the day. This improvement comes about due to the geometry of the corner-cube-cell, which captures internal reflections and can exploit multi-channel maximum power point tracking. The corner-cube-cell also shows substantial improvements in generated electrical energy density during sun-rise and sun-set due to its protrusion out of the two-dimensional (horizontal) plane. These performance improvements result in a 6.10% increase in semi-annual generated electrical energy density over a conventional flat-lying solar cell. The introduced technology has strong prospects for realizing low-cost and high-performance solar power system implementations. iv Lay Summary Governments around the world are striving to develop clean energy solutions to lessen their impact on the environment while simultaneously addressing the increase in global energy demand. Solar technology has emerged as a major contender; however, it has also been hampered by its fundamental challenge of cost versus performance. The proposed work recognizes this challenge and introduces a corner-cube-cell capable of being uniformly arrayed for scalability. It is shown that the corner-cube-cell facilitates low-cost implementations, due to its compatibility with standard silicon solar cells, and enables improved performance. This improved performance manifests as a minimum semi-annual increase in generated electrical energy per square meter of 6.10%. The annual increase is due to daily improvements of up to 8.29% with corner-cube-cell implementations over regular flat-lying solar cell implementations. The introduced technology has strong prospects for realizing low-cost and high-performance solar power system implementations. v Preface The theory developed in Chapter 3 of this work has been partially published in a conference proceeding. Westgate, T. M., et al. "A smart solar energy collecting device." Physics, Simulation, and Photonic Engineering of Photovoltaic Devices VII. Vol.10527. International Society for Optics and Photonics, 2018. I developed the simulation that is presented in the conference proceeding and wrote the manuscript. The manuscript was reviewed by Holzman, J. F. and Boivin, A. B. The concept was that of Holzman, J. F. Iterations of the device were developed by Veerman, B. W. D., Bergen, M. H., Jin, X., Born, B., and Bernier, M. vi Table of Contents Abstract ........................................................................................................................................................ iii Lay Summary ............................................................................................................................................... iv Preface .......................................................................................................................................................... v Table of Contents ......................................................................................................................................... vi List of Tables ............................................................................................................................................... ix List of Figures ............................................................................................................................................... x List of Abbreviations ............................................................................................................................... xxiii Acknowledgements .................................................................................................................................. xxiv Dedication ................................................................................................................................................. xxv Chapter 1 Introduction .................................................................................................................................. 1 1.1. Motivation .......................................................................................................................................... 2 1.2. Background ........................................................................................................................................ 3 1.3. Scope of this Thesis ........................................................................................................................... 7 Chapter 2 The Solar Cell (Preliminaries) ...................................................................................................... 9 2.1. Background ...................................................................................................................................... 10 2.2. Optical and Electrical Properties of Semiconductors ....................................................................... 14 2.2.1. Optical Properties of Semiconductors ....................................................................................... 15 2.2.2. Electrical Properties of Semiconductors ................................................................................... 22 2.3. Conversion of a Semiconductor into a Solar Cell ............................................................................ 25 vii 2.4. Optical and Electrical Performance Characteristics of a Solar Cell................................................. 27 2.4.1. Optical Performance Characteristics of a Solar Cell ................................................................. 28 2.4.2. Electrical Performance Characteristics of a Solar Cell ............................................................. 34 2.4.3. Practical Considerations ............................................................................................................ 43 2.5. Summary .......................................................................................................................................... 47 Chapter 3 The Corner-Cube-Cell ................................................................................................................ 48 3.1. Background ...................................................................................................................................... 50 3.2. Optical Performance Characteristics of a CC-cell ........................................................................... 52 3.2.1. Illumination Cases within Octant I ........................................................................................... 59 3.2.2. Illumination Cases within Octant II .......................................................................................... 62 3.2.3. Illumination Cases within Octant III ......................................................................................... 74 3.2.4. Illumination Cases within Octant IV......................................................................................... 76 3.2.5. Illumination Cases within Octant V .......................................................................................... 88 3.2.6. Illumination Cases within Octant VI....................................................................................... 100 3.2.7. Illumination Cases within Octant VII ..................................................................................... 102 3.2.8. Illumination Cases within Octant VIII .................................................................................... 103 3.2.9. Summary of Illumination Cases .............................................................................................. 105 3.3. Electrical Performance Characteristics of a CC-cell ...................................................................... 113 3.4. Experimental Implementation of a CC-cell ................................................................................... 130 3.5. Summary ........................................................................................................................................ 142 Chapter 4 Conclusion ................................................................................................................................ 145 viii 4.1. Summary ........................................................................................................................................ 146 4.2. Future Work ................................................................................................................................... 147 Bibliography ............................................................................................................................................. 148 Appendices ................................................................................................................................................ 152 Appendix A ........................................................................................................................................... 152 Appendix B ........................................................................................................................................... 159 Appendix C ........................................................................................................................................... 161 ix List of Tables Table 3.1. Summary of optical performance characteristics of a CC-cell within octant I. Each captured incident optical power density expression is a function of the directional cosines and specific illumination case condition. The captured incident optical power densities in column four make use of the normal(,,), min(,,), mid(,,), and max(,,) functions. ............................................................................................................. 60 Table 3.2. Summary of the captured incident optical power density of SC-x for each illumination case within octants I to VIII. ................................................................................................ 106 Table 3.3. Summary of all possible captured incident optical power densities generalized for any solar cell family given the specific illumination cases. Each captured incident optical power density is a function of a specific group of illumination case conditions as well as the solar cell family. The captured incident optical power densities in column four are a function of the normal(,,), min(,,), mid(,,), and max(,,) functions. These functions find the normal, minimum-valued, middle-valued, and maximum-valued directional cosine components associated with the illumination case condition, respectively. ......................................................................................................................... 109 x List of Figures Figure 1.1. Modelled and physical representations of the (a) four-sided pyramid,27 (b) solar tower,31 and (c) tubular solar29 structures. ............................................................................................. 6 Figure 2.1. The current-voltage curve of an ideal diode (on the left, with operation in quadrants I, II, III, and IV) and its corresponding circuit symbol (on the right). ........................................... 10 Figure 2.2. The current-voltage curve of an ideal photodiode (on the left) and its symbol (on the right). Quadrant IV of the current-voltage curve has the photodiode generate light-induced electrical power, i.e., drive current out of its positive terminal, and so it is within this quadrant that the photodiode acts as a solar cell. The symbol on the right portion of the figure is simply a diode with two light rays being incident upon it. ................................ 12 Figure 2.3. Plane of incidence for light incident upon an interface. The top medium has a complex refractive index of ñ1(); the bottom medium has a complex refractive index value of ñ2(). Light is incident upon the interface at an incident angle of , as defined off the interface normal (represented by the dashed black line). The light then reflects off of the interface at an angle of reflection, r, being equal to the incident angle. ......................... 16 Figure 2.4. Refractive index, n(), (blue curve) and extinction coefficient, (), (red curve) of Si as a function of wavelength.43 .................................................................................................... 17 Figure 2.5. Experimental setup that is used to measure the reflectance of a sample. A top view is seen in (a); a side profile is seen in (b). The laser diode directs light through an optical filter. The light is then modulated by a chopper blade at a 1-kHz modulation rate. The light then travels through two polarizers to impart the desired s- or p-polarization state. The optically filtered, modulated, and polarized light is then directed off the sample’s surface and onto a photodetector (Thorlabs DET36A/M) to produce a photocurrent signal. The photocurrent signal is passed through a bandpass filter that passes 300 Hz xi to 3 kHz and is amplified by a current preamplifier (Stanford Research Systems SR570). The current preamplifier outputs a sufficiently large voltage signal that is displayed on an oscilloscope (Tektronics TDS 2024B). ............................................................................. 19 Figure 2.6. Reflectance of an air-Si interface for a 532-nm beam with the Si surface being non-textured. The measured reflectance, R(i), is shown as a function of the incident angle, i, for s-polarized (solid blue), p-polarized (solid red), and unpolarized (solid green) light. The theoretical Fresnel reflectance is shown for the three states of polarization as dashed curves in the respective colours. ................................................................................ 21 Figure 2.7. Photoconduction within a semiconductor’s crystal lattice. The photon (red arrow) in (a) is absorbed by an electron, freeing an electron (solid circle) and leaving a hole (hollow circle) as shown in (b). ........................................................................................................... 22 Figure 2.8. The five main stages of forming a Si solar cell. (a) The first stage has a Si wafer that has been diced and made ready for processing. (b) The second stage follows from chemical etching of the wafer’s surface. (c) The third stage follows from the diffusion of dopants into the wafer, with the p- and n- type dopants shown here in red and blue, respectively. (d) The fourth stage follows from the deposition of film of SiNx:H that acts as an AR coating and passivates the surface. (e) The fifth stage follows from the deposition of silver and aluminum contacts on the top and bottom surfaces of the wafer, respectively. ................................................................................................................................................ 25 Figure 2.9. The AM 0 (solid red curve) and AM 1.5 (solid blue curve) solar spectral irradiance and the 5777-K blackbody spectral irradiance (solid black curve) shown normalized as a function of the wavelength, . The O3 and H2O molecular absorption bands are labeled in (a) and the spectrum up to the bandgap energy of Si is shown in (b). This air mass data is from the National Renewable Energy Laboratory (NREL).48 .................................... 30 xii Figure 2.10. Coordinates and angles in the Global Coordinate Frame. The frame has primed coordinates, x’, y’, and z’, where the x’-axis faces South, the y’-axis faces East, and the z’-axis faces up. Incident light (red arrow) is defined within the Global Coordinate Frame by its solar azimuth angle, , and solar zenith angle, . ............................................. 31 Figure 2.11. Captured incident optical power density of a textured Si solar cell and non-textured Si wafer as a function of the time of day. The textured Si solar cell is assumed to have a reflectance of zero; the non-textured-Si wafer is assumed to have a reflectance defined by equation (2.4). The STC incident optical power density is 1000 W/m2, and the results are shown for the summer solstice, June 21, at a latitude of 23.45°N, for which the sun is directly overhead at noon. .................................................................................................. 33 Figure 2.12. Ultrafast pump-probe transmittance setup. A top view is seen in (a), and a side profile is seen in (b). The setup has a pump laser beam (in red) and probe laser beam (in yellow). The beams are redirected with mirrors, overlapped with a beamsplitter, and focused onto the sample by a microscope objective. The probe laser beam having transmitted through the sample is re-collimated by a microscope objective, focused by a lens, and measured by a photodetector. ............................................................................... 37 Figure 2.13. Normalized transmittance of the probe laser beam through the textured Si solar cell and non-textured Si wafer as a function of the pump-probe delay time, t. The pump laser pulse fluence is 345.8 µJ/cm2 and its arrival time is 50 ps. Results for the textured Si solar cell and non-textured Si wafer are shown as purple and orange curves, respectively, with SEM images of the two samples’ surfaces shown in the insets. ............... 38 Figure 2.14. Simulated I-V and P-V curves of the XS156B3-210R+ series solar cell. In (a), the simulated I-V (solid blue) curve is shown along with datasheet values for the STC current and voltage. The open-circuit voltage indicated on the I-V curve differs from the datasheet value due to the approximation used to obtain Isat. The Maximum Power xiii Point (MPP) lies at the knee of the I-V curve. In (b), the simulated (solid green) P-V curve is shown. This power curve clearly shows the MPP at its local maximum. ................. 41 Figure 2.15. Reflectance of a protected-and-textured Si solar cell for a 532-nm beam. A glass layer is bonded onto the Si solar cell via NOA 60 to protect its surface. The measured reflectance, R(i), is shown as a function of the incident angle, i, for s-polarized (solid blue), p-polarized (solid red), and unpolarized (solid green) light. The theoretical Fresnel reflectance is shown for the three states of polarization as dashed curves in the respective colours. .................................................................................................................. 45 Figure 2.16. Generated electrical power density of a protected-and-textured Si solar cell (in blue) and an unprotected-and-textured Si solar cell (in red), with the protective glass layer having been bonded with NOA 60. The generated electrical power density of the protected-and-textured Si solar cell is approximately 5% less than that of the unprotected-and-textured Si solar cell at noon. The solar cell used in this experimentally-seeded simulation has an optical to electrical conversion efficiency of 26.3%. ................................ 46 Figure 3.1. The CC-cells arranged in a close-packed array. The array is shown as (a) a photograph having textured-and-protected Si solar cells, with a side length of 7.9 cm, and (b) a schematic having its Si solar cells shaded in colours that correspond to three solar cell families: the SC-x family is in green, the SC-y family is in blue, and the SC-z family is in red. The Global Coordinate Frame, on the left side of the array, shows an incident light ray at a solar zenith angle of θ and solar azimuth angle of ϕ split into its directional cosine components of α', β', and γ'. The Body Coordinate Frame, on the right side of the array, shows the directional cosine components of the incident light ray transformed to the Global Coordinate Frame as α, β, and γ. .......................................................................... 53 Figure 3.2. Depiction of the eight octants within the Global Coordinate Frame. All of the octants are depicted in order from octant I in (a) to octant VIII in (h). .................................................... 58 xiv Figure 3.3. Depiction of octant I, which is enclosed by the positive x-axis, positive y-axis, and positive z-axis. ........................................................................................................................ 59 Figure 3.4. Depiction of octant II, which is enclosed by the negative x-axis, positive y-axis, and positive z-axis. ........................................................................................................................ 62 Figure 3.5. Depiction of a CC-cell illuminated by light incident from octant II given the cases –α>β>γ>0 and –α>γ>β>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The light rays in (a) have a larger directional cosine along the y-axis, β, which causes SC-z to be illuminated more than SC-y. The light rays in (b) have a large directional cosine along the z-axis, γ, which causes SC-y to be illuminated more than SC-z. ............................................................................................................................... 63 Figure 3.6. Depiction of a CC-cell being illuminated by light incident from octant II given the illumination cases β>–α>γ>0 and β>γ>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). .......................................................................... 66 Figure 3.7. Depiction of a CC-cell being illuminated by light incident from octant II given the illumination cases γ>–α>β>0 and γ>β>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). .......................................................................... 70 Figure 3.8. Depiction of octant III, which is enclosed by the negative x-axis, negative y-axis, and positive z-axis. ........................................................................................................................ 74 Figure 3.9. Depiction of a CC-cell being illuminated by light incident from octant III given the illumination cases γ>–α>–β>0 and γ>–β>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. ..................................................................................... 75 xv Figure 3.10. Depiction of octant IV, which is enclosed by the positive x-axis, negative y-axis, and positive z-axis. ........................................................................................................................ 76 Figure 3.11. Depiction of a CC-cell being illuminated by light incident from octant VI given the illumination cases α>–β>γ>0 and α>γ>–β>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). .......................................................................... 78 Figure 3.12. Depiction of a CC-cell being illuminated by light incident from octant IV given the cases –β>α>γ>0 and –β>γ>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The light ray in (a) has a larger projection along the x-axis due to its larger α directional cosine, which causes SC-z to be illuminated more than SC-x. The light ray in (b) has a larger projection along the z-axis due to its larger γ directional cosine, which causes SC-x to be illuminated more than SC-z. .............................................. 82 Figure 3.13. Depiction of a CC-cell being illuminated by light incident from octant IV given the illumination cases γ>α>–β>0 and γ>–β>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). .......................................................................... 85 Figure 3.14. Depiction of octant V, which is enclosed by the positive x-axis, positive y-axis, and negative z-axis. ....................................................................................................................... 88 Figure 3.15. Depiction of a CC-cell being illuminated by light incident from octant V given the illumination cases α>β>–γ>0 and α>–γ>β>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). .......................................................................... 90 xvi Figure 3.16. Depiction of a CC-cell being illuminated by light incident from octant V given the illumination cases β>α>–γ>0 and β>–γ>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). .......................................................................... 95 Figure 3.17. Depiction of a CC-cell being illuminated by light incident from octant V given the cases –γ>α>β>0 and –γ>β>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The light ray in (a) has a larger projection along the x-axis due to its larger α directional cosine, which causes SC-y to be illuminated more than SC-x. The light ray in (b) has a larger projection along the y-axis due to its larger β directional cosine, which causes SC-x to be illuminated more than SC-y. .............................................. 98 Figure 3.18. Depiction of octant VI, which is enclosed by the negative x-axis, positive y-axis, and negative z-axis. ..................................................................................................................... 100 Figure 3.19. Depiction of a CC-cell being illuminated by light incident from octant VI given the illumination cases β>–α>–γ>0 and β>–γ>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. ................................................................................... 101 Figure 3.20. Depiction of octant VII, which is enclosed by the negative x-axis, negative y-axis, and negative z-axis. ..................................................................................................................... 102 Figure 3.21. Depiction of octant VIII, which is enclosed by the positive x-axis, negative y-axis, and negative z-axis. ..................................................................................................................... 103 Figure 3.22. Depiction of a CC-cell being illuminated by light incident from octant VIII given the illumination cases γ>–α>–β>0 and γ>–β>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. ................................................................................... 104 Figure 3.23. The percent improvement in captured incident optical power density of the CC-cell compared to that of the flat-cell. This comparison uses the reflectance data gathered in xvii Chapter 2 for a protected-and-textured Si solar cell, the equations in Table 3.3, and the solar zenith and solar azimuth angles varied over (most) possible angles, i.e., 0°< θ <85° and -180°< ϕ <180°. The results suggest a median improvement in captured incident optical power density of roughly 3.88%, across all the displayed angles, for the CC-cell in comparison to the flat-cell over all chosen angles. .......................................................... 112 Figure 3.24. Three representative views of the CC-cell in an arrayed implementation for the first deployment, which illuminates the CC-cell with a high level of symmetry. The CC-cell array is viewed along the direction of illumination for (a) a view of the CC-cell array at t1 = 06:22, corresponding to illumination along  = 76.00° and  = 109.80°, (b) a view of the CC-cell array at t2 = 12:00, corresponding to illumination along  = 0.00° and  = 0.00°, and (c) a view of the CC-cell array at t3 = 14:22, corresponding to illumination along  = 32.48° and  = -97.25°. ........................................................................................ 116 Figure 3.25. Generated electrical power density and generated electrical energy density of the flat-cell and CC-cell versus time of day for the first deployment, which illuminates the CC-cell with a high level of symmetry. Results are shown for (a) the generated electrical power density of the flat-cell and the SC-x, SC-y, and SC-z families in the CC-cell, and (b) the generated electrical power density and generated electrical energy density of the flat-cell and CC-cell. In (a), the results for the flat-cell are denoted by the solid black curve, while the results for the CC-cell array are denoted by the green dashed/solid curve for SC-x labeled PEL-x, the blue dashed/solid curve for SC-y labeled PEL-y, and the red dashed/solid curve for SC-z labeled PEL-z. In (b), the results for the generated electrical power density (labeled PEL-flat) and generated electrical energy density (rising to the right) of the flat-cell are denoted by solid/dashed black curves, respectively. The results for the generated electrical power density (labeled PEL-CC) and generated electrical energy density (rising to the right) of the CC-cell for the multi- and mono-xviii MPPT implementations are denoted by solid/dashed orange and purple curves, respectively. Vertical delineations are shown at times of t1 = 06:22, t2 = 12:00, and t3 = 14:22. ................................................................................................................................... 118 Figure 3.26. Three representative views of the CC-cell array for the second deployment, which has a low level of symmetry for illumination. The CC-cell array is viewed along the direction of illumination for (a) a view of the CC-cell array at t1 = 09:08, corresponding to illumination along  = 53.63° and  = 57.88°, (b) a view of the CC-cell array at t2 = 12:00, corresponding to illumination along  = 35.61° and  = 0.00°, and (c) a view of the CC-cell array at t3 = 17:07, corresponding to illumination along  = 79.47° and = -81.77°. ................................................................................................................................. 120 Figure 3.27. Generated electrical power density and generated electrical energy density of the flat-cell and CC-cell versus time of day for the second deployment, which illuminated the CC-cell with a low level of symmetry. Results are shown for (a) the generated electrical power density of the flat-cell and the SC-x, SC-y, and SC-z families in the CC-cell, and (b) the generated electrical power density and generated electrical energy density of the flat-cell and CC-cell. In (a), the results for the flat-cell are denoted by the solid black curve, while the results for the CC-cell are denoted by the green dashed/solid curve for SC-x labeled PEL-x, the blue dashed/solid curve for SC-y labeled PEL-y, and the red dashed/solid curve for SC-z labeled PEL-z. In (b), the results for the generated electrical power density (labeled PEL-flat) and generated electrical energy density (rising to the right) of the flat-cell are denoted by solid/dashed black curves, respectively. The results for the generated electrical power density (labeled PEL-CC) and generated electrical energy density (rising to the right) of the CC-cell for the multi- and mono-MPPT implementations are denoted by solid/dashed orange and purple curves, respectively. Vertical delineations are shown at times of t1 = 09:08, t2 = 12:00, and t3 = 17:07. .............. 121 xix Figure 3.28. The generated electrical energy density of the mono-MPPT flat-cell as a function of day of the year and latitude. The mono-MPPT flat-cell has an annual average generated electrical energy density of approximately 993 W·hr/m2, while considering all latitudes, with the highest performance culminating near the summer solstice at latitudes in the range of 30°-40°. .................................................................................................................. 123 Figure 3.29. The generated electrical energy density of the mono-MPPT CC-cell as a function of day of the year and latitude. The mono-MPPT CC-cell has an annual average generated electrical energy density of about 1,061 W·hr/m2, while considering all latitudes, with the highest performance culminating around the summer solstice at latitudes around 22°-60°. ................................................................................................................................ 124 Figure 3.30. The generated electrical energy density of the multi-MPPT CC-cell as a function of day of the year and latitude. The multi-MPPT CC-cell has an annual average generated electrical energy density of about 1,088 W·hr/m2, while considering all latitudes, with the highest performance culminating around the summer solstice at latitudes around 20°-66°. ................................................................................................................................ 125 Figure 3.31. Solar zenith angles at latitudes of 23°, 35°, and 66° during the summer solstice, June 21, represented as solid red, blue, and green curves, respectively. Here it is observed that the length of a day increases the further north one travels during the summer solstice. ...... 126 Figure 3.32. The summed generated electrical energy density of the mono-MPPT flat-cell, mono-MPPT CC-cell, and multi-MPPT CC-cell over the course of a year and comparisons between all three implementations for latitudes from the equator to 66°. The solid purple, orange, and black curves, linked to the left vertical axis, show the summed generated electrical energy densities of the multi-MPPT CC-cell, mono-MPPT CC-cell, and mono-MPPT flat-cell, respectively. The dashed blue, red, and green curves, linked to the right vertical axis, show the percent improvement between the multi-MPPT CC-xx cell over the mono-MPPT flat-cell, the mono-MPPT CC-cell over the mono-MPPT flat-cell, and the multi-MPPT CC-cell over the mono-MPPT CC-cell, respectively. ................ 129 Figure 3.33. A schematic of the measurement tool connected to one solar cell. The four sections of the measurement tool are outlined in four unique colours. The green section shows the summing amplifier. Its negative terminal is attached to the voltage output terminals on the NI DAQ units, Vout1 and Vout2. The voltage output of the summing amplifier is negative. The blue section shows the unity gain amplifier. This section is used to invert the negative voltage to incrementally vary the gate voltage of the n-channel MOSFET. The orange section shows the variable resistive load. The variable resistive load is made up of a power resistor bank with an equivalent resistance of 0.2 and a MOSFET, which has gate subjected to a varying voltage. The red section shows the instrumentation amplifier. This section is used to indirectly measure the current generated by the solar cell that is attached to the variable resistive load with the NI DAQ units. The current is inferred by taking the output voltage of the instrumentation amplifier, Vcur, and dividing it by 0.2 . The output voltage of the solar cell, Vpv, is measured directly with the NI DAQ units. The voltage rails, VCC and VEE, used to power all of the operational amplifiers are supplied by a DC Power Supply (BK Precision 1673). ................................................................................................................................... 132 Figure 3.34. Photographs of the setup for experimental implementation of a CC-cell showing the (a) overall setup and (b) positioning of the halogen lamp. The device under test (DUT) can be the CC-cell, as is shown, or the flat-cell. The DUT collects optical power from the halogen lamp. The measurement tool, powered by the DC power supply, and NI DAQ units collect the voltage and current data of each solar cell within the DUT. The DUT is centred on the origin of the Global Coordinate Frame and the halogen lamp is placed xxi 60 cm away at varying solar zenith () and azimuth () angles. The DUT is attached to an optical breadboard for stability. ....................................................................................... 134 Figure 3.35. Normalized spectral power density of the halogen lamp used in the experimental implementation of the CC-cell. The spectral power density of the halogen lamp is shifted to the near infrared compared to that of the solar spectral power density. ............... 135 Figure 3.36. A representative scan of the generated electrical power of the solar cell families making up the CC-cell (solid curves) and flat-cell (dashed curve). The measured values for SC-x are shown as green circles connected by a solid green curve, the measured values for SC-y are shown as blue circles connected by a solid green curve, the measured values for SC-z are shown as red circles connected by a solid red curve, and the measured average of the three solar cells making up the flat-cell is shown as black triangles connected by a dashed black curve. The halogen lamp used as the illumination source and was positioned over top of the CC-cell and flat-cell starting at an azimuth angle of 90° and a zenith angle of 75° and ending at an azimuth angle of -90° and a zenith angle of 75° (with respect to the Global Coordinate Frame). This is represented as 75°E and 75°W, respectively. .............................................................................................................. 137 Figure 3.37. The experimentally-seeded simulation performance improvements (solid black curves) and the measured performance improvements (purple circles) in generated electrical power density of each solar cell family as compared to its flat-cell counter-part. The agreement between the measured and experimentally-seeded simulation results is clear. The performance improvement of SC-x, seen in (a), shows a general increase as the illumination source sweeps from East to West. The performance improvement of SC-y, seen in (b), shows a general decrease as the illumination source sweeps from East to West. The performance improvement of SC-z, seen in (c), shows a nearly constant trend as the illumination source sweeps from East to West. The measured performance improvements are seen with error bars representing the acceptable ±1% random error xxii tolerance. The tolerance was set after taking the measurements of the system. The halogen lamp used as the illumination source was positioned over top of each structure starting with an azimuth angle of 90° and a zenith angle of 75°, i.e., 75°E, and ending with an azimuth angle of -90° and a zenith angle of 75°, i.e., 75°E, with respect to the Global Coordinate Frame. For the experimentally-seeded simulation, the solar zenith angle sweeps from 90°E to 90°W to capture the entire performance. ................................. 140 xxiii List of Abbreviations Abbreviations Definitions AM Air Mass AR Antireflection DC Direct Current GW Giga Watt MPPT Maximum Power Point Tracking PV Photovoltaic Si Silicon STC Standard Test Condition USD United States Dollar W Watt xxiv Acknowledgements There are a number of people that I would like to thank as, without their support, the work presented in this Thesis would not have been possible. I would like to thank my supervisor, Dr. Jonathan F. Holzman. It is truly his encouragement, understanding, mentorship, and patience that has made this work possible. His uncompromising professional integrity, selfless leadership, and remarkable scholarship will continue to influence me for the rest of my life. I would also like to express my gratitude to Dr. Wilson Eberle and Dr. Liwei Wang for serving as my committee members. I appreciate their time and constructive comments over the years. I would also like to give a special thanks to Dr. Jake Bobowski for his willingness to serve as my University Examiner. I wish to acknowledge my colleagues Adrian Boivin, Dr. Brandon Born, Mark Bergen, Ilija Hristovski, and Dr. Xian Jin for their expertise, assistance, insight, and friendship in and out of the laboratory. I extend my gratitude for the technical support given to me by David Arkinstall, Tim Giesbrecht, Durwin Bossy, Marc Nadeau, and Emily Zhang. I would also like to give thanks to the School of Engineering for providing research facilities and funding opportunities to carry out my studies. Finally, thanks are owed to my parents, Tim and Jackie Westgate. I continue to follow my dreams because of your love, support, and encouragement. xxv Dedication To my mother for all of her love and support and to my father for all of his advice and inspiration 1 Chapter 1 Introduction Solar energy has been harnessed by humans in a variety of ways throughout history. A simple example is the drying and preserving of food, which dates back to humankind’s earliest existence. Today we harness solar energy in more complicated ways, whether it is through the combustion of fossil fuels, the rotation of hydro/wind turbines, or the capture of light by solar cells. Almost all energy we use on Earth is either directly or indirectly solar energy. Technology capable of harvesting solar energy and converting it into electrical energy has existed for decades. In 1941, Bell Laboratories was the first to describe the function of a solar cell, which used silicon (Si) as its fundamental building block.1 This grand demonstration was followed by research efforts that focused on improving the conversion efficiency of Si solar cells and scaling the technology up for cost-effective commercialization. Today, Si-based solar cell technology makes up approximately 90% of the roughly 500 GW of installed solar power generation on Earth.2,3 Silicon-based solar technology not only represents the largest portion of installed capacity on Earth but also represents the largest portion of the solar energy market. Despite the large success of Si-based solar cell technology, there are still concerns surrounding its economic viability and its suitability to meet global demands for energy. 2 1.1. Motivation Conventional sources of energy generation, such as fossil fuels, will not last forever, and the ability of fossil fuels to meet the energy demands of the planet have been the subject of ongoing debate among scientists and governments.3–5 At the same time, fossil fuels are yielding great concern due to their damaging by-products, which take the form of greenhouse gases. The reduction and capture of greenhouse gases has become a major focus for governments across the globe, and particularly those participating in the Paris Climate Agreement,4 which would have its participants limit their generation of greenhouse gases. Greenhouse gases are (almost) universally understood to be the major contributor to climate change,6,7 and many governments are seeking means to transition away from fossil fuels for their energy generation. This is transforming the landscape of global energy generation. Clean energy technologies are now being eagerly pursued, and solar technology has emerged as a major contender.8 However, the growing interest in solar technology has revealed a challenge for its widespread deployment—in that its demands of cost and performance are often at odds.9,10 A solar technology that meets these demands of low cost and high performance would allow the economic development of a clean energy technology in high-cost and low-area deployment zones (i.e., high-density urban environments). Researchers across the globe now recognize this challenge and are looking for an effective solar technology to meet these concurrent demands. 3 1.2. Background An effective solar technology, as defined in this work, would offer both low cost (for its installation) and high performance (for its conversion of optical power to electrical power). The pursuit of such a technology is reflected in its metric of net cost per unit power ($/W), which has been the traditional means of comparison between different solar technologies.11 This metric not only allows solar technologies to be compared on a relative scale, amongst each other, but it also provides a means for comparison of solar power generation systems to traditional grid-tied power generation systems. For solar power generation systems, the net cost per unit power takes into consideration raw material refining, wafer production, solar cell production, module production, and installation costs,12 all of which can be targeted for cost reductions. (The costs and values stated throughout this thesis are all quoted in United States Dollars (USD).) For example, the cost of Si-based solar modules fell from $4/W in 2008 to roughly $1/W in 2012,11 due to an increase in infrastructure, but it has since remained close to 2012 levels, with negligible reductions seen in 2017. Unfortunately, and despite the dramatic decrease in module cost, the median cost of a fully-installed grid-tied residential solar power system in the United States has continued to hover at approximately $4/W over the past 4 years. Clearly, further innovation is needed to bring down the cost of the fully-installed grid-tied solar power systems if nations are to increase their reliance on solar power and meet their goals for alternative energy. The United States, for example, has a SunShot goal of reaching $1.50/W for fully-installed grid-tied residential solar power systems by 2020.13 Solar cell production is a particularly interesting field of research that has traditionally targeted the development of an effective solar technology, with both low cost and high performance, through refinements to the manufacturing process and increases to the efficiencies of solar cells. Of these two targets, increased efficiencies of the solar cells are particularly advantageous because they can support both lower installation costs and reduced footprints for the installations. This can make solar power systems attainable for low-income populations and enable installations in dense urban environments where the area is limited (mainly to rooftops). To this end, the average module efficiency of solar cells has risen from 4 12.7% in 2002 to 17.4% in 2017,11 and further increases are being eagerly pursued through developments in solar materials and structures. The first, and most common, approach to realizing an effective solar technology has focused on developments of solar materials. Examples include various forms of multi-junction,14 thin film,15 bio-,16,17 and nano-18,19 photovoltaic (PV) materials, which can promise either excellent solar conversion efficiencies or low costs. Unfortunately, these materials typically only exhibit one of these ideal traits. For this reason over 90% of solar installations to date apply conventional Si-based PV technology,2 which exhibits only average cost and performance. The second approach to realizing an effective solar technology has focused on developments of solar structures that can minimize losses due to reflected optical power. This increases the captured incident optical power density, for heightened performance, while allowing for the use of conventional Si-based PV technology, for lower costs. The solar structures resulting from this approach can be broadly categorized into microscopic and macroscopic antireflective structures. Microscopic antireflective structures have features that are comparable to the wavelength of incident light. The first and most common form of a microscopic antireflective structure is the thin film. The quarter-wavelength thin film, for example, uses interference to reduce reflective losses and increase the captured incident optical power density. However, such interference only functions over a narrowed spectrum with limited directionality.20 (The graded-index thin film is an alternative that exhibits improved directionality, although it still suffers from a narrowed spectrum.21) A second form of microscopic antireflective structure is the textured surface.22–25 The textured surface is a broadband and omni-directional solution for antireflection because it promotes light scattering over the active region of solar cells. While such a process does increase the captured incident optical power density of the solar cell, it also increases surface recombination,26 to the detriment of the electrical efficiency, and increases capital investment to the manufacturing process.12 Macroscopic antireflective structures have features that are far larger than the wavelength of incident light, and they have been the focus of recent attention. They are similar to their microscopic 5 counterparts in that they control reflections to increase the captured incident optical power density. However, they differ from their microscopic counterparts in that their large physical features can be realized in virtually any three-dimensional form. One particularly innovative solar structure, put forward by Xia et al., is based upon four-sided inverted pyramids.27 The architecture captures internal reflections to give a 260% increase in generated electrical power density. At the same time, it can be effectively scaled up in area, to increase the generated electrical power, because four-sided inverted pyramids can be arrayed across the horizontal plane with a unity packing density. Unfortunately, from a geometrical standpoint, four-sided pyramids cannot be implemented with the square solar cells that are manufactured via conventional Si-based PV fabrication processes. Thus, the researchers implemented polymer-based PV technology that can realize the necessary (isosceles) triangular solar cells—albeit with a lower efficiency and thus a lower generated electrical power density compared to conventional Si-based PV technology. A second innovative solar structure, put forward by Bernardi et al., is the solar tower.28 The solar tower is a vertical structure that is constructed from square solar cells, making it compatible with conventional Si-based PV technology. The researchers used a genetic algorithm to orient each solar cell for maximal capture of internal reflections. The solar tower showed an increase in the captured incident optical power density of up to 20 times that of the standard flat-cell. Unfortunately, the solar tower cannot be effectively arrayed across the horizontal plane, to scale up the generated electrical power, because shadowing by neighbouring solar towers would diminish the performance. A third innovative solar structure, put forward by Yu et al., is the tubular solar cell29. The group used a thin film amorphous Si absorbing layer that was deposited onto the inner surface of a tubular glass substrate. The tubular structure had improved omni-directionality compared to a similar flat-lying amorphous Si solar cell for an improved conversion efficiency of 9.86%. This was comparable to the high-performance amorphous Si solar cell devices at that time, which exhibited conversion efficiencies of 10.2 ± 0.3%.30 The tubular structure exploits the transparency of the amorphous Si thin film by promoting light scattering throughout the structure. The tubular solar cell was found to suffer from a high reflectivity whenever incident light was at a glancing angle to its surface. To mitigate this, researchers incorporated an antireflective nanocone layer, which dramatically decreased the reflectivity at glancing angles of incidence. 6 Unfortunately, the tubular solar cell can only be realized with flexible solar cell materials or materials that can be deposited into intricate shapes. It cannot be realized with conventional Si-based PV technology. Figure 1.1. Modelled and physical representations of the (a) four-sided pyramid,27 (b) solar tower,31 and (c) tubular solar29 structures. Given the above motivations and pursuits for an effective solar technology, with both low cost and high performance, this work introduces a new macroscopic solar array structure. The solar structure can support a unity packing density and can be implemented with conventional Si-based PV technology for reduced cost. Moreover, the solar structure takes the shape of a macroscopic corner-cube-cell (CC-cell) that can capture internal reflections and minimize the reflected optical power loss. This leads to increased efficiency and improved performance. 7 1.3. Scope of this Thesis This thesis consists of four chapters that explore the theoretical and experimental characteristics of the CC-cell as a new solar structure. Chapter 1 presents the motivation and background for this study of solar energy generation. The cost and performance of conventional PV technology is discussed—with emphasis on potential means for lowering the cost or heightening the performance. Of these means, the work on macroscopic antireflective structures is highlighted, and the CC-cell is introduced as a macroscopic solar structure that has the potential to function as an effective solar technology with both low cost and high performance. Chapter 2 presents a detailed background for the optical and electrical performance characteristics of conventional Si solar cells. First, a review is given on the optical and electrical properties of semiconductors and the practical considerations of fabricating solar cells. Second, the specific characteristics of Si, including its advantages and disadvantages, are discussed. Third, the steps that must be taken to transform a semiconductor into a solar cell are discussed. Fourth, the optical and electrical performance characteristics of Si solar cells are analysed through experimentally-seeded simulation. Chapter 3 presents full-scale theoretical analyses of the proposed (Si-based) CC-cell and the conventional Si solar cell—along with experimental results from their implementation and operation. First, a general overview of the advantages of the CC-cell is discussed. Second, the optical performance characteristics of the CC-cell are derived in full detail and a comparison is made between its optical performance characteristics to those of a conventional Si solar cell. Third, the electrical performance characteristics of the CC-cell are investigated via experimentally-seeded simulations of the structure under two representative illumination conditions. The CC-cell is seen here to outperform the conventional Si solar cell in terms of its generated electrical energy density over the course of a day. Fourth, experimental results are presented from the implementation and operation of the CC-cell and conventional Si solar cell. The CC-cell is again seen to outperform the conventional Si solar cell in terms of its generated electrical energy density. 8 Chapter 4 gives a brief summary of this thesis and concludes with recommendations for future work on (and the burgeoning potential of) the proposed CC-cell technology. 9 Chapter 2 The Solar Cell (Preliminaries) In this chapter, a theoretical foundation is laid to facilitate the discussion of solar cells and their optical and electrical performance characteristics. The theoretical foundation is then experimentally tested and the data is seen to be consistent with the theory. The data is then used in experimentally-seeded, simulation-based solar cell characterizations. The upcoming sections show a logical progression from a semiconductor ingot to a protected-and-textured Si solar cell. A broad discussion covering the basics of semiconductors and their application to the PV industry is presented as a background in section 2.1. The optical and electrical properties of semiconductors are then discussed in section 2.2. A brief review of the optical properties of semiconductors is presented in the section 2.2.1, and a brief review of the electrical properties of semiconductors is presented in the section 2.2.2. From there, section 2.3 presents an overview of the steps pertaining to the transition from a semiconductor to a fully-functioning solar cell. This transition is made through the viewpoint of the PV industry, detailing the structural changes necessary to transform a semiconductor into a solar cell. Section 2.4 presents the optical and electrical performance characteristics of a conventional Si solar cell. The optical performance characteristics are discussed in section 2.4.1 and the electrical performance characteristics are discussed in section 2.4.2. Section 2.4 is then concluded by section 2.4.3, which looks into the key practical considerations that are often disregarded in solar cell efficiency calculations—but are important in the proposed work. Chapter 2 is then summarized in section 2.5 with all key pieces of preliminary information for the prior sections. 10 2.1. Background One of the most common electrical devices used in electronics is the diode. The basic function of a diode is to let current flow preferentially in one direction. However, a diode does not let current flow through it unless a specific electric potential is applied across its anode and cathode, being equivalent to its positive and negative terminals, respectively. This specific electric potential is referred to as the threshold voltage. Once the threshold voltage of a diode is crossed, the current rapidly increases with small increases to the voltage. Thus, a diode is a nonlinear and voltage-sensitive device. The current and voltage characteristics of the diode can be seen by way of the current-voltage curve in Figure 2.1. The forward-biased current and voltage relationship is seen in quadrants I and IV of Figure 2.1. Alternatively, if the polarity of the electric potential across the anode and cathode is switched, the diode prohibits current flow. This reversed-biased condition is seen on the left side of the voltage axis, in quadrants II and III of Figure 2.1. If the voltage across the terminals of a diode in a reversed-biased condition become very large, it can enter a reverse-breakdown condition. Figure 2.1. The current-voltage curve of an ideal diode (on the left, with operation in quadrants I, II, III, and IV) and its corresponding circuit symbol (on the right). 11 While in this reverse-breakdown, the diode conducts in much the same way as it does while in a forward-biased condition. A diode also exhibits an extremely small leakage current while reverse-biased. The symbol of a diode, with its direction of current flow, is shown on the right of Figure 2.1. Diodes get their unusual functionality through the use of semiconductors. Semiconductors exhibit properties that are intermediate between those of conductors, which permit substantial current flow, and insulators, which prevent current flow. In this way, a semiconductor exhibits a threshold voltage at which it can permit the flow of current. This would imply that a diode has an internal electric potential that must first be overcome. This internal electric potential is created by implanting semiconductors with dopants to form a p-n junction. The characteristics of the p-n junctions and their threshold voltages are defined by the semiconductor and its dopant concentrations. Commercially available diodes typically use Si as the semiconductor, which yields a threshold voltage of roughly 0.7 V. For typical circuit applications, diodes are encased by an opaque protective layer to shield them from contaminants and ensure longevity. However, if the protective layer were made to be transparent to visible light, the operating characteristics of the diode could be tuned. For instance, the aforementioned threshold voltage could be made a function of the optical illumination, whereby the incident photons increase the magnitude of the reverse saturation current and thus increase the threshold voltage. Such a device is referred to as a photodiode. Photodiodes have similar current and voltage characteristics to diodes. The two devices do have some minor but important differences, however, which can be seen in Figure 2.2. The main difference is the response to light in the case of photodiodes. The light-induced current of the photodiode shifts down the original (dark) current-voltage curve of the diode along the negative current axis, effectively increasing the magnitude of the reverse saturation current. Thus, the illuminated photodiode can output a current from its positive terminal. Such operation allows the photodiode to act as a source, which is in stark contrast to simple passive components for which the current enters the positive terminal. Ultimately, the photodiode can capture optical power that is incident upon its surface and generate electrical power from its terminals. This is the key principle of a solar cell! A solar cell is a (typically large) photodiode that has been engineered 12 to have broadband optical absorption with a high optical-to-electrical power conversion efficiency. Such solar cells operate in quadrant IV of Figure 2.2, because it is within this quadrant that they generate light-induced electrical power. Figure 2.2. The current-voltage curve of an ideal photodiode (on the left) and its symbol (on the right). Quadrant IV of the current-voltage curve has the photodiode generate light-induced electrical power, i.e., drive current out of its positive terminal, and so it is within this quadrant that the photodiode acts as a solar cell. The symbol on the right portion of the figure is simply a diode with two light rays being incident upon it. Economics dictates that solar cell technologies that differ in conversion efficiency and processing techniques have differing costs, and so consumers must pay a premium for higher conversion efficiency devices or emerging technologies. At present, Si-based PV technologies make up 90% of the global market.2 This use of Si for solar cells is due to three key factors. The first key factor is that Si is a readily available material found in abundance in the Earth’s crust.32–34 The second key factor is that the infrastructure and methodologies for refining feedstock Si are well established and can produce highly pure (99.9999%) Si for use by the PV industry.35 The third key factor is the well-understood, albeit only adequate in value, conversion efficiency of Si-based solar cell technology. All of these factors have led to a Si-based solar industry founded upon availability, honed processing techniques, and years of research. Such factors have also allowed Si-based solar cell technology to encroach upon its theoretical efficiency limit.36–38 13 Ultimately, a solar cell’s conversion efficiency depends upon the optical and electrical properties of the semiconductor used for its p-n junction. The upcoming section 2.2 briefly reviews such optical and electrical properties of semiconductors, with a focus on their characteristics of relevance to solar cells. 14 2.2. Optical and Electrical Properties of Semiconductors A solar cell has a maximum theoretical optical-to-electrical conversion efficiency that is dictated by the optical and electrical properties of its underlying semiconductor. In the upcoming sections, the optical and electrical properties of semiconductors are explored with an emphasis on Si due to its prevalence throughout the PV industry. The optical properties of semiconductors are discussed first in section 2.2.1. The reflectance of the surface of Si is then investigated experimentally and found to agree with values found in literature. This agreement is used as a basis to confidently simulate the reflectance of other, more complicated, structures. The electrical properties of semiconductors are discussed in section 2.2.2. Mechanisms that govern electrical inefficiencies are briefly discussed and an ideal model is used to describe the electrical functionality of a solar cell is introduced. 15 2.2.1. Optical Properties of Semiconductors Light interacts with materials through various manifestations. For instance, light will be strongly reflected off of an aluminum mirror but will be weakly reflected off of a piece of glass. The reflection is a result of the material properties that affect both the speed and absorption of light. Light travels through free-space at a speed of 299,792,458 m/s with no absorption. However, when light travels through materials with a higher optical density than free-space, with finite absorption, the speed of light (or more precisely the phase velocity) and power are decreased. The decrease in the speed of light is due to the material’s heightened refractive index, which grows above unity. The decrease in the power is due to the material’s finite extinction coefficient, which rises above zero. The refractive index and extinction coefficient can be quantified together as real and imaginary components of a complex refractive index, respectively. Knowledge of this complex refractive index allows materials to be characterized in terms of their reflectance and transmittance. (The terms reflectance and transmittance are used in the remainder of this work, rather than reflectivity and transmissivity, because the former terms can be defined for any interface while the latter terms are defined for only a perfectly flat interface.) Both components of the complex refractive index typically vary with the wavelength, , of light. Thus, the wavelength-dependent complex refractive index can be expressed as ñ(λ) = n(λ) – jκ(λ), (2.1) where n(λ) is the refractive index, and κ(λ) is the extinction coefficient. If media such as free-space and a semiconductor form an interface, we can use the complex refractive indices of the two media to determine the amount of optical power that will be reflected or transmitted at the interface. Two pieces of information are needed to do this. The first piece of information is the angle at which the light is incident upon the interface, which is called the incident angle. The second piece of information that is needed is the wavelength-dependent complex refractive indices of both media. The first piece of information can be found through simple observations of the system, while the second 16 piece of information is either found in the literature or experimentally determined. In this work, the wavelength-dependent complex refractive index is first found in the literature and then experimentally verified via measurements of reflectance. This process establishes confidence in the experimental setup and methods before characterizing more complicated interfaces. The abovementioned verification can be carried out by measuring the reflected optical power, and thus reflectance, for an air-Si interface. The light is incident from air onto Si with the incident angle being varied and monitored as an independent variable in the setup. (In the remainder of this work, the Si can be assumed to be monocrystalline, unless otherwise stated, but great care will be paid in identifying the state of texturing on its surface.) A simple depiction of the air-Si interface is seen in Figure 2.3. The figure depicts the plane of incidence, as the plane orthogonal to the interface while containing the surface normal and propagating ray vectors. A ray of light, shown as a red arrow, strikes the interface at an incident angle of θi, defined off of the normal to the interface, seen as a dashed black line. It reflects off of the interface at an angle of reflection of θr, being equal to θi, according to the law of reflection.39 Figure 2.3. Plane of incidence for light incident upon an interface. The top medium has a complex refractive index of ñ1(); the bottom medium has a complex refractive index value of ñ2(). Light is incident upon the interface at an incident angle of , as defined off the interface normal (represented by the dashed black line). The light then reflects off of the interface at an angle of reflection, r, being equal to the incident angle. The complex refractive index of air and Si can be obtained from the literature.40–43 The complex refractive index of air is assumed to be wavelength-independent with a refractive index of 1.000 and an extinction coefficient of zero. These assumptions are reliable to three decimal places given the low 17 dispersion and absorption of air for the solar spectral irradiance, i.e., the sun’s power spectral density in units of W·m-2·nm-1, and the typical propagation length of a benchtop experiment. For larger scales, however, with propagation lengths spanning the thickness of the atmosphere, water vapour can exhibit noticeable absorption for the solar spectral irradiance. This will be discussed later with respect to the impact of atmospheric absorption on solar cells. In contrast, the complex refractive index of Si has non-negligible wavelength dependencies that should be considered for reflectance. This wavelength dependence can be seen in Figure 2.4 as a varying refractive index and extinction coefficient for Si. Given the knowledge of the wavelength-dependent refractive index and extinction coefficient of Si, the theoretical reflectance of the air-Si interface can be calculated as a function of the incident angle via the Fresnel reflectance equations, which define the reflectance as the ratio of reflected optical power to incident optical power. However, the Frensel reflectance equations have an added complexity in that they are polarization-dependent, with separate expressions being defined for both s- and p-polarization states. This is important because light striking the surface of the Earth is typically unpolarized and can thus be treated as having independent and equally intense s- and p-polarization states. Figure 2.4. Refractive index, n(), (blue curve) and extinction coefficient, (), (red curve) of Si as a function of wavelength.43 18 The categorizing of light into s- or p-polarization depends upon the orientation of its oscillating electric field with respect to the aforementioned plane of incidence. When the electric field oscillates transverse to the plane of incidence, it is s-polarized. When the electric field oscillates within the plane of incidence, it is p-polarized. With these definitions, the Fresnel equations for the reflectance of s- and p-polarized light striking an interface between two media can be averaged to form an expression for the reflectance of unpolarized light. The resulting unpolarized reflectance, being the ratio of reflected optical power to incident optical power, is given by R(ñ1(λ), ñ2(λ), θi) = 12||ñ1(λ) cos θi – ñ2(λ)√1 – ((ñ1(λ)ñ2(λ))2(1 – cos2 θi))ñ1(λ) cos θi + ñ2(λ)√1 – ((ñ1(λ)ñ2(λ))2(1 – cos2 θi))||2 +12||ñ1(λ)√1 – ((ñ1(λ)ñ2(λ))2(1 – cos2 θi)) – ñ2(λ) cos θiñ1(λ)√1 – ((ñ1(λ)ñ2(λ))2(1 – cos2 θi)) + ñ2(λ) cos θi||2 (2.2) where the first term is for the s-polarization component of unpolarized light, the second term is for the p-polarization component of unpolarized light, ñ1(λ) is the complex refractive index of the medium from which light is incident, ñ2(λ) is the complex refractive index of the medium into which light is transmitted, and θi is the incident angle defined off the normal. An experimental investigation of the above reflectance can be carried out for an air-Si interface if the appropriate polarization-sensitive optics are used. The polarizing optics, called polarizers, allow the reflectance of s- and p-polarized light to be measured independently. The polarizers can be seen in Figure 2.5 along with the rest of the experimental setup. The setup starts with a laser diode emitting collimated 532-nm narrowband light that is passed through an optical filter. The optical filter ensures that the passed light has a sufficiently narrow spectrum around the centre wavelength of 532 nm. It is of particular 19 importance for this 532-nm laser diode, because such diodes use frequency doubling to convert 1064-nm laser light into 532-nm light, and there is often 1064-nm light remaining in the 532-nm beam. Following the optical filter, the light passes through a spinning chopper blade that modulates the light at a frequency of 1 kHz. The optical modulation allows the signal to later be extracted via electronic filtering from the ambient light signal within the room, which is not modulated at this frequency. The light having passed through the chopper blade then passes through two polarizers that apply the desired s- or p-polarization state onto the light. The desired polarization state is selected by rotating these polarizers’ mounts. Two polarizers are used to ensure that the extinction ratio between the orthogonal polarization states is sufficiently large. Figure 2.5. Experimental setup that is used to measure the reflectance of a sample. A top view is seen in (a); a side profile is seen in (b). The laser diode directs light through an optical filter. The light is then modulated by a chopper blade at a 1-kHz modulation rate. The light then travels through two polarizers to impart the desired s- or p-polarization state. The optically filtered, modulated, and polarized light is then directed off the sample’s 20 surface and onto a photodetector (Thorlabs DET36A/M) to produce a photocurrent signal. The photocurrent signal is passed through a bandpass filter that passes 300 Hz to 3 kHz and is amplified by a current preamplifier (Stanford Research Systems SR570). The current preamplifier outputs a sufficiently large voltage signal that is displayed on an oscilloscope (Tektronics TDS 2024B). For optimal balancing of the polarizations, the laser diode is mounted with its polarization axis at 45° to the horizontal plane. This produces equal powers for the s- and p-polarization states exiting the polarizers. The polarized light then passes through a lens that weakly focuses the light into a beam that reflects off the sample’s surface and onto a distant photodiode (Thorlabs DET36A/M). The photodiode generates the desired photocurrent signal in proportion to the incident optical power. The optically-induced photocurrent signal is then electrically filtered through a passband of 300 Hz to 3 kHz using a bandpass filter and is amplified by a current preamplifier (Stanford Research Systems SR570). The current preamplifier converts the photocurrent signal into a sufficiently large voltage signal, which is displayed on an oscilloscope (Tektronics TDS 2024B). The reflectance experiment described above was implemented for the measurement of the angular-dependent reflectance of an air-Si interface, with the Si surface being non-textured. The significance of the 532-nm narrowband light source will be discussed later in more detail, however, for now it will suffice to say that 532-nm light is representative of the solar spectral irradiance when weighted by the wavelength-dependent complex refractive index of Si. Figure 2.6 shows the theoretically-calculated angular-dependent reflectance of this air-Si interface at a wavelength of 532 nm for the s- and p-polarization states as dashed blue and red curves, respectively, as well as the reflectance of the unpolarized light as a dashed green curve, having been calculated from (2.2). Figure 2.6 also shows the experimentally-measured angular-dependent reflectance of this air-Si interface at a wavelength of 532 nm for the s- and p-polarization states as solid blue and red curves, respectively, as well as the reflectance of the unpolarized light as a solid green curve. The experimental values were collected between 15° to 85° and measured in 2.5° increments. The collecting of experimental values for extreme angles was hampered by practical issues in the experimental 21 setup. The (unavoidable) 15° lower limit was due to the obstruction of the photodiode’s housing, while the (unavoidable) 85° upper limit was due to the combined challenge of the diameter of the collimated light beam and the finite thickness of the Si sample. The experimental values that were collected have an average absolute percent error of 0.71% with respect to the theoretical values. The overall results show excellent agreement between theory and experiment—and validate the accuracy of the experimental setup for the upcoming analyses. Ultimately, we must note that light that is not reflected off of the interface is transmitted through the surface and into the semiconductor, where it has the potential to be converted to electrical energy. The next section discusses the basics of this conversion and its exploitation for solar energy collection. Figure 2.6. Reflectance of an air-Si interface for a 532-nm beam with the Si surface being non-textured. The measured reflectance, R(i), is shown as a function of the incident angle, i, for s-polarized (solid blue), p-polarized (solid red), and unpolarized (solid green) light. The theoretical Fresnel reflectance is shown for the three states of polarization as dashed curves in the respective colours. 22 2.2.2. Electrical Properties of Semiconductors Light being incident upon a semiconductor can enable current flow. More specifically, incident photons can be absorbed by electrons within the semiconductor’s crystal lattice. The energized electrons are then free to move within the semiconductor. This process is called photoconduction and it is depicted in Figure 2.7a, where the large black circles depict atoms and the smaller black circles depict electrons. The incident photon, seen as the red arrow, enters the lattice and is absorbed, giving rise to an electron that is free to move throughout the crystal lattice. This freeing of an electron also leaves behind the “lack of an electron”, which is called a hole and is seen in Figure 2.7b as a small hollow circle. The electron and hole are then free to move as charge-carriers randomly throughout the lattice, with zero average displacement. However, if the lattice is subject to an electric field, via an external bias or a p-n junction, the charge-carriers will exhibit a net displacement parallel to the field with a characteristic drift velocity. The drifting charge-carriers establish an electric current, which can be exploited as work in an external circuit. Figure 2.7. Photoconduction within a semiconductor’s crystal lattice. The photon (red arrow) in (a) is absorbed by an electron, freeing an electron (solid circle) and leaving a hole (hollow circle) as shown in (b). The above photoconduction process in a semiconductor is contingent upon multiple processes. First, the incident photons must be capable of entering the semiconductor. The likelihood of this is quantified by the aforementioned reflectance of the semiconductor surface. Specifically, the surface must have a sufficiently low reflectance to allow the photons to enter the semiconductor. Second, the photons 23 that have entered the semiconductor must undergo absorption. The likelihood of this absorption is quantified by a parameter called the quantum efficiency.38 The quantum efficiency is close to unity if the semiconductor is a direct bandgap semiconductor, as opposed to an indirect bandgap semiconductor, and the energy of the photon is greater than the bandgap. This is because it is only sufficiently energetic photons that can free the charge-carriers for motion. Third, the photo-generated charge-carriers must be capable of drifting through the biased semiconductor and be collected by electrodes. The propensity for these charge-carriers to drift through the semiconductor and be collected by the electrodes is defined by both the charge-carrier mobility and lifetime of the semiconductor. In general, the presence of impurities or surface states, which introduce surface recombination, impede such drifting and collection and are best avoided for solar cell applications. Further details on these processes are given in section 2.4.2. For photodiodes or solar cells, the electrodes that capture the charge-carriers are often implemented as transparent conducting oxide films or as small silver electrodes on the top and as a large aluminum electrode on the bottom. Ultimately, this sandwiched electrode-semiconductor-electrode structure can have a photodiode or solar cell yield the desired electrical current within an external circuit. The current and voltage characteristics of a photodiode or solar cell can be visualized according to the current-voltage curve in Figure 2.2 of section 2.1 with one small but important difference. The tradition for solar cells is to define the direction of positive current flow opposite to that for photodiodes. Thus, Figure 2.2 of section 2.1, being for a photodiode, would be flipped vertically for a solar cell, and the region of interest for electrical solar power generation would be in the top right corner (quadrant I) of the new figure. As such, the (negated) ideal diode equation is used to define the relationship between the output current of the solar cell and the voltage across its terminals. The solar cell’s output current is defined by Ipv = Iph – Isat( exp(Vpv Vt⁄ ) – 1 ), (2.3) where Iph is the photocurrent, Isat is the reverse saturation current, Vpv is the operating voltage across the terminals, and Vt = kBT/q is the thermal voltage, given kB as Boltzmann’s constant, T as the temperature, and q as the charge of an electron. Equation (2.3) is an ideal equation suitable for simulating the solar cell 24 output current assuming standard test conditions and identical crystalline solar cells. There are also non-ideal alternatives that introduce series and shunt resistances. However, these non-ideal models are suboptimal when generalizing the efficiencies of solar structures as series and shunt resistances are specific to individual solar cells. The use of the ideal model ensures a general comparison of solar structures. Ultimately, the optical and electrical properties of semiconductors can be exploited for effective collection of solar energy—but this can only be done with appropriate processing of the semiconductors. The process of converting a semiconductor into a solar cell is described in the upcoming section 2.3. 25 2.3. Conversion of a Semiconductor into a Solar Cell The process of converting a semiconductor into a solar cell has several stages, each of which is critical to the performance of the solar cell. The five main stages in the process of converting an ingot of single-crystal (monocrystalline) Si semiconductor into a conventional Si solar cell are depicted in Figure 2.8. Figure 2.8a shows a Si wafer in the first stage, having been sliced from an ingot and made ready for further processing. The second stage, shown in Figure 2.8b, follows from chemical etching of the surface of the Si wafer. The chemical etching removes saw damage from slicing of the Si wafer out of the ingot and also textures the surface. The chemical etch is anisotropic, so it textures the surface of the Si wafer, with microscopic four-sided pyramids whose locations and sizes are random. This micron-scale texturing increases the optical efficiency of the solar cell by promoting the scattering of light off of the top surface, leading to increased light propagation through the active region. The third stage, shown in Figure 2.8c, follows from the diffusion of dopants into the Si wafer’s crystal lattice in a well-controlled manner to form the p-n junction. Typically, for a Si-based solar cell, the dopants are boron, which creates a p-type region with an excess of holes, and phosphorous, which creates an n-type region with an excess of electrons. Figure 2.8. The five main stages of forming a Si solar cell. (a) The first stage has a Si wafer that has been diced and made ready for processing. (b) The second stage follows from chemical etching of the wafer’s surface. (c) The third stage follows from the diffusion of dopants into the wafer, with the p- and n- type dopants shown here in red and blue, respectively. (d) The fourth stage follows from the deposition of film of SiNx:H that acts as an AR coating and passivates the surface. (e) The fifth stage follows from the deposition of silver and aluminum contacts on the top and bottom surfaces of the wafer, respectively. 26 The regions are layered to create the desired p-n junction with its built-in electric field—within which the photo-generated electrons and holes can form a current. In Figure 2.8c, the blue particles being infused into the top surface are elemental phosphorus and the red particles being infused into the bottom are elemental boron. The fourth stage, shown in Figure 2.8d, follows from deposition of a thin film on the front surface of the solar cell that acts as an antireflection (AR) layer. The thin film is typically a quarter-wavelength-thick film of hydrogenated silicon nitride (SiNx:H) that is designed to prevent the reflection of wavelengths in the range of 550 nm to 700 nm,20 while simultaneously passivating the surface. The use of a quarter-wavelength-thick film here has an AR effect because it introduces destructive interference in the direction of reflected light and constructive interference in the direction of the incident light; the use of SiNx:H passivates the surface by filling vacant bonds that would otherwise promote electron-hole recombination and thus reduce the charge-carrier lifetime.44 Surface passivation will be discussed in more detail in section 2.4.2. The fifth stage, shown in Figure 2.8e, follows from the patterning of contacts as charge-carrier collectors, which are ultimately used to draw the current out of the p-n junction. The front contacts of Si solar cells are typically made of silver, although a uniform layer of transparent conducting oxide can also be used. As discussed in section 2.2.2, care is necessary here to ensure that the junction between the Si wafer and the metal contact does not form a rectifying contact. An Ohmic contact is needed, instead, to allow for the free flow of electrons and holes across the junction and thereby increase the electrical efficiency of the solar cell. The contacts on the backside of Si solar cells are typically aluminum, with silver pads being included to aid soldering. Following this fifth stage, the Si wafer is co-fired to turn the layered materials from the individual steps into a cohesive solar cell. The solar cell is then tested and sorted based upon its performance characteristics for optical to electrical power conversion. The next section gives information on these performance characteristics. 27 2.4. Optical and Electrical Performance Characteristics of a Solar Cell The optical and electrical performance characteristics of a solar cell govern its ability to convert optical power to useable electrical power. In this section, the theoretical models used to describe the optical and electrical performance characteristics of solar cells are developed and then applied within experimentally-seeded simulations. The optical performance characteristics of a Si solar cell are described in section 2.4.1. A detailed description of the electrical performance characteristics then follow in section 2.4.2. Lastly, section 2.4.3 describes the practical (and often overlooked) considerations on the optical and electrical performance of a solar cell. 28 2.4.1. Optical Performance Characteristics of a Solar Cell The optical performance characteristics of a Si solar cell are largely dependent upon three factors. The three factors are the incident solar spectral irradiance (being the power spectral density in units of W·m-2·nm-1), the wavelength-dependent complex refractive index of Si, and the structural means for mitigating reflections off of the solar cell’s surface. These factors together are analysed in this work to define the optical performance characteristics of a (conventional) textured Si solar cell and compare these characteristics to those of a (simple) non-textured Si wafer. With respect to the first factor of the optical performance characteristics, being the incident solar spectral irradiance, we consider the sun. The sun is a broadband light source that closely mimics a blackbody, with a predictable power spectral density distribution. Figure 2.9a shows the normalized solar spectral irradiance outside the Earth’s atmosphere, called Air Mass (AM) 0, displayed as the red curve, along with the normalized spectral irradiance of a 5777-K blackbody radiator as the black curve. The two curves are similar in form. The solar spectral irradiance will decrease, however, upon propagating through the atmosphere, due to absorption and scattering, and this is quantified by an increased AM rating. The AM rating of light passing through the Earth’s atmosphere can be approximated by the secant of the solar zenith angle, being the angle between the normal to the surface of the Earth and a vector towards the sun. Thus, the AM 1 rating is for direct illumination through the atmosphere with light parallel to the normal to the surface of the Earth. We can also consider the AM 1.5 solar spectral irradiance, for a solar zenith angle of θ ≈ 48°, because it gives an average level of the solar spectral irradiance over the continental United States and is often used globally to test and sort solar cells. This solar spectral irradiance, displayed in Figure 2.9a as the blue curve, shows that Ozone (O3) leads to the absorption of high-energy ultraviolet light, at wavelengths below 400 nm, while water vapor (H2O) manifests itself as absorption bands for the lower-energy visible and infrared light.45,46 With respect to the second factor of the optical performance characteristics, being the wavelength-dependent complex refractive index, we can combine our knowledge of the wavelength-dependent complex 29 refractive index of Si and the AM 1.5 solar spectral irradiance. Specifically, by applying the Fresnel reflectance equation (2.2) for unpolarized light, a weighted average can be formed for the complex refractive index of Si across all wavelengths of the AM 1.5 spectrum that have energies above the 1.12-eV bandgap energy of Si.47 This truncated and weighted spectrum is used because it is only the wavelengths within the AM 1.5 spectrum being shorter than 1120 nm that have the potential to be absorbed and converted into electrical energy by a Si solar cell. Figure 2.9b shows this spectral region of interest for the normalized AM 0 and AM 1.5 solar spectral irradiance curves along with the spectral irradiance curve of the 5777-K blackbody radiator. Given this spectrum, an effective refractive index can be defined for Si to characterize its reflectance. This is done by first calculating the magnitude of the complex refractive index and assuming that it represents a purely real refractive index, n(λ). This is a good assumption given that the absorption at an air-Si interface (i.e., within /4 of the surface) is small. This is equivalent to stating that the incident optical power that is not reflected at the surface is transmitted into the bulk, which is a reasonable assumption. Next, we use the new n(λ) distribution for Si together with the AM 1.5 solar spectral irradiance to define an effective refractive index across the spectrum of interest. In doing this, it is assumed that the incident light is close to normal incidence, i.e., the incident angle has θi ≈ 0°. The values of n(λ) are then weighted across the AM 1.5 spectrum and averaged to form a wavelength-independent effective refractive index of neff ≈ 4.06 for Si. This effective refractive index is then used in the Fresnel reflectance equation (2.2), with ?̃?1(λ) ≈ 1 for air and ?̃?2(λ) ≈ neff = 4.06 for Si, to form a wavelength-independent and incident-angle-dependent expression for the reflectance of unpolarized light, in the form of R(θi) = 12|cos θi – neff√1 – (1 – cos2 θi) neff2⁄cos θi + neff√1 – (1 – cos2 θi) neff2⁄|2+12|√1 – (1 – cos2 θi) neff2⁄ – neff cos θi√1 – (1 – cos2 θi) neff2⁄ + neff cos θi|2. (2.4) 30 Figure 2.9. The AM 0 (solid red curve) and AM 1.5 (solid blue curve) solar spectral irradiance and the 5777-K blackbody spectral irradiance (solid black curve) shown normalized as a function of the wavelength, . The O3 and H2O molecular absorption bands are labeled in (a) and the spectrum up to the bandgap energy of Si is shown in (b). This air mass data is from the National Renewable Energy Laboratory (NREL).48 Note that the refractive index of Si is equal to this effective refractive index, neff = 4.06, at a wavelength of 555 nm. A narrowband laser source at this wavelength is not readily available, but the 532-nm laser in the experimental setup of section 2.2.1 can be used to approximate it. The percent difference between Si’s reflectance at 555 nm and 532 nm is only 1.6%, and so this is deemed to be a good approximation. The incident angle, θi, used in the above analyses and in this last expression has been discussed as an incident angle on an arbitrary surface. However, for solar cells, it is necessary to link this incident angle to the sun’s positioning in the sky. The positioning of the sun in the sky is defined with respect to two angles.49 The first angle is defined off the normal to the surface of the Earth and is called the solar zenith angle, θ. It is the same angle that is used to define the AM rating of light. The second angle is defined within the plane that is tangent to the surface of the earth and is called the solar azimuth angle, ϕ. These angles are shown with respect to the surface of the Earth in the Global (primed) Coordinate Frame in Figure 2.10. For the remainder of this work the Si solar cell is assumed to lie flat on the Earth's surface, to be consistent with 31 standardized test conditions for solar cells.50 This flat orientation allows us to equate the incident angle, θi, and solar zenith angle, θ, and state that R(θi) = R(θ) in (2.4). Figure 2.10. Coordinates and angles in the Global Coordinate Frame. The frame has primed coordinates, x’, y’, and z’, where the x’-axis faces South, the y’-axis faces East, and the z’-axis faces up. Incident light (red arrow) is defined within the Global Coordinate Frame by its solar azimuth angle, , and solar zenith angle, . With respect to the third factor of the optical performance characteristics, being structural means for mitigating reflection, we note that the reflectance results in Figure 2.6 of section 2.2.1 were acquired for 532-nm light being incident onto a non-textured air-Si interface, while the description in section 2.3 of conventional Si solar cells suggested that their Si surfaces are typically textured, to increase light scattering and promote absorption. This is an important difference. The optical power reflected off of the textured surface exhibits diffuse reflection, which is far weaker in magnitude than specular reflection and does not adhere to the law of reflection. It can have light reflect at angles that differ greatly from the incident angle. As such, to quantify diffuse reflection, the experimental setup in section 2.2.1 was used. The diffuse reflectance was measured over a wide range of incident angles for the textured Si solar cell and found to be negligible. Thus, we can ultimately state that R(θ) is defined by (2.4) for the non-textured Si wafer, with  = i and neff = 4.06, and R(θ) = 0 for the textured Si solar cell. A comparison of the optical performance characteristics of the textured Si solar cell and non-textured Si wafer can only be made with knowledge of the diurnal solar trajectory under standard test 32 conditions (STC). This is done by defining the metric of captured incident optical power density, while the structures lie flat, as POP-flat(θ) = PSTC(1 – R(θ)) cos(θ) , (2.5) where PSTC is the STC incident optical power density of 1000 W/m2, (1 – R(θ)) sets the total incident optical power density that is captured by (i.e., transmitted into) the surface, and the solar zenith angle, θ, changes throughout the day based upon the sun’s positioning in the sky. The solar zenith angle appears via a cos(θ) term because the fraction of captured incident optical power is set by the component normal to the horizontal surface. For this same reason, the solar azimuth angle, ϕ, does not play a role and does not appear in this equation. For the purposes of comparison, Figure 2.11 shows the captured incident optical power density for the textured Si solar cell (red curve) and non-textured Si wafer (blue curve) during the summer solstice, June 21, at a latitude of 23.45°N. Such a day has the sun directly overhead, with θ = 0°, at noon. The optical performance characteristics of the textured Si solar cell seen here are clearly superior to those of the non-textured Si wafer. Besides the improvement in captured incident optical power density at noon, when the incident angle is small, there are more subtle improvements seen during the early and late times of the day, when the incident angle is large. Ultimately, it can be said that texturing on the surface of Si solar cells is an effective way of mitigating reflection and promoting the capture of incident optical power. However, it must also be noted that this texturing simultaneously degrades the electrical performance. This issue will be explained in the upcoming section 2.4.2. 33 Figure 2.11. Captured incident optical power density of a textured Si solar cell and non-textured Si wafer as a function of the time of day. The textured Si solar cell is assumed to have a reflectance of zero; the non-textured-Si wafer is assumed to have a reflectance defined by equation (2.4). The STC incident optical power density is 1000 W/m2, and the results are shown for the summer solstice, June 21, at a latitude of 23.45°N, for which the sun is directly overhead at noon. 34 2.4.2. Electrical Performance Characteristics of a Solar Cell The electrical performance characteristics of solar cells are largely dependent upon three factors. The three factors are the amount of light that is transmitted into the solar cell, the electrical performance loss due to surface recombination, and the use of electrical optimization. We can use these factors together to analyse the electrical performance characteristics of a textured Si solar cell and compare these characteristics to those of a non-textured Si solar cell. For the remainder of this work, a non-textured Si solar cell is approximated by (and assumed to have the same optical performance characteristics as) a non-textured Si wafer. This approximation is justified as the optical performance characteristics of a solar cell are largely defined by the semiconductor material used to make the solar cell as discussed in section 2.2.1. With respect to the first factor of the electrical performance characteristics, being the amount of light that is transmitted into the solar cell, we recall that the photocurrent comes about from sufficiently energetic photons being absorbed by the semiconductor, to form mobile charge-carriers, which are then accelerated by the built-in electric field of the p-n junction. As such, the number of mobile charge-carriers will be proportional to the number of incident photons that are captured—and so the photocurrent, Iph, will be proportional to the captured incident optical power density, POP-flat(θ), shown in (2.5). At the same time, the photocurrent from the solar cell will change as its temperature changes. Such dependence is typically quantified by an empirical relation comparing the photocurrent at the temperature of interest to the photocurrent at the STC temperature of 300 K.51,52 This empirical relationship can be merged with the aforementioned knowledge that the photocurrent scales in proportion to the captured incident optical power density, POP-flat(θ), to express the overall photocurrent from the solar cell as Iph = [Isc + α(T – TSTC)]POP-flat( )PSTC. (2.6) Here, Isc is the short-circuit current produced by the Si solar cell at STC when Vpv = 0 in (2.3). This value can be found on the manufacturer’s datasheet for the solar cell of interest. The temperature dependence comes about by the term α(T – TSTC), where α is the temperature coefficient of the Si solar cell, T is the 35 ambient temperature, and TSTC is the ambient temperature at STC. The values for α and TSTC are both available on the manufacturer’s datasheet. The angle-dependent captured incident optical power density appears via POP-flat(θ) PSTC⁄ term, where POP-flat(θ) is the solar-zenith-angle-dependent captured incident optical power density from (2.5) and PSTC is the incident optical power density at STC, i.e., 1000 W m2⁄ , that generates the short-circuit current, Isc. For the remainder of this work, the ambient temperature is made equal to the ambient temperature at STC, which allows (2.6) to be relieved of temperature dependence. This is a common practice for simulation-based studies found in the literature.53 As such, (2.5) and (2.6) can be merged to express the photocurrent as Iph = Isc(1 – R(θ)) cos(θ) . (2.7) It follows that the photocurrent of a textured Si solar cell and that of a non-textured Si solar cell depend upon the solar zenith angle in the same manner as the curves of captured incident optical power density in Figure 2.11. However, such a comparison ignores electrical inefficiencies caused by the increased surface area and recombination of textured Si solar cells. With respect to the second factor of the electrical performance characteristics, the electrical performance loss due to surface recombination, we recall that the presence of surface states can lessen the lifetime of charge-carriers within the semiconductor. This happens because the textured surface introduces trapping sites that readily capture holes. This yields hastened surface recombination of the photo-generated charge-carriers at the trapping sites, a decreased charge-carrier lifetime, and ultimately leads to fewer charge-carriers being collected at the electrodes. Surface recombination, and its reduction in the optical-to-electrical conversion efficiency, is a noted problem in the literature for textured Si solar cells.26,54 An ultrafast pump-probe transmittance55 setup can be used to characterize the effects of surface recombination within the textured Si solar cell and non-textured Si solar cell. For these measurements, the non-textured Si solar cell is approximated by a simple non-textured Si wafer. The setup probes the density of charge-carriers within the solar cells and defines their lifetimes—with a femtosecond time resolution. 36 The setup works by applying pump laser beam pulses to the sample, to photo-generate charge-carriers, and applying probe laser beam pulses to the same sample, to see the effects of the pump-induced charge-carriers on the transmission of the probe laser beam. In this way, the power of the probe laser beam transmitted through the sample can be monitored as a function of the delay time between the pump and probe laser pulses. This forms a time-resolved trace of the charge-carrier density within the sample as a function of the delay time. Such a technique can yield results on a femtosecond timescale, given laser pulses with a femtosecond pulse duration. The ultrafast pump-probe transmittance55 setup that is used in this work is shown in Figure 2.12. It has an erbium-doped pulsed fibre laser (Toptica Photonics Inc.) that produces the synchronized pulses for the pump and probe laser beams with 100-fs pulse durations at a repetition rate of 90 MHz. The pump and probe laser beams have wavelengths of 775 nm and 1550 nm, respectively. As such, the 775-nm pump laser beam has a photon energy (1.6 eV) that is above the bandgap energy of Si and is thus absorbed within it, while the 1550-nm probe laser beam has a photon energy (0.6 eV) that is below the bandgap energy of Si and is thus transmitted through it. This lets the pump laser pulses photo-generate charge-carriers in the Si, leading to an increase in its extinction coefficient and a corresponding decrease in the transmitted power of the probe laser beam—as defined by Drude theory for semiconductor plasmas.56 A time-resolved trace of the transmitted power of the probe laser beam is built up as a function of the time delay between the pump and probe laser pulses by varying the delay between the pump and probe laser pulses via a mechanically-actuated delay stage. The zero-time on the delay stage corresponds to the simultaneous arrival of the pump and probe laser pulses on the sample, while a shortening of the length of the delay stage has the probe pulse measure the pump-beam-induced changes to the sample at later times. The transmitted power of the probe laser beam is measured during this process with an InGaAs photodetector (Thorlabs DET20C/M). To improve the signal-to-noise ratio of the system, the pump laser beam is modulated at a rate of 1 kHz using a beam chopper (SR540, Stanford Research Systems Inc.), and the probe signal is filtered around this frequency by way of a transimpedance amplifier (SR570, Stanford Research Systems) and a lock-in 37 amplifier (SR830, Stanford Research Systems). This electronic filtering and processing rejects a great deal of the ambient noise that is present in the environment. Figure 2.12. Ultrafast pump-probe transmittance setup. A top view is seen in (a), and a side profile is seen in (b). The setup has a pump laser beam (in red) and probe laser beam (in yellow). The beams are redirected with mirrors, overlapped with a beamsplitter, and focused onto the sample by a microscope objective. The probe laser beam having transmitted through the sample is re-collimated by a microscope objective, focused by a lens, and measured by a photodetector. Representative results from the ultrafast pump-probe transmittance setup are shown in Figure 2.13 for samples taking the form of a textured Si solar cell (purple curve) and non-textured Si wafer (orange 38 curve). The figure shows the normalized transmittance of the probe laser beam through each sample versus the delay time, t, between the pump and probe laser pulses, for a pump laser pulse fluence of 345.8 µJ/cm2 and arrival time at 50 ps. The figure insets show scanning electron microscope (SEM) images of the samples’ surfaces. Figure 2.13. Normalized transmittance of the probe laser beam through the textured Si solar cell and non-textured Si wafer as a function of the pump-probe delay time, t. The pump laser pulse fluence is 345.8 µJ/cm2 and its arrival time is 50 ps. Results for the textured Si solar cell and non-textured Si wafer are shown as purple and orange curves, respectively, with SEM images of the two samples’ surfaces shown in the insets. It is clear from the results that the textured Si solar cell exhibits more rapid recombination of charge-carriers, in comparison to that of the non-textured Si wafer. After 1000 ps, the charge-carrier density for the textured Si solar cell, being proportional to the displayed change in the transmittance, is only 64% of that for the non-textured Si wafer. This difference is due to the increased surface area of the textured Si solar cell, which leads to hastened surface recombination and a shorter charge-carrier lifetime. Ultimately, the reduced charge-carrier lifetime can have a major impact in the collection of charge-carriers at the electrodes and the optical-to-electrical conversion efficiency of the solar cell. 39 With respect to the third factor of the electrical performance characteristics, being the use of electrical optimization, we must first consider the I-V characteristics of a Si solar cell. The I-V characteristics of a Si solar cell can be modelled by combining (2.3) and (2.7) to form a solar zenith and solar cell voltage dependent ideal solar cell output current equation, Ipv(θ, Vpv) = Isc(1 – R(θ)) cos(θ) – Isat( exp( Vpv Vt⁄ ) – 1 ). (2.8) The above equation can be used to simulate realistic I-V curves of Si solar cells if their manufacturer’s datasheet is available. As an example, the I-V curve for an XS156B3-210R+ series solar cell by Motech Industries Inc. can be established using the manufacturer’s datasheet (henceforth referred to as “the datasheet”) in Appendix B. The datasheet provides values for the short-circuit current, Isc, the open-circuit voltage, Voc, the maximum power current, Imp, and maximum power voltage, Vmp, of a solar cell at STC. The open-circuit voltage is the operating voltage of the solar cell while it outputs zero current. The maximum power current and voltage are the respective operating current and voltage of the solar cell while maximum power is delivered from the solar cell to the load it serves. The reverse saturation current, Isat, is not usually given on a manufacturer’s datasheet, although it can be estimated using the aforementioned values that are found on it. To estimate Isat, we rearrange (2.8) to give Isat = Isc(1 – R(θ)) cos(θ) – Ipvexp (VpvVt) – 1. (2.9) In (2.9), Ipv and Vpv can be equal to Imp and Vmp, respectively, or 0 A and Voc, respectively. The choice of one over the other can be made according to the information that is available on the manufacturer’s datasheet and/or knowledge of the operating conditions, i.e., maximum power operation or open-circuit operation. When setting θ = 0° and using Ipv = Imp and Vpv = Vmp, the calculation more accurately estimates the Isat of the solar cell while it is transferring maximum power to a load at STC. When setting θ = 0°and using Ipv = 0 A and Vpv = Voc, the calculation more accurately estimates Isat of the solar cell while it is in an open-circuit condition at STC. In this work, accurately estimating Isat of the solar cell while it is delivering 40 maximum power to a load at STC is deemed to be more important, and both Imp and Vmp are available on the datasheet, so the first of the two choices is used. Navigating the datasheet to the values for the 19% efficient XS156B3-210R+ series solar cell shows values of Isc = 9.09 A, Imp = 8.59 A, and Vmp = 0.540 V at STC, which equates to a value of Isat ≈ 379 pA. The reverse saturation current of a Si solar cell is typically extremely small and is consistent with this estimate.57 Using the aforementioned values found on the datasheet, Isat, and (2.8), an I-V curve for the XS156B3-210R+ series solar cell is simulated, as seen in Figure 2.14a. Figure 2.14a also depicts all of the datasheet values along with the open-circuit voltage, Voc, which deviates here from the value found on the datasheet. This deviation has a negligible impact, as discussed in more detail later. Another point of interest is the intersection labelled as the Maximum Power Point (MPP) with corresponding values of current and voltage of Imp and Vmp, respectively. The MPP represents the operating point at which maximum power is transferred from the solar cell to the load. It can be seen more clearly in Figure 2.14b by examining the power-voltage (P-V) curve (solid green curve) alongside the I-V curve (solid blue curve). The MPP represents the local maximum of the P-V curve and can be found mathematically by using a first derivative test. In practice, there are many different techniques for finding and tracking the MPP of a solar cell.58–62 The techniques are typically programmed into microcontrollers, which are integrated into the solar power system as Maximum Power Point Tracking (MPPT) units. The number of MPPT units used in a solar power system can vary depending on the application. For larger arrayed implementations, such as an industrial solar power plant, there may be one MPPT unit dedicated to a grouping of solar modules. Conversely, for a smaller arrayed implementation, such as a residential installation, there may be one MPPT unit for each solar module. In this work, the MPP is considered according to a metric of generated electrical power density. To carry out such an investigation, the solar cell output current must first be defined as a current density, along with the short-circuit current and reverse saturation current. This can be done by simply dividing both sides of (2.8) by the area of the solar cell, A, to give 41 Jpv(θ, Vpv) = Jsc(1 – R(θ)) cos(θ) – Jsat(exp(Vpv Vt⁄ ) – 1). (2.10) By multiplying (2.10) by the solar cell voltage, the generated electrical power density for a Si solar cell is found to be PELEC-flat(θ, Vpv) = VpvJpv(θ, Vpv) = Vpv (Jsc(1 – R(θ)) cos(θ) – Jsat(exp(Vpv Vt⁄ ) – 1)) . (2.11) Figure 2.14. Simulated I-V and P-V curves of the XS156B3-210R+ series solar cell. In (a), the simulated I-V (solid blue) curve is shown along with datasheet values for the STC current and voltage. The open-circuit voltage indicated on the I-V curve differs from the datasheet value due to the approximation used to obtain Isat. The Maximum Power Point (MPP) lies at the knee of the I-V curve. In (b), the simulated (solid green) P-V curve is shown. This power curve clearly shows the MPP at its local maximum. It should be noted that the generated electrical power density defined in (2.11) assumes that the surface of the Si solar cell is perfectly passivated to mitigate surface recombination. While this is not the case for textured Si solar cells, as was shown in Figure 2.13, there are cases of Si solar cells in the research environment displaying near perfect surface passivation. Such Si solar cells can have both good optical and electrical performance characteristics—although their manufacturing is not currently scalable or cost-42 effective. As an example, Yoshikawa et al. demonstrated a Si solar cell with a conversion efficiency of 26.3%.38 This is a vast improvement from manufactured (conventional) Si solar cells, which have a conversion efficiency typically ranging from 18-20%. The Si solar cell with a conversion efficiency of 26.3% was created with a textured surface and an extremely well passivated surface. In general, for manufactured (conventional) Si solar cells with less effective surface passivation, it can be said that the benefits of texturing in terms of improved optical performance characteristics outweigh the sacrifices in electrical performance characteristics, which come about from more rapid surface recombination. For this reason, texturing is typically applied to manufactured Si solar cells. At the same time, manufactured Si solar cells also have certain practical considerations that should be noted (and are often overlooked). These considerations are discussed in the next section. 43 2.4.3. Practical Considerations Solar cells that are implemented in practical settings rarely exhibit the characteristics claimed on their datasheets. This is due to two main reasons. The first reason is the non-unity packing density of solar modules. The second reason is the unavoidable optical power losses associated with the protective layer that is used on the solar cell to prolong its life. In this section, these two reasons are explored and the optical power losses are experimentally tested. The non-unity packing density of solar modules is due to the housing of the module itself as well as the shape of the solar cells within it. The housing of a solar module has aluminum edging to both protect the solar cells from moisture and contain the modules together. However, this aluminum edging leads to a non-unity packing density for arrayed implementations. The shape of the solar cell causes issues for the packing density because Si solar cells are formed from cylindrical Si ingots, via the Czochralski process, which yield wafers that only approximate squares. The exact dimensions of the wafers are determined through an optimization that takes into consideration the price of feedstock Si as well as the price of solar cells. The resulting shape covers the area with less efficiency than a true square. The inefficiencies together decrease the generated electrical power density by a few percent when the generated electrical power becomes normalized with respect to the installation (ground coverage) area, rather than the area of silicon, as was done earlier in this work. The unavoidable optical power losses associated with the protective layer on a solar module, having been added to ensure its longevity, can be discussed at the level of a single solar cell. This protective layer is typically made of glass, which is bonded to the solar cell with an index-matching adhesive. This leads to two considerations. The first consideration is the choice of material. The glass that is used should be sufficiently inexpensive, and in this work standard glass with a thickness of 2 mm is used for the protective layer. The index matching adhesive that is used here is an ultraviolet-curable polymer called Norland Optical Adhesive (NOA) 60. The manufactures datasheet of NOA 6063 indicates that all of the light within the AM 1.5 spectrum can pass through the adhesive once it is cured, with the exception of a small fraction 44 of the ultraviolet light below a wavelength of 380 nm. With this in mind, an ultraviolet transmission experiment was performed by passing 365-nm and 405-nm light through samples of the protective glass with and without the NOA 60 adhesive. These two wavelengths span a considerable range of the ultraviolet portion of the AM 1.5 spectrum and can form a good representation for the ability of ultraviolet light to pass through the protective glass and NOA 60 adhesive. The results, in Appendix C, show that the protective glass and NOA 60 adhesive have negligible effects on the ultraviolet transmission. The second consideration is the effect each material might have on the overall optical performance characteristics of the solar cell. Glass is transparent to most light within the visible spectrum and has a refractive index that is roughly 1.5.64 Admittedly, glass can block transmission of sufficient high energy light, but such light has wavelengths well below those of the AM 1.5 spectrum. Glass also absorbs a small amount of visible light, which has prompted the solar industry to use low-iron glass in lieu of standard glass. The standard glass used in the transmission experiment in Appendix C is not low-iron glass, but it is deemed to be sufficiently close to (and a good representation of the worse-case scenario for) the low-iron glass covering a solar cell. The effects of the glass and NOA 60 adhesive are explored experimentally via reflectance measurements in this work using the setup introduced in section 2.2.1 and depicted in Figure 2.5. Light that is reflected off of the protected-and-textured Si solar cell, as a specular reflection, can be measured and the resulting reflectance curves can be formed for subsequent experimentally-seeded simulations. Figure 2.15 shows such results for 532-nm light being reflected off of the surface of a textured Si solar cell having a protective layer of glass and an adhesive layer of NOA 60. The figure shows the theoretical angle-dependent Fresnel reflectance curves for s-polarized (dashed blue curve), p-polarized (dashed red curve), and unpolarized (dashed green curve) 532-nm light reflecting off the surface of glass with a refractive index of 1.5. The added layer of NOA 60 is not explicitly taken into account for the estimate due to its similar refractive index, being approximately 1.56, to that of glass. In essence, it is treated as an extension of the glass. The theoretical curves are overlaid with their experimentally measured counterparts for s-polarized (solid blue curve), p-polarized (solid red curve), and unpolarized (solid green 45 curve) 532-nm light reflecting off of the protected-and-textured Si solar cell. Overall, the theoretical and experimental curves are seen to be in good agreement, indicating that light reflecting off of the surface of the protected-and-textured Si solar cell can be quantified and well predicted. Figure 2.15. Reflectance of a protected-and-textured Si solar cell for a 532-nm beam. A glass layer is bonded onto the Si solar cell via NOA 60 to protect its surface. The measured reflectance, R(i), is shown as a function of the incident angle, i, for s-polarized (solid blue), p-polarized (solid red), and unpolarized (solid green) light. The theoretical Fresnel reflectance is shown for the three states of polarization as dashed curves in the respective colours. The predictable reflectance of the protected-and-textured Si solar cell enables its incorporation, with confidence, into experimentally-seeded simulations. The experimentally-seeded simulations of the protected-and-textured Si solar cell can then be compared to that of an unprotected-and-textured Si solar cell to understand the extent to which the protection impacts the generated electrical power density. With this in mind, Figure 2.16 shows the generated electrical power density of a protected-and-textured Si solar cell (solid blue curve) and an unprotected-and-textured Si solar cell (solid red curve) on June 21. The two curves are similar in form, although the protected-and-textured Si solar cell has a generated electrical power density at noon that is approximately 5% lower than that of the unprotected-and-textured Si solar cell. This 46 difference is due to specular reflection off the top surface of the protective glass layer, which also exists at earlier and later times of day to varying extents. Figure 2.16. Generated electrical power density of a protected-and-textured Si solar cell (in blue) and an unprotected-and-textured Si solar cell (in red), with the protective glass layer having been bonded with NOA 60. The generated electrical power density of the protected-and-textured Si solar cell is approximately 5% less than that of the unprotected-and-textured Si solar cell at noon. The solar cell used in this experimentally-seeded simulation has an optical to electrical conversion efficiency of 26.3%. 47 2.5. Summary In summary, Chapter 2 provided background knowledge that is needed to understand the physics behind the remainder of this work. The background outlined the logical progression from a semiconductor ingot to a protected-and-textured Si solar cell. Two metrics were defined in this process to describe the function and performance of a solar cell. The captured incident optical power density, being the first metric, defines the ability of a solar cell to pass light into its semiconductor (to potentially undergo absorption). A solar cell that is better at passing incident light into its semiconductor has higher probabilities for its charge-carriers to be collected by its electrodes. This directly affects the electrical performance of the solar cell. The generated electrical power density, being the second metric, defines the amount of electrical power a solar cell can deliver to an external load. Both of these metrics were used to compare different solar cells. The captured incident optical power density metric was used to declare that texturing is a good option to maximize the optical performance of a solar cell. The generated electrical power density metric was used to declare that, despite the electrical inefficiencies of texturing, a solar cell with superior optical performance characteristics will usually demonstrate superior electrical performance characteristics. Ultimately, the two metrics were used to rule that the standard implementation of a solar cell, being the protected-and-textured Si solar cell, results in a relatively small (less than 5%) decrease in optical and electrical performance due to specular reflection at the surface of the protective layer. It is important to note that the above characteristics are exclusive to independent solar cells lying flat on the surface of the Earth. It remains to be seen, however, if different structural implementations of the protected-and-textured Si solar cells can be created to outperform the flat implementation described in this chapter. The upcoming Chapter 3 takes up this challenge and introduces a new implementation. 48 Chapter 3 The Corner-Cube-Cell In this chapter, the adoption of a macroscopic, corner-cube-shaped antireflective structure for the improvement of solar energy collection is explored. The structure is implemented with conventional protected-and-textured Si solar cells in the form of a corner-cube-cell (CC-cell). The reflectance data collected in Chapter 2 is used in experimentally-seeded simulations of the complex optical and electrical performance characteristics of the CC-cell and benefits are seen over those of a simple flat-cell. The CC-cell is built, tested, and compared to the flat-cell. The term "CC-cell" is used throughout this chapter to refer to this corner-cube structure, with its three mutually-orthogonal textured-and-protected Si solar cells, while the term "flat-cell" is used to refer to one isolated textured-and-protected Si solar cell. A broad discussion detailing the benefits of the corner-cube-shaped antireflective structure over other macroscopic antireflective structures is first presented as a background in section 3.1. This CC-cell is then analysed in the following sections with regards to optical and electrical performance characteristics. The optical performance characteristics of a CC-cell are developed and discussed in section 3.2. The development of the optical performance characteristics is broken into nine sections: sections 3.2.1 to 3.2.8 show the optical performance characteristics of a CC-cell for each of the octants within the three-dimensional Cartesian coordinate system, and section 3.2.9 summarizes the developed theory by presenting a comprehensive simulation of the optical performance characteristics. The optical performance characteristics of the CC-cell differ greatly from the simple flat-cell and are seen as an increase in the captured incident optical power density. The electrical performance characteristics of the CC-cell are then developed in section 3.3. Two Maximum Power Point Tracking (MPPT) implementations for the CC-cell are discussed in section 3.3.1, both of which offer unique advantages. An experimentally-seeded simulation of the electrical performance characteristics of the CC-cell is presented in section 3.3.2 and improvements 49 in the generated electrical energy density are seen to exist for all days of the year across the globe. A physical implementation of the CC-cell is presented in section 3.4. The development of the measurement tool to probe the voltage and current characteristics of the CC-cell and flat-cell is presented in section 3.4.1. The physical CC-cell is then presented in section 3.4.2, and the electrical performance characteristics of the both the CC-cell and flat-cell are compared and discussed. Chapter 3 is then summarized in section 3.5 with all key findings regarding the performance of the CC-cell. 50 3.1. Background Macroscopic antireflective structures offer a possible route for the realization of an effective solar technology due to their ability to increase the captured incident optical power density and thus the generated electrical power density. One such structure, the corner-cube, has been shown to capture internal reflections,65 allowing light to interact with multiple surfaces within the structure, while enabling implementations with a perfect packing density across the horizontal plane. This lets the CC-cell offer improved optical and electrical performance characteristics when compared to the simple flat-cell. In a way, the macroscopic antireflective structure of the CC-cell lets it gain the benefits of textured Si solar cell surfaces, via increased absorption, but it does this without promoting surface recombination. The CC-cell offers two improvements to the optical performance characteristics, which manifest themselves as increases in the captured incident optical power density. The first improvement is the ability of the CC-cell to capture internal reflections. When the CC-cell is implemented with a solar cell that partially reflects incident light, the specular and/or diffuse reflection becomes incident onto the adjacent solar cells, which leads to a subsequent opportunity for absorption. In this work, only specular reflectance is considered because the level of diffuse reflection is much lower, as shown by and discussed in the experiments of Chapter 2. The specular reflectance is quantified in this work to define the optical performance characteristics of the CC-cell, and the results are presented as a table of algebraic expressions. The second improvement of the CC-cell is the omni-directional nature of the corner-cube-shape, which allows it to capture light from many directions while it is arrayed in the horizontal plane. These ideas are explored in more detail in section 3.2 where the optical performance characteristics of a CC-cell are developed and explored. The CC-cell also offers two improvements to the electrical performance characteristics, which manifest themselves as increases in the generated electrical power density. The first improvement is the ability of the CC-cell to generate electrical power during early mornings and late evenings, i.e., as the sun rises and sets. In such conditions, the flat-cell suffers from high reflective losses due to the large incident 51 angles of light. The second improvement relates to the electrical optimization of the CC-cell. For a flat-cell, the electrical optimization comes in the form of one MPPT unit. For a CC-cell, the electrical optimization can be implemented with a single MPPT unit, with electrical optimization of the entire structure as a whole, or with three MPPT units, with electrical optimization of each solar cell within the CC-cell. This latter case allows the three solar cells within the CC-cell to undergo independent electrical optimization. The nature of such electrical optimization is discussed in more detail in section 3.3. 52 3.2. Optical Performance Characteristics of a CC-cell The proposed solar structure is based upon a CC-cell. The CC-cell is constructed from three mutually-orthogonal square solar cells, to enable compatibility with conventional Si-based PV technology, and it is arrayed across the horizontal plane, to enable the captured optical power to be scaled up with the area. For the theoretical and experimental analyses in this work the square solar cells are implemented with protected-and-textured conventional Si solar cells. An array of CC-cells is illustrated in Figure 3.1. Figure 3.1a shows a photograph of the CC-cell array. Figure 3.1b shows a coloured rendering of the structure with the solar cells within the CC-cells grouped in three families. The family for solar cell x (SC-x) is shown in green, the family for solar cell y (SC-y) is shown in blue, and the family for solar cell z (SC-z) is shown in red. For uniform illumination of the CC-cell array at any incident angle, the solar cells within each family will capture equal optical powers while the solar cells within differing families will, in general, capture different optical powers. It is important to note that the CC-cell array covers the overall deployment area with a perfect packing density, assuming that there are negligible inactive areas for contacts and edges. This perfect packing density enhances the captured incident optical power density of the CC-cell array. To establish a benchmark for quantifying the captured incident optical power density of the CC-cell array, its results are contrasted to those of the flat-cell array, which can also cover the overall deployment area in a horizontal plane with a perfect packing density. The ensuing analyses compare the captured incident optical power densities of a single CC-cell and a single flat-cell, with the captured incident optical powers normalized by the respective deployment areas in the horizontal plane, as such an approach makes the results applicable to arrayed implementations. The optical performance characteristics of the CC-cell are quantified by its captured incident optical power density for the varying illumination conditions occurring over the day. A ray-tracing model is applied for this by way of two coordinate frames. The Global Coordinate Frame, discussed in Chapter 2 and shown on the left side of Figure 3.1b, is denoted by the primed coordinates x', y', and z'. It has its x'y'-plane on the 53 surface of the Earth, in the northern hemisphere, with its x'-axis directed South, its y'-axis directed East, and its z'-axis directed up. The angular orientation of an incident light ray from the sun is defined on this Global Coordinate Frame by the solar zenith angle, θ, and solar azimuth angle, ϕ.49 The Body Coordinate Frame, shown on the right side of Figure 3.1b, is denoted by the unprimed coordinates x, y, and z. The Body Coordinate Frame is shown attached to a representative CC-cell, with its x, y, and z axes normal to SC-x, SC-y, and SC-z, respectively. The central axis of symmetry of the CC-cell is parallel to the z'-axis of the Global Coordinate Frame. Figure 3.1. The CC-cells arranged in a close-packed array. The array is shown as (a) a photograph having textured-and-protected Si solar cells, with a side length of 7.9 cm, and (b) a schematic having its Si solar cells shaded in colours that correspond to three solar cell families: the SC-x family is in green, the SC-y family is in blue, and the SC-z family is in red. The Global Coordinate Frame, on the left side of the array, shows an incident light ray at a solar zenith angle of θ and solar azimuth angle of ϕ split into its directional cosine components of α', β', and γ'. The Body Coordinate Frame, on the right side of the array, shows the directional cosine components of the incident light ray transformed to the Global Coordinate Frame as α, β, and γ. 54 An incident light ray is shown in the Global Coordinate Frame, on the left side of Figure 3.1b, with a solar zenith angle of θ and solar azimuth angle of ϕ. The ray is projected onto the x', y', and z' axes to define respective directional cosines of α' = cosϕsinθ, (3.1) β'= sinϕsinθ, (3.2) γ' = cosθ. (3.3) A coordinate transformation is then used to define the directional cosines of α, β, and γ, along the respective x, y, and z axes of the Body Coordinate Frame. The coordinate transformation matrix is [αβγ]= [1 √6⁄ – 1 √2⁄ 1 √3⁄1 √6⁄ 1 √2⁄ 1 √3⁄–√2 √3⁄ 0 1 √3⁄] [α'β'γ'] . (3.4) With the above definitions, a given incident light ray with a solar zenith angle of θ and solar azimuth angle of ϕ (in the Global Coordinate Frame on the left side of Figure 3.1b) is redefined in terms of the directional cosines α, β, and γ (in the Body Coordinate Frame on the right side of Figure 3.1b). The benchmark for the ensuing analysis is set by the optical performance characteristics of the flat-cell. The captured incident optical power density for it was seen in section 2.4.1, and the developed expression (2.5) can be used directly here to define the captured incident optical power density of the flat-cell. The analysis of the CC-cell is carried out in terms of captured incident optical power densities. The captured incident optical power densities for SC-x, SC-y, and SC-z are defined as the incident optical power transmitted into the surfaces of each of the three solar cells divided by the area A of the solar cell. The captured incident optical power density for the entire CC-cell is then defined as the total incident optical power transmitted into the surfaces of the three solar cells divided by the deployment area of the CC-cell, √3A, being the area of the CC-cell projected in the x'y'-plane. Such an analysis follows that of our prior work,65 although the analysis is broadened in this work to characterize reflectance and transmittance for all 55 solar zenith angles, 0° < θ < 90°, and all solar azimuth angles, –180° < ϕ < 180°. This broadened analysis is carried out to encompass the effects of shadowing between the mutually orthogonal solar cells within an individual CC-cell. The captured incident optical power density for SC-x in the CC-cell is denoted by POP-x(α, β, γ). It is formed from primary, secondary, and tertiary incident optical power densities. The primary incident optical power density is the arriving optical power that directly illuminates SC-x divided by the area of the solar cell, A. It is denoted by δPOP-x(α, β, γ). The secondary incident optical power densities are the optical powers that illuminate SC-x following reflections off neighbouring solar cells divided by the area of the solar cell, A. For reflection off SC-y and onto SC-x or reflection off SC-z and onto SC-x, the secondary incident optical power densities are denoted by δPOP-yx(α, β, γ) and δPOP-zx(α, β, γ), respectively. The tertiary incident optical power densities are the optical powers that illuminate SC-x following reflections off two neighbouring solar cells divided by the area of the solar cell, A. For reflection off SC-z then SC-y and onto SC-x or reflection off SC-y then SC-z and onto SC-x, the tertiary incident optical power densities are denoted by δPOP-zyx(α, β, γ) and δPOP-yzx(α, β, γ), respectively. With these definitions, the captured incident optical power density for SC-x is POP-x(α, β, γ) = (1 – Rα)( δPOP-x(α, β, γ)+ δPOP-yx(α, β, γ)+ δPOP-zx(α, β, γ)+ δPOP-zyx(α, β, γ)+ δPOP-yzx(α, β, γ)) . (3.5) The factor of (1 – Rα) in (3.5) is the transmittance of the SC-x surface. It is a function of the angular-dependent reflectance, Rα, which is defined by the Fresnel reflectance expression (2.2) in Chapter 2 for unpolarized light striking the surface at an incident angle of cos-1(α). The dependence of the reflectance on  is denoted here by a simple subscript, rather than parentheses, to provide a more compact notation. The captured incident optical power density for SC-y in the CC-cell is denoted by POP-y(α, β, γ). It is also the result of multiple incident optical power densities. The primary incident optical power density, 56 δPOP-y(α, β, γ), is the arriving optical power that directly illuminates SC-y divided by the area of the solar cell, A. The secondary incident optical power densities, δPOP-zy(α, β, γ) and δPOP-xy(α, β, γ), are the optical powers that illuminate SC-y following reflections off neighbouring solar cells divided by the area of the solar cell, A. The tertiary incident optical power densities, δPOP-xzy(α, β, γ) and δPOP-zxy(α, β, γ), are the optical powers that illuminate SC-y following reflections off two neighbouring solar cells divided by the area of the solar cell, A. With these definitions, the captured incident optical power density for SC-y is POP-y(α, β, γ) = (1 – Rβ)( δPOP-y(α, β, γ)+ δPOP-zy(α, β, γ)+ δPOP-xy(α, β, γ)+ δPOP-xzy(α, β, γ)+ δPOP-zxy(α, β, γ)) . (3.6) The factor of (1 – Rβ) in (3.6) is the transmittance of the SC-x surface. It is a function of the angular-dependent reflectance, Rβ, which is defined by the Fresnel reflectance expression (2.2) in Chapter 2 for unpolarized light striking the surface at an incident angle of cos-1(β). The dependence of the reflectance on  is denoted here by a simple subscript, rather than parentheses, to provide a more compact notation. The captured incident optical power density for SC-z in the CC-cell is denoted by POP-z(α, β, γ). It is also the result of multiple incident optical power densities. The primary incident optical power density, δPOP-z(α, β, γ), is the arriving optical power that directly illuminates SC-z divided by the area of the solar cell, A. The secondary incident optical power densities, δPOP-xz(α, β, γ) and δPOP-yz(α, β, γ), are the optical powers that illuminate SC-z following reflections off neighbouring solar cells divided by the area of the solar cell, A. The tertiary incident optical power densities, δPOP-xyz(α, β, γ) and δPOP-yxz(α, β, γ), are the optical powers that illuminate SC-z following reflections off two neighbouring solar cells divided by the area of the solar cell, A. With these definitions, the captured incident optical power density for SC-z is 57 POP-z(α, β, γ) = (1 – Rγ)( δPOP-z(α, β, γ)+ δPOP-xz(α, β, γ)+ δPOP-yz(α, β, γ)+ δPOP-xyz(α, β, γ)+ δPOP-yxz(α, β, γ)) . (3.7) The factor of (1 – Rγ) in (3.7) is the transmittance of the SC-x surface. It is a function of the angular-dependent reflectance, Rγ, which is defined by the Fresnel reflectance expression (2.2) in Chapter 2 for unpolarized light striking the surface at an incident angle of cos-1(γ). The dependence of the reflectance on  is denoted here by a simple subscript, rather than parentheses, to provide a more compact notation. To investigate the optical performance of the CC-cell, the Global Coordinate Frame is split into eight octants as shown in Figure 3.2. Each octant can be differentiated by its unique combination of components being parallel or anti-parallel to the x-, y-, and z-axes. Each octant is then subject to an analysis of six distinct illumination cases, which arise due to the six unique combinations that can be made with the three directional cosines components. Each illumination case produces three expressions, one for each solar cell family within the CC-cell, for the captured incident optical power density of SC-x, SC-y, and SC-z. These expressions are derived with the following sections, in order, from octant I to VIII. 58 Figure 3.2. Depiction of the eight octants within the Global Coordinate Frame. All of the octants are depicted in order from octant I in (a) to octant VIII in (h). 59 3.2.1. Illumination Cases within Octant I Octant I is the octant enclosed by the positive x-axis, positive y-axis, and positive z-axis as seen in Figure 3.3. All light incident on the CC-cell within this octant has its directional cosines projected along the positive x-, y-, and z-axes. This means that incident light for this octant can undergo a maximum of three reflections within the CC-cell. In general, the number of positive axes upon which the directional cosines are projected defines the maximum number of reflections that can occur. This is because any projections along negative axes will have the incident light propagate away from the CC-cell. After the light has undergone the three internal reflections, it is directed outward from the CC-cell in an antiparallel direction to the incident light. Figure 3.3. Depiction of octant I, which is enclosed by the positive x-axis, positive y-axis, and positive z-axis. The analysis of octant I has been investigated in previous work.65 A compressed version of the previous findings is seen in Table 3.1. The equations presented in Table 3.1 depend solely on illumination case conditions for each solar cell family, which can be summarized into only five equations that can be applied given all illumination case conditions within octant I. The captured incident optical power density expressions in column four make use of the normal(α,β,γ), min(α,β,γ), mid(α,β,γ), and max(α,β,γ) functions. The function normal(α,β,γ) outputs the directional cosine, , , or , corresponding to the surface normal for the solar cell of interest, i.e., it outputs , , or  when defining an expression for SC-x, SC-y, or SC-z, respectively. The functions min(α,β,γ), mid(α,β,γ), and max(α,β,γ) output the directional cosines 60 Table 3.1. Summary of optical performance characteristics of a CC-cell within octant I. Each captured incident optical power density expression is a function of the directional cosines and specific illumination case condition. The captured incident optical power densities in column four make use of the normal(,,), min(,,), mid(,,), and max(,,) functions. POP-x Illumination Case Conditions POP-y Illumination Case Conditions POP-z Illumination Case Conditions Expression for POP-x, having normal(,,) =  Expression for POP-y, having normal(,,) =  Expression for POP-z, having normal(,,) =  α>β>γ>0 α>γ>β>0 β>α>γ>0 β>γ>α>0 γ>α>β>0 γ>β>α>0 =PSTC(1–Rnormal(α,β,γ))normal(α,β,γ) +PSTC(1–Rnormal(α,β,γ))Rmin(α,β,γ) (min(α,β,γ)–min(α,β,γ)mid(α,β,γ)2max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmid(α,β,γ) (mid(α,β,γ)–min(α,β,γ)mid(α,β,γ)2max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmin(α,β,γ)Rmid(α,β,γ) (min(α,β,γ)mid(α,β,γ)max(α,β,γ)) 2β>γ>β>α>0 2γ>β>γ>α>0 2α>γ>α>β>0 2γ>α>γ>β>0 2α>β>α>γ>0 2β>α>β>γ>0 =PSTC(1–Rnormal(α,β,γ))normal(α,β,γ) +PSTC(1–Rnormal(α,β,γ))Rmid(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmax(α,β,γ) (min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmid(α,β,γ)Rmax(α,β,γ) (2 min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)–min(α,β,γ)max(α,β,γ)2mid(α,β,γ)) 2α>γ>α>β>0 2α>β>α>γ>0 2β>γ>β>α>0 2β>α>β>γ>0 2γ>β>γ>α>0 2γ>α>γ>β>0 =PSTC(1–Rnormal(α,β,γ))normal(α,β,γ) +PSTC(1–Rnormal(α,β,γ))Rmin(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmax(α,β,γ) (mid(α,β,γ)–min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmin(α,β,γ)Rmax(α,β,γ) (2 min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)–min(α,β,γ)max(α,β,γ)2mid(α,β,γ)) γ>2α>α>β>0 β>2α>α>γ>0 γ>2β>β>α>0 α>2β>β>γ>0 β>2γ>γ>α>0 α>2γ>γ>β>0 =PSTC(1–Rnormal(α,β,γ))normal(α,β,γ) +PSTC(1–Rnormal(α,β,γ))Rmin(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmax(α,β,γ) (mid(α,β,γ)–min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmin(α,β,γ)Rmax(α,β,γ) (3min(α,β,γ)mid(α,β,γ)2max(α,β,γ)) γ>2β>β>α>0 β>2γ>γ>α>0 γ>2α>α>β>0 α>2γ>γ>β>0 β>2α>α>γ>0 α>2β>β>γ>0 =PSTC(1–Rnormal(α,β,γ))normal(α,β,γ) +PSTC(1–Rnormal(α,β,γ))Rmid(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmax(α,β,γ) (min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)) +PSTC(1–Rnormal(α,β,γ))Rmid(α,β,γ)Rmax(α,β,γ) (3min(α,β,γ)mid(α,β,γ)2max(α,β,γ)) 61 , , or  that hold the minimum, middle, and maximum values, respectively, for the given illumination case conditions. 62 3.2.2. Illumination Cases within Octant II Octant II is the octant enclosed by the negative x-axis, positive y-axis, and positive z-axis as seen in Figure 3.4. Thus, incident light can be reflected by a maximum of two surfaces within the CC-cell. Figure 3.4. Depiction of octant II, which is enclosed by the negative x-axis, positive y-axis, and positive z-axis. The four light vectors that are used in the analysis of this octant are r⃑SC-y = + αx̂ – βŷ – γẑ, (3.8) r⃑SC-z = + αx̂ – βŷ – γẑ, (3.9) r⃑SC-yz= + αx̂ + βŷ – γẑ, (3.10) and r⃑SC-zy= + αx̂ – βŷ + γẑ. (3.11) Equations (3.8) – (3.11) can be thought of as the light vector directly incident on SC-y, directly incident on SC-z, incident on SC-z after having been reflected off of SC-y, and incident on SC-y after having been reflected off of SC-z, respectively. The six illumination cases within this octant can be grouped into pairs that result in similar captured incident optical power densities. The first pair consists of the two illumination cases containing large absolute values for their α directional cosine, i.e., –α>β>γ>0 and –α>γ>β>0. An illustration of these two illumination cases is seen in Figure 3.5. Figure 3.5a shows the areas (in yellow) on the CC-cell that are illuminated given the –α>β>γ>0 case. Figure 3.5b shows the areas on the CC-cell that are illuminated given the –α>γ>β>0 case. The red line 63 being partially covered by SC-x is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.5a has a larger projection along the y-axis due to its larger β directional cosine, while the light ray in Figure 3.5b has a larger projection along the z-axis due to its larger γ directional cosine. Figure 3.5. Depiction of a CC-cell illuminated by light incident from octant II given the cases –α>β>γ>0 and –α>γ>β>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The light rays in (a) have a larger directional cosine along the y-axis, β, which causes SC-z to be illuminated more than SC-y. The light rays in (b) have a large directional cosine along the z-axis, γ, which causes SC-y to be illuminated more than SC-z. Light that is incident onto the CC-cell interacts with either SC-y or SC-z and is then reflected outward from the CC-cell. This pair produces only primary captured incident optical power densities before exiting the CC-cell altogether making only (3.8) and (3.9) relevant for this analysis. The large value of α indicates that most of the light is directed onto the CC-cell along the x-direction. The negative value of α indicates that the light is incident onto the CC-cell from the negative x-direction. As a consequence, there is a large amount of shadowing that occurs from light hitting the back of SC-x. This means the light can barely get by the solar cell in the yz-plane, and it surrounds the extremities of SC-y and SC-z. If β is larger, SC-z will be illuminated slightly more than SC-y. If γ is larger, SC-y will be illuminated slightly more than SC-z. The illuminated areas on SC-y and SC-z are simply triangles and are expressed as A⃑⃑SC-y=a2γ2α ŷ (3.12) 64 and A⃑⃑SC-z=a2β2α ẑ. (3.13) By using (3.8), (3.9), (3.12), and (3.13) with the STC incident optical power density, the primary captured incident optical power densities for SC-y and SC-z can be calculated given uniform illumination by finding the projection of the light vector onto the area normal-vectors, giving δPOP-y(α, β, γ) = PSTCr⃑SC-y ∙ A⃑⃑SC-yA = PSTC (–βγ2α) (3.14) and δPOP-z(α, β, γ) = PSTCr⃑SC-z ∙ A⃑⃑SC-zA = PSTC (–βγ2α) . (3.15) The second pair consists of the two illumination cases containing large absolute values for their β directional cosine, i.e., β>–α>γ>0 and β>γ>–α>0. An illustration of these two cases is seen in Figure 3.6. This pair has light incident onto the CC-cell interact with either SC-y or SC-z and then reflect onto a neighbouring cell. The light then reflects outward from the CC-cell. This pair produces both primary and secondary captured incident optical power densities making (3.8) – (3.11) relevant for the analysis. The primary captured incident optical power density contributions (in yellow) can be seen in Figure 3.6a and Figure 3.6b. The red line being partially covered by SC-x is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.6a has a larger projection along the x-axis due to its larger α directional cosine, while the light ray in Figure 3.6b has a larger projection along the z-axis due to its larger γ directional cosine. Illuminated areas on SC-y and SC-z for the primary captured incident optical power density contributions are seen in Figure 3.6a and Figure 3.6b. For both cases, SC-x is completely shaded and receives no secondary illumination, which can be predicted due to α being negative. Additionally, the β directional cosine is the largest and thus produces a large projection along the y-axis. This large projection manifests itself as a right-angled triangle, seen in the shadowing on SC-z as well as a large illumination on both SC-y and SC-z. These two manifestations can be further described by ray-tracing made from the top 65 right corner of SC-x onto the xz-plane. The large β directional cosine pushes this point inward toward the z-axis resulting in a large illuminated area on SC-y. The illuminated area on SC-y is a combination of a trapezoid and a rectangle and is calculated using a summed area. The illuminated area on SC-z is a trapezoid and is calculate using the formula for a trapezoid. The respective illuminated area normal-vectors for SC-y and SC-z are expressed as A⃑⃑SC-y = a2 (1 – αβ + αγ2β2) ŷ (3.16) and A⃑⃑SC-z = a2 (1 – α2β) ẑ. (3.17) By combining (3.8), (3.9), (3.16), and (3.17) with the STC optical power density, the primary captured incident optical power densities for SC-y and SC-z can be calculated given uniform illumination by finding the projection of the light vector onto the illumination area normal-vectors, giving δPOP-y(α, β, γ) = PSTCr⃑SC-y ∙ A⃑⃑SC-yA = PSTC (β + α – αγ2β) (3.18) and δPOP-z(α, β, γ) = PSTCr⃑SC-z ∙ A⃑⃑SC-zA = PSTC (γ + αγ2β) . (3.19) Illuminated areas on SC-y and SC-z for the secondary captured incident optical power density contributions (in dark yellow) are seen in Figure 3.6c - Figure 3.6f. The secondary contributions are more complicated in that they are subject to a comparative condition, namely knowing whether β>2γ or β<2γ. This analysis uses (3.10) and (3.11) as light vectors that have been reflected off of SC-y onto SC-z and SC-z onto SC-y, respectively. 66 Figure 3.6. Depiction of a CC-cell being illuminated by light incident from octant II given the illumination cases β>–α>γ>0 and β>γ>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). Light being reflected off of SC-z onto SC-y for both illumination cases is depicted in Figure 3.6c. The illuminated area on SC-y is simply a rectangle that stretches toward the z-axis the larger β becomes. Light 67 being reflected off of SC-y onto SC-z for both illumination cases, depicted in Figure 3.6d-Figure 3.6f, is more involved and is subject to the aforementioned comparative condition. Given the condition β>2γ, seen in Figure 3.6d, light being reflected off of SC-y for both illumination cases produces an illuminated area on SC-z in the shape of a trapezoid. Given the condition β<2γ, seen in Figure 3.6e and Figure 3.6f, the illuminated area on SC-z differs depending on the illumination case. For β>–α>γ>0, the illuminated area is a small triangle. This is due to α being large, ultimately causing the light to exit the CC-cell quickly out toward the positive x-axis. For β>γ>–α>0, the illuminated area is larger and takes an aggregate form of a trapezoid and a rectangle. This is due to γ being large, ultimately weighting the negative z-direction heavily over the positive x-direction. These illuminated area normal-vectors are expressed as A⃑⃑SC-zy = a2 (γβ – αγβ2) ŷ, (3.20) A⃑⃑SC-yz = a2 (1 – 3α2β) ẑ, (3.21) A⃑⃑SC-yz = a2 (β2α + α2β – 1) ẑ, (3.22) and A⃑⃑SC-yz = a2 (1 + α2β – 2αγ + αβ2γ2) ẑ, (3.23) for the illumination conditions depicted in Figure 3.6d-Figure 3.6f, respectively. The secondary captured incident optical power density contributions for SC-y and SC-z can be calculated for uniform illumination using (3.10), (3.11), (3.20) – (3.23), and the STC incident optical power density. A key difference between primary and secondary captured incident optical power densities is the inclusion of the incident-angle-dependent reflectance. The incident angle of light normal to any solar cell within the CC-cell can be found by taking the inverse cosine of the directional cosine normal to the solar cell. The incident angle of light found this way is defined off of the surface normal of the solar cell and can be used either with the reflectance data gathered in Chapter 2 or the Fresnel equations39 for unpolarized light. The secondary 68 captured incident optical power density contributions given the illumination case of β>–α>γ>0 are expressed as δPOP-zy(α, β, γ) = PSTCr⃑SC-zy ∙ A⃑⃑SC-zyA = RγPSTC (γ + αγβ) (3.24) and δPOP-yz(α, β, γ) = PSTCr⃑SC-yz ∙ A⃑⃑SC-yzA= RβPSTC( γ + 3αγ2β , β<2γ–βγ2α – αγ2β – γ , β<2γ) . (3.25) Equations (3.24) and (3.25) use the directional cosines γ and β in their reflectance calculation due to light having reflected off of SC-z onto SC-y in the case of (3.24) and due to light having reflected off of SC-y onto SC-z in the case of (3.25). The secondary captured incident optical power density contributions given the illumination case of β>γ>–α>0 are expressed as δPOP-zy(α, β, γ) = PSTCr⃑SC-zy ∙ A⃑⃑SC-zyA = RγPSTC (γ + αγβ) (3.26) and δPOP-yz(α, β, γ) = PSTCr⃑SC-yz ∙ A⃑⃑SC-yzA= RβPSTC( γ + 3αγ2β , β<2γγ – αγ2β + 2α – αβ2γ , β<2γ) . (3.27) Equations (3.26) and (3.27) use the directional cosines γ and β in their reflectance calculation due to light having reflected off of SC-z onto SC-y in the case of (3.26) and due to light having reflected off of SC-y onto SC-z in the case of (3.27). The third pair consists of the two illumination cases containing large absolute values for their γ directional cosine, γ>–α>β>0 and γ>β>–α>0. An illustration of these two cases is seen in Figure 3.7. This pair has light incident onto the CC-cell interact with either SC-y or SC-z and then reflect onto a neighbouring 69 cell. The light then reflects outward from the CC-cell. This pair produces both primary and secondary captured incident optical power densities making (3.8) – (3.11) relevant for the analysis. The primary captured incident optical power density contributions (in yellow) can be seen in Figure 3.7a and Figure 3.7b. The red line being partially covered by SC-x is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.7a has a larger projection along the x-axis due to its larger α directional cosine while the light ray in Figure 3.7b has a larger projection along the y-axis due to its larger β directional cosine. Illuminated areas on SC-y and SC-z for the primary captured incident optical power density contributions are seen in Figure 3.7a and Figure 3.7b. For both cases, SC-x is completely shaded and receives no secondary illumination, which can be predicted due to α being negative, in that it is constantly moving away from SC-x after it passes by the yz-plane. Additionally, the γ directional cosine is the largest and thus produces a large projection, which is projected along the z-axis. This large projection manifests itself as a right-angled triangle seen in the shadowing on SC-y as well as a large illumination on both SC-y and SC-z. These two manifestations can be further described by the ray trace made from the top left corner of SC-x onto the xz-plane. The large γ directional cosine pushes this point inward toward the z-axis resulting in a large illuminated area on SC-z. The illuminated area on SC-z is a combination of a trapezoid and a rectangle and is calculated using an aggregate formula for the area. The illuminated area on SC-y is a trapezoid and is calculate using the formula for its area. The illuminated area normal-vectors for SC-y and SC-z are expressed as A⃑⃑SC-y = a2 (1 – α2γ) ŷ (3.28) and A⃑⃑SC-z = a2 (1 – αγ + αβ2γ2) ẑ. (3.29) 70 Figure 3.7. Depiction of a CC-cell being illuminated by light incident from octant II given the illumination cases γ>–α>β>0 and γ>β>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). 71 By combining (3.8), (3.9), (3.28), and (3.29) with the STC incident optical power density, the primary captured incident optical power densities for SC-y and SC-z can be calculated given uniform illumination by finding the projection of the light vector onto the illumination area normal-vector, δPOP-y(α, β, γ) = PSTCr⃑SC-y ∙ A⃑⃑SC-yA = PSTC (β + αβ2γ) (3.30) and δPOP-z(α, β, γ) = PSTCr⃑SC-z ∙ A⃑⃑SC-zA = PSTC (γ + α – αβ2γ) . (3.31) Illuminated areas on SC-y and SC-z for the secondary captured incident optical power density contributions (in dark yellow) are seen in Figure 3.7c - Figure 3.7f. The secondary contributions are more complicated in that they are subject to a comparative condition, namely knowing whether γ>2β or γ<2β. This analysis uses (3.10) and (3.11) as light vectors that have been reflected off of SC-y onto SC-z and SC-z onto SC-y, respectively. Light being reflected off of SC-y onto SC-z for both illumination cases is depicted in Figure 3.7c. The illuminated area on SC-z is simply a rectangle that stretches toward the y-axis the larger γ becomes. Light being reflected off of SC-z onto SC-y for both illumination cases, depicted in Figure 3.7d-Figure 3.7f, is more involved and is subject to the aforementioned comparative condition. Given the condition γ>2β, seen in Figure 3.7d, light being reflected off of SC-z for both illumination cases produces an illuminated area on SC-y in the shape of a trapezoid. Given the condition γ<2β, seen in Figure 3.7e and Figure 3.7f, the illuminated area on SC-y differs depending on the illumination case. For γ>–α>β>0, the illuminated area is a small triangle. This is due to the light ray vector having a large α component, ultimately causing the light to exit the CC-cell quickly, out toward the positive x-axis. In the case of γ>β>–α>0, the illuminated area is larger and takes an aggregate form of a trapezoid and a rectangle. This is due to the light ray vector having a large β component, ultimately weighting the negative y-direction heavily over the positive x-direction. These illuminated area normal-vectors can be expressed as 72 A⃑⃑SC-yz = a2 (βγ – αβγ2) ẑ, (3.32) A⃑⃑SC-zy = a2 (1 – 3α2γ) ŷ, (3.33) A⃑⃑SC-zy = a2 (γ2α + α2γ – 1) ŷ, (3.34) and A⃑⃑SC-zy = a2 (1 + α2γ – 2αβ + αγ2β2) ŷ, (3.35) for the illumination conditions depicted in Figure 3.7d-Figure 3.7f, respectively. The secondary captured incident optical power density contributions for SC-y and SC-z can be calculated for uniform illumination using (3.10), (3.11), (3.32) – (3.35), and the STC optical power density. A key difference between primary and secondary captured incident optical power densities is the inclusion of the incident-angle-dependent reflectance. The incident angle of light normal to any solar cell within the CC-cell can be found by taking the inverse cosine of the directional cosine normal to the solar cell. The incident angle of light found this way is defined off of the surface normal of the solar cell and can be used either with the reflectance data gathered in Chapter 2 or the Fresnel equations39 for unpolarized light. The secondary captured incident optical power density contributions given the illumination case of γ>–α>β>0 are expressed as δPOP-yz(α, β, γ) = PSTCr⃑SC-yz ∙ A⃑⃑SC-yzA = RβPSTC (β + αβγ) (3.36) and δPOP-zy(α, β, γ) = PSTCr⃑SC-zy ∙ A⃑⃑SC-zyA= RγPSTC( β + 3αβ2γ , γ>2β–βγ2α – αβ2γ – β , γ<2β) . (3.37) Equations (3.36) and (3.37) use the directional cosines β and γ in their reflectance calculation due to light having reflected off of SC-y onto SC-z in the case of (3.36) and due to light having reflected off of SC-z 73 onto SC-y in the case of (3.37). The secondary captured incident optical power density contributions given the illumination case of γ>β>–α>0 are expressed as δPOP-yz(α, β, γ) = PSTCr⃑SC-yz ∙ A⃑⃑SC-yzA = RβPSTC (β + αβγ) (3.38) and δPOP-zy(α, β, γ) = PSTCr⃑SC-zy ∙ A⃑⃑SC-zyA = RγPSTC( β + 3αβ2γ , γ>2ββ – αβ2γ + 2α – αγ2β , γ<2β) . (3.39) Equations (3.38) and (3.39) use the directional cosines β and γ in their reflectance calculation due to light having reflected off of SC-y onto SC-z in the case of (3.38) and due to light having reflected off of SC-z onto SC-y in the case of (3.39). 74 3.2.3. Illumination Cases within Octant III Octant III is the octant enclosed by the negative x-axis, negative y-axis, and positive z-axis as seen in Figure 3.8. Thus, incident light can be reflected by a maximum of one surface within the CC-cell. Figure 3.8. Depiction of octant III, which is enclosed by the negative x-axis, negative y-axis, and positive z-axis. The single light vector that is used in the analysis of this octant is r⃑SC-z = + αx̂ + βŷ – γẑ. (3.40) Equation (3.40) can be thought of as a light vector directly incident onto SC-z. Once light is reflected off of SC-z it exits the CC-cell resulting in a single chance for light to be captured by the CC-cell. This single surface interaction can be predicted due to the two negative directional cosines α and β, which indicate that light is traveling away from both SC-x and SC-y. Only the positive γ directional cosine yields a surface interaction due to light traveling toward SC-z. The analysis within this octant only requires investigation into a single pair of illumination cases, γ>–α>–β>0 and γ>–α>–β>0. This can be further seen through the analysis of the square illumination area in Figure 3.9. For –α>–β>γ>0 and –α>γ>–β>0, the value of the innermost corner of the square illumination area always has its x-component larger than the side length of SC-z. For –β>–α>γ>0 and –β>γ>–α>0, the value of the inner most corner of the square illumination area always has its y-component larger than the side length of SC-z. For γ>–α>–β>0 and γ>–α>–β>0, the value of the inner most corner of the square 75 illumination area always has its x- and y-components smaller than the side length of SC-z. The square illuminated area created on SC-z is expressed as A⃑⃑SC-z=a2 (1 – βγ – αγ + αβγ2) ẑ. (3.41) Combining (3.40) and (3.41) with the STC optical power density, the captured incident optical power density for SC-z is calculated as δPOP-z(α, β, γ) = PSTCr⃑SC-z ∙ A⃑⃑SC-zA=PSTC (γ + β + α + αβγ) . (3.42) Figure 3.9. Depiction of a CC-cell being illuminated by light incident from octant III given the illumination cases γ>–α>–β>0 and γ>–β>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. 76 3.2.4. Illumination Cases within Octant IV Octant IV is the octant enclosed by the positive x-axis, negative y-axis, and positive z-axis as seen in Figure 3.10. Thus, incident light can be reflected by a maximum of two surfaces within the CC-cell. Figure 3.10. Depiction of octant IV, which is enclosed by the positive x-axis, negative y-axis, and positive z-axis. The four vectors of light that are used in the analysis of this octant are r⃑SC-x = – αx̂ + βŷ – γẑ, (3.43) r⃑SC-z = – αx̂ + βŷ – γẑ, (3.44) r⃑SC-xz= + αx̂ + βŷ – γẑ, (3.45) and r⃑SC-zx= – αx̂ + βŷ + γẑ. (3.46) Equations (3.43) – (3.46) can be thought of as the light vector directly incident on SC-x, directly incident on SC-z, incident on SC-z after having been reflected off of SC-x, and incident on SC-x after having been reflected off of SC-z. The six illumination cases within this octant can be grouped into pairs that result in similar captured incident optical power densities. The first pair consists of the two illumination cases containing large absolute values for their α directional cosine, α>–β>γ>0 and α>γ>–β>0. An illustration of these two illumination cases is seen in Figure 3.11. This pair has light incident onto the CC-cell interact with either SC-x or SC-z and then reflect onto a neighbouring cell. The light is then reflected outward from the CC-cell. This pair produces both 77 primary and secondary captured incident optical power densities making (3.43) – (3.46) relevant for the analysis. The primary captured incident optical power density contributions (in yellow) can be seen in Figure 3.11a and Figure 3.11b. The red arrow being partially covered by SC-y is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.11a has a larger projection along the y-axis due to its larger β directional cosine while the light ray in Figure 3.11b has a larger projection along the z-axis due to its larger γ directional cosine. Illuminated areas on SC-x and SC-z for the primary captured incident optical power density contributions are seen in Figure 3.11a and Figure 3.11b. For both cases, SC-y is completely shaded and receives no secondary illumination, which can be predicted due to β being positive, in that it is constantly moving away from SC-y after it passes by the xz-plane. Additionally, the α directional cosine is the largest and thus produces a large projection, which is projected along the x-axis. This large projection manifests itself as a right-angled triangle seen in the shadowing on SC-z as well as a large illumination on both SC-x and SC-z. These two manifestations can be further described by the ray trace made from the top left corner of SC-y onto the yz-plane. The large α directional cosine pushes this point inward toward the y-axis resulting in a large illuminated area on SC-x. The illuminated area on SC-x is a combination of a trapezoid and a rectangle and is calculated using an aggregate area formula. The illuminated area on SC-z is a trapezoid and is calculate using the formula for its area. The illuminated area normal-vectors for SC-x and SC-z are expressed as A⃑⃑SC-x = a2 (1 – βα + βγ2α2) x̂ (3.47) and A⃑⃑SC-z = a2 (1 – β2α) ẑ. (3.48) 78 Figure 3.11. Depiction of a CC-cell being illuminated by light incident from octant VI given the illumination cases α>–β>γ>0 and α>γ>–β>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). 79 By combining (3.43), (3.44), (3.47), and (3.48) with the STC optical power density, the primary captured incident optical power densities for SC-x and SC-z can be calculated given uniform illumination by finding the projection of the light rays onto the illumination area normal-vector, δPOP-x(α, β, γ) = PSTCr⃑SC-x ∙ A⃑⃑SC-xA = PSTC (α + β – βγ2α) (3.49) and δPOP-z(α, β, γ) = PSTCr⃑SC-z ∙ A⃑⃑SC-zA = PSTC (γ + βγ2α) . (3.50) Illuminated areas on SC-x and SC-z for the secondary captured incident optical power density contributions (in dark yellow) are seen in Figure 3.11c - Figure 3.11f. The secondary contributions are more complicated in that they are subject to a comparative condition, namely knowing whether α>2γ or α<2γ. This analysis uses (3.45) and (3.46) as light vectors that have been reflected off of SC-x onto SC-z and SC-z onto SC-x respectively. Light being reflected off of SC-z onto SC-x for both illumination cases is depicted in Figure 3.11c. The illuminated area on SC-x is simply a rectangle that stretches toward the z-axis the larger α becomes. Light being reflected off of SC-x onto SC-z for both illumination cases, depicted in Figure 3.11d - Figure 3.11f, is more involved and is subject to the aforementioned comparative condition. Given the condition α>2γ, seen in Figure 3.11d, light being reflected off of SC-y for both illumination cases produces an illuminated area on SC-z in the shape of a trapezoid. Given the condition α<2γ, depicted in Figure 3.11e and Figure 3.11f, the illuminated area on SC-y differs depending on the illumination case. For α>–β>γ>0, the illuminated area is a small triangle. This is due to β being large, ultimately causing the light to exit the CC-cell quickly out toward the positive y-axis. For α>γ>–β>0, the illuminated area is larger and takes an aggregate form of a trapezoid and a rectangle. This is due to γ being large, ultimately weighting the negative z-direction heavily over the positive y-direction. These illuminated area normal-vectors can be expressed as 80 A⃑⃑SC-zx = a2 (γα – βγα2) x̂, (3.51) A⃑⃑SC-xz = a2 (1 – 3β2α) ẑ, (3.52) A⃑⃑SC-xz = a2 (α2β + β2α – 1) ẑ, (3.53) and A⃑⃑SC-xz = a2 (1 + β2α – 2βγ + αβ2γ2) ẑ, (3.54) for the illumination conditions depicted in Figure 3.11d - Figure 3.11f, respectively. The secondary captured incident optical power density contributions for SC-x and SC-z can be calculated for uniform illumination using (3.45), (3.46), (3.51) – (3.54), and the STC optical power density. A key difference between primary and secondary captured incident optical power densities is the inclusion of the incident-angle-dependent reflectance. The incident angle of light normal to any solar cell within the CC-cell can be found by taking the inverse cosine of the directional cosine normal to the solar cell. The incident angle of light found this way is defined off of the surface normal of the solar cell and can be used either with the reflectance data gathered in Chapter 2 or the Fresnel equations39 for unpolarized light. The secondary captured incident optical power density contributions given the illumination case of α>–β>γ>0 are expressed as δPOP-zx(α, β, γ) = PSTCr⃑SC-zx ∙ A⃑⃑SC-zxA = RαPSTC (γ + βγα) (3.55) and δPOP-xz(α, β, γ) = PSTCr⃑SC-xz ∙ A⃑⃑SC-xzA= RγPSTC( γ + 3βγ2α , α>2γ–αγ2β – βγ2α – γ , α<2γ) . (3.56) Equations (3.55) and (3.56) use the directional cosines α and γ in their reflectance calculation due to light having reflected off of SC-x onto SC-z in the case of (3.55) and due to light having reflected off of SC-z 81 onto SC-x in the case of (3.56). The secondary captured incident optical power density contributions given the illumination case of α>γ>–β>0 are expressed as δPOP-zx(α, β, γ) = PSTCr⃑SC-zx ∙ A⃑⃑SC-zxA = RαPSTC (γ + βγα) (3.57) and δPOP-xz(α, β, γ) = PSTCr⃑SC-xz ∙ A⃑⃑SC-xzA = RγPSTC( γ + 3βγ2α , α>2γγ – βγ2α + 2β – αβ2γ , α<2γ) . (3.58) Equations (3.57) and (3.58) use the directional cosines α and γ in their reflectance calculation due to light having reflected off of SC-x onto SC-z in the case of (3.57) and due to light having reflected off of SC-z onto SC-x in the case of (3.58). The second pair consists of the two illumination cases containing large absolute values for their β directional cosine, –β>α>γ>0 and –β>γ>α>0. An illustration of these two illumination cases is seen in Figure 3.12. Figure 3.12a shows the areas on the CC-cell that are illuminated given the –β>α>γ>0 case. Figure 3.12b shows the areas on the CC-cell that are illuminated given the –β>γ>α>0 case. The red arrow being partially covered by SC-y is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.12a has a larger projection along the x-axis due to its larger α directional cosine while the light ray in Figure 3.12b has a larger projection along the z-axis due to its larger γ directional cosine. 82 Figure 3.12. Depiction of a CC-cell being illuminated by light incident from octant IV given the cases –β>α>γ>0 and –β>γ>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The light ray in (a) has a larger projection along the x-axis due to its larger α directional cosine, which causes SC-z to be illuminated more than SC-x. The light ray in (b) has a larger projection along the z-axis due to its larger γ directional cosine, which causes SC-x to be illuminated more than SC-z. Light that is incident onto the CC-cell interacts with either SC-x or SC-z and is then reflected outward from the CC-cell. This pair produces only primary captured incident optical power densities (in yellow) before exiting the CC-cell altogether making only (3.43) and (3.44) relevant for this analysis. The large value of β indicates that most of the light is directed onto the CC-cell along the y-direction. The negative value of β indicates that the light is incident onto the CC-cell from the negative y-direction. As a consequence, there is a large amount of shadowing that occurs from light hitting the back of SC-y. This light can barely get by the solar cell in the xz-plane and hugs the extremities of SC-x and SC-z. If α is larger, SC-z will be illuminated slightly more than SC-x. If γ is larger, SC-x will be illuminated slightly more than SC-z. The illuminated areas on SC-x and SC-z are simply triangles and are expressed as A⃑⃑SC-x=a2γ2β x̂ (3.59) and A⃑⃑SC-z=a2α2β ẑ. (3.60) 83 By using (3.43), (3.44), (3.59), and (3.60) with the STC optical power density, the primary captured incident optical power densities for SC-x and SC-z can be calculated given uniform illumination by finding the projection of the light vector onto the illumination area normal-vector, δPOP-x(α, β, γ) = PSTCr⃑SC-x ∙ A⃑⃑SC-xA = PSTC (–αγ2β) (3.61) and δPOP-z(α, β, γ) = PSTCr⃑SC-z ∙ A⃑⃑SC-zA = PSTC (–αγ2β) . (3.62) The third pair consists of the two illumination cases containing large absolute values for their γ directional cosine, γ>α>–β>0 and γ>–β>α>0. An illustration of these two cases is seen in Figure 3.13. This pair has light incident onto the CC-cell interact with either SC-x or SC-z and then reflect onto a neighbouring cell. The light is then reflects outward from the CC-cell. This pair produces both primary and secondary captured incident optical power densities making (3.43) – (3.46) relevant for the analysis. The primary captured incident optical power density contributions can be seen in Figure 3.13a and Figure 3.13b. The red arrow that is being partially covered by SC-y is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.13a has a larger projection along the x-axis due to its larger α directional cosine while the light ray in Figure 3.13b has a larger projection along the z-axis due to its larger β directional cosine. Illuminated areas on SC-x and SC-z for the primary captured incident optical power density (in yellow) contributions are seen in Figure 3.13a and Figure 3.13b. For both cases, SC-y is completely shaded and receives no secondary illumination, which can be predicted due to β being negative, in that it is constantly moving away from SC-y after it passes by the xz-plane. Additionally, the γ directional cosine is the largest and thus produces a large projection, which is projected along the z-axis. This large projection manifests itself as a right-angled triangle seen in the shadowing on SC-x as well as a large illumination on both SC-x and SC-z. These two manifestations can be further described by the ray trace made from the top left corner of SC-x onto the xz-plane. The large γ directional cosine pushes this point 84 inward toward the z-axis resulting in a large illuminated area on SC-z. The illuminated area on SC-z is a combination of a trapezoid and a rectangle and is calculated using an aggregate area formula. The illuminated area on SC-x is a trapezoid and is calculate using the formula for its area. The illuminated area normal-vectors for SC-x and SC-z are expressed as A⃑⃑SC-x = a2 (1 – β2γ) x̂ (3.63) and A⃑⃑SC-z = a2 (1 – βγ + αβ2γ2) ẑ. (3.64) By combining (3.43), (3.44), (3.63), and (3.64) with the STC optical power density, the primary captured incident optical power densities for SC-y and SC-z can be calculated given uniform illumination by finding the projection of the light rays onto the illumination area normal-vector, δPOP-x(α, β, γ) = PSTCr⃑SC-x ∙ A⃑⃑SC-xA = PSTC (α + αβ2γ) (3.65) and δPOP-z(α, β, γ) = PSTCr⃑SC-z ∙ A⃑⃑SC-zA = PSTC (γ + β – αβ2γ) . (3.66) Illuminated areas on SC-x and SC-z for the secondary captured incident optical power density contributions (in dark yellow) are seen in Figure 3.13c - Figure 3.13f. The secondary contributions are more complicated in that they are subject to a comparative condition, namely knowing whether γ>2α or γ<2α. This analysis uses (3.45) and (3.46) as light vectors that have been reflected off of SC-x onto SC-z and SC-z onto SC-x, respectively. Light being reflected off of SC-x onto SC-z for both illumination cases is depicted in Figure 85 3.13c. The illuminated area on SC-z is simply a rectangle that stretches toward the x-axis the larger γ becomes. Figure 3.13. Depiction of a CC-cell being illuminated by light incident from octant IV given the illumination cases γ>α>–β>0 and γ>–β>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). 86 Light being reflected off of SC-z onto SC-x for both illumination cases, depicted in Figure 3.13d - Figure 3.13f, is more involved and is subject to the aforementioned comparative condition. Given the condition γ>2α, seen in Figure 3.13d, light being reflected off of SC-z for both illumination cases produces an illuminated area on SC-x in the shape of a trapezoid. Given the condition γ<2α, seen in Figure 3.13e and Figure 3.13f, the illuminated area on SC-z differs depending on the illumination case. For γ>–β>α>0, the illuminated area is a small triangle. This is due to β being large, ultimately causing the light to exit the CC-cell quickly, out toward the positive y-axis. For γ>α>–β>0, the illuminated area is larger and takes an aggregate form of a trapezoid and a rectangle. This is due to α being large, ultimately weighting the negative x-direction heavily over the positive y-direction. These illuminated area normal-vectors can be expressed as A⃑⃑SC-xz = a2 (αγ – αβγ2) ẑ, (3.67) A⃑⃑SC-zx = a2 (1 – 3β2γ) x̂, (3.68) A⃑⃑SC-zx = a2 (1 + β2γ – 2βα + βγ2α2) x̂, (3.69) and A⃑⃑SC-zx = a2 (γ2β + β2γ – 1) x̂, (3.70) for the illumination conditions depicted in Figure 3.13d - Figure 3.13f, respectively. The secondary captured incident optical power density contributions for SC-x and SC-z can be calculated for uniform illumination using (3.45), (3.46), (3.67) – (3.70), and the STC optical power density. A key difference between primary and secondary captured incident optical power densities is the inclusion of the incident-angle-dependent reflectance. The incident angle of light normal to any solar cell within the CC-cell can be found by taking the inverse cosine of the directional cosine normal to the solar cell. The incident angle of light found this way is defined off of the surface normal of the solar cell and can be used either with the reflectance data 87 gathered in Chapter 2 or the Fresnel equations39 for unpolarized light. The secondary captured incident optical power density contributions given the illumination case of γ>α>–β>0 are expressed as δPOP-xz(α, β, γ) = PSTCr⃑SC-xz ∙ A⃑⃑SC-xzA = RαPSTC (α + αβγ) (3.71) and δPOP-zx(α, β, γ) = PSTCr⃑SC-zx ∙ A⃑⃑SC-zxA= RγPSTC( α + 3αβ2γ , γ>2αα – αβ2γ + 2β – βγ2α , γ<2α) . (3.72) Equations (3.71) and (3.72) use the directional cosines β and γ in their reflectance calculation due to light having reflected off of SC-y onto SC-z in the case of (3.71) and due to light having reflected off of SC-z onto SC-y in the case of (3.72). The secondary captured incident optical power density contributions given the illumination case of γ>–β>α>0 are expressed as δPOP-xz(α, β, γ) = PSTCr⃑SC-xz∙A⃑⃑SC-xzA = RαPSTC (α + αβγ) (3.73) and δPOP-zx(α, β, γ) = PSTCr⃑SC-zx∙A⃑⃑SC-zxA = RγPSTC( α + 3αβ2γ , γ>2α–αγ2β – αβ2γ – α , γ<2α) . (3.74) Equations (3.73) and (3.74) use the directional cosines α and γ in their reflectance calculation due to light having reflected off of SC-x onto SC-z in the case of (3.73) and due to light having reflected off of SC-z onto SC-x in the case of (3.74). 88 3.2.5. Illumination Cases within Octant V Octant V is the octant enclosed by the positive x-axis, positive y-axis, and negative z-axis as seen in Figure 3.14.Thus, incident light can be reflected by a maximum of two surfaces within the CC-cell. Figure 3.14. Depiction of octant V, which is enclosed by the positive x-axis, positive y-axis, and negative z-axis. The four light vectors that are used in the analysis of this octant are r⃑SC-x = – αx̂ – βŷ + γẑ, (3.75) r⃑SC-y = – αx̂ – βŷ + γẑ, (3.76) r⃑SC-xy= + αx̂ – βŷ + γẑ, (3.77) and r⃑SC-yx= – αx̂ + βŷ + γẑ. (3.78) Equations (3.75) – (3.78) can be thought of as the light vector directly incident on SC-x, directly incident on SC-y, incident on SC-y after having been reflected off of SC-x, and incident on SC-x after having been reflected off of SC-y, respectively. The six illumination cases within this octant can be grouped into pairs that result in similar captured incident optical power densities. The first pair consists of the two illumination cases containing large absolute values for their α directional cosine, α>β>–γ>0 and α>–γ>β>0. An illustration of these two illumination cases is seen in Figure 3.15.This pair has light that is incident onto the CC-cell interact with either SC-x or SC-y and then 89 reflect onto a neighbouring cell. The light is then reflects outward from the CC-cell. This pair produces both primary and secondary captured incident optical power densities making (3.75) – (3.78) relevant for the analysis. The primary captured incident optical power density contributions (in yellow) can be seen in Figure 3.15a and Figure 3.15b. The red arrow being partially covered by SC-z is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.15a has a larger projection along the y-axis due to its larger β directional cosine while the light ray in Figure 3.15b has a larger projection along the z-axis due to its larger γ directional cosine. Illuminated areas on SC-x and SC-y for the primary captured incident optical power density contributions are seen in Figure 3.15a and Figure 3.15b. For both cases, SC-z is completely shaded and receives no secondary illumination, which can be predicted due to γ being negative, in that it is constantly moving away from SC-z after it passes by the xy-plane. Additionally, the α directional cosine is the largest and thus produces a large projection, which is projected along the x-axis. This large projection manifests itself as a right-angled triangle seen in the shadowing on SC-y as well as a large illumination on both SC-x and SC-y. These two manifestations can be further described by the ray trace made from the left-most corner of SC-z onto the xy-plane. The large α directional cosine pushes this point inward toward the y-axis resulting in a large illuminated area on SC-x. The illuminated area on SC-x is a combination of a trapezoid and a rectangle and is calculated using an aggregate area formula. The illuminated area on SC-y is a trapezoid and is calculate using the formula for its area. The illuminated area normal-vectors for SC-x and SC-y are expressed as A⃑⃑SC-x = a2 (1 – γα + βγ2α2) x̂ (3.79) and A⃑⃑SC-y = a2 (1 – γ2α) ŷ. (3.80) 90 Figure 3.15. Depiction of a CC-cell being illuminated by light incident from octant V given the illumination cases α>β>–γ>0 and α>–γ>β>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). 91 By combining (3.75), (3.76), (3.79), and (3.80) with the STC optical power density, the primary captured incident optical power densities for SC-x and SC-z can be calculated given uniform illumination by finding the projection of the light vector onto the illumination area normal-vector, δPOP-x(α, β, γ) = PSTCr⃑SC-x ∙ A⃑⃑SC-xA = PSTCa2 (α + γ – βγ2α) (3.81) and δPOP-y(α, β, γ) = PSTCr⃑SC-y ∙ A⃑⃑SC-yA = PSTCa2 (β + βγ2α) . (3.82) Illuminated areas on SC-x and SC-y for the secondary captured incident optical power density contributions (in dark yellow) are seen in Figure 3.15c - Figure 3.15f. The secondary contributions are more complicated in that they are subject to a comparative condition, namely knowing whether α>2β or α<2β. This analysis uses (3.77) and (3.78) as light vectors that have been reflected off of SC-x onto SC-y and SC-y onto SC-x, respectively. Light being reflected off of SC-y onto SC-x for both illumination cases is depicted in Figure 3.15c. The illuminated area on SC-x is simply a rectangle that stretches toward the y-axis the larger α becomes. Light being reflected off of SC-x onto SC-y for both illumination cases, depicted in Figure 3.15d - Figure 3.15f, is more involved and is subject to the aforementioned comparative condition. Given the condition α>2β , seen in Figure 3.15d, light being reflected off of SC-x for both illumination cases produces an illuminated area on SC-y in the shape of a trapezoid. Given the condition α<2β, seen in Figure 3.15e and Figure 3.15f, the illuminated area on SC-x differs depending on the illumination case. For α>–γ>β>0, the illuminated area is a small triangle. This is due to α being large, ultimately causing the light to exit the CC-cell quickly out toward the positive x-axis. For α>β>–γ>0, the illuminated area is larger and takes an aggregate form of a trapezoid and a rectangle. This is due to β being large, ultimately weighting the negative y-direction heavily over the positive x-direction. These illuminated area normal-vectors can be expressed as 92 A⃑⃑SC-yx = a2 (βα – βγα2) x̂, (3.83) A⃑⃑SC-xy = a2 (1 – 3γ2α) ŷ, (3.84) A⃑⃑SC-xy = a2 (1 + γ2α – 2βγ + αγ2β2) ŷ, (3.85) and A⃑⃑SC-xy = a2 (α2γ + γ2α – 1) ŷ, (3.86) for the illumination conditions depicted in Figure 3.15d - Figure 3.15f, respectively. The secondary captured incident optical power density contributions for SC-x and SC-z can be calculated for uniform illumination using (3.77), (3.78), (3.83) – (3.86), and the STC optical power density. A key difference between primary and secondary captured incident optical power densities is the inclusion of the incident-angle-dependent reflectance. The incident angle of light normal to any solar cell within the CC-cell can be found by taking the inverse cosine of the directional cosine normal to the solar cell. The incident angle of light found this way is defined off of the surface normal of the solar cell and can be used either with the reflectance data gathered in Chapter 2 or the Fresnel equations39 for unpolarized light. The secondary captured incident optical power density contributions given the illumination case of α>β>–γ>0 are expressed as δPOP-yx(α, β, γ) = PSTCr⃑SC-yx ∙ A⃑⃑SC-yxA = RβPSTC (β + βγα) (3.87) and δPOP-xy(α, β, γ) = PSTCr⃑SC-xy ∙ A⃑⃑SC-xyA= RαPSTC( β + 3βγ2α , α>2β–αβ2γ – βγ2α – β , α<2β) . (3.88) Equations (3.87) and (3.88) use the directional cosines α and γ in their reflectance calculation due to light having reflected off of SC-x onto SC-z in the case of (3.87) and due to light having reflected off of SC-z 93 onto SC-x in the case of (3.88). The secondary captured incident optical power density contributions given the illumination case of α>–γ>β>0 are expressed as δPOP-yx(α, β, γ) = PSTCr⃑SC-yx ∙ A⃑⃑SC-yxA = RβPSTC (β + βγα) (3.89) and δPOP-xy(α, β, γ) = PSTCr⃑SC-xy ∙ A⃑⃑SC-xyA = RαPSTC( β + 3βγ2α , α>2ββ – βγ2α + 2γ – αγ2β , α<2β) . (3.90) Equations (3.89) and (3.90) use the directional cosines α and β in their reflectance calculation due to light having reflected off of SC-x onto SC-y in the case of (3.89) and due to light having reflected off of SC-y onto SC-x in the case of (3.90). The second pair consists of the two illumination cases containing large absolute values for their β directional cosine, β>α>–γ>0 and β>–γ>α>0. An illustration of these two cases is seen in Figure 3.16.This pair has light that is incident onto the CC-cell interact with either SC-x or SC-z and then reflect onto a neighbouring cell. The light is then reflected outward from the CC-cell. This pair produces both primary and secondary captured incident optical power densities making (3.75) – (3.78) relevant for the analysis. The primary captured incident optical power density contributions (in yellow) can be seen in Figure 3.16a and Figure 3.16b. The red arrow being partially covered by SC-z is representative of a single light ray incident onto the CC-cell. The light ray in Figure 3.16a has a larger projection along the x-axis due to its larger α directional cosine while the light ray in Figure 3.16b has a larger projection along the z-axis due to its larger γ directional cosine. Illuminated areas on SC-x and SC-y for the primary captured incident optical power density contributions are seen in Figure 3.16a and Figure 3.16b. For both cases, SC-z is completely shaded and receives no secondary illumination, which can be predicted due to γ being negative, in that it is constantly moving away from SC-z after it passes by the xy-plane. Additionally, the β directional cosine is the largest and thus produces a large projection, which is projected along the y-axis. This large projection 94 manifests itself as a right angled triangle seen in the shadowing on SC-x as well as a large illumination on both SC-x and SC-y. These two manifestations can be further described by the ray trace made from the corner of SC-z furthest from the origin onto the xz-plane. The large γ directional cosine pushes this point inward toward the x-axis resulting in a large illuminated area on SC-y. The illuminated area on SC-y is a combination of a trapezoid and a rectangle and is calculated using an aggregate formula for the area. The illuminated area on SC-x is a trapezoid and is calculate using the formula for the area of a trapezoid. The illuminated area normal-vectors for SC-x and SC-y are expressed as A⃑⃑SC-x = a2 (1 – γ2β) x̂ (3.91) and A⃑⃑SC-y = a2 (1 – γβ + αγ2β2) ŷ. (3.92) By combining (3.75), (3.76), (3.91), and (3.92) with the standard test condition optical power density, the primary captured incident optical power densities for SC-y and SC-z can be calculated given uniform illumination by finding the projection of the light vector onto the illumination area normal-vector, δPOP-x(α, β, γ) = PSTCr⃑SC-x ∙ A⃑⃑SC-xA = PSTC (α + αγ2β) (3.93) And δPOP-y(α, β, γ) = PSTCr⃑SC-y ∙ A⃑⃑SC-yA = PSTC (β + γ – αγ2β) . (3.94) Illuminated areas on SC-x and SC-y for the secondary captured incident optical power density contributions (in dark yellow) are seen in Figure 3.16c - Figure 3.16f. The secondary contributions are more complicated in that they are subject to a comparative condition, namely knowing whether β>2α or β<2α. This analysis uses (3.93) and (3.94) as light vectors that have been reflected off of SC-x onto SC-y and SC-y onto SC-x, respectively. 95 Figure 3.16. Depiction of a CC-cell being illuminated by light incident from octant V given the illumination cases β>α>–γ>0 and β>–γ>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The primary captured incident optical power densities can be seen in (a) and (b). The secondary captured incident optical power densities can be seen in (c), (d), (e), and (f). Light being reflected off of SC-x onto SC-y for both illumination cases is depicted in Figure 3.16c. The illuminated area on SC-y is simply a rectangle that stretches toward the x-axis the larger β becomes. Light 96 being reflected off of SC-y onto SC-x for both illumination cases, depicted in Figure 3.16d - Figure 3.16f, is more involved and is subject to the aforementioned comparative condition. Given the condition β>2α , seen in Figure 3.16d, light being reflected off of SC-y for both illumination cases produces an illuminated area on SC-x in the shape of a trapezoid. Given the condition β<2α, seen in Figure 3.16e and Figure 3.16f, the illuminated area on SC-y differs depending on the illumination case. For β>–γ>α>0, the illuminated area is a small triangle. This is due to γ being large, ultimately causing the light to exit the CC-cell quickly, out toward the positive z-axis. For β>α>–γ>0, the illuminated area is larger and takes an aggregate form of a trapezoid and a rectangle. This is due to α being large, ultimately weighting the negative x-direction heavily over the positive z-direction. These illuminated area normal-vectors can be expressed as A⃑⃑SC-xy = a2 (αβ – αγβ2) ŷ, (3.95) A⃑⃑SC-yx = a2 (1 – 3γ2β) x̂, (3.96) A⃑⃑SC-yx = a2 (1 + γ2β – 2γα + βγ2α2) x̂, (3.97) And A⃑⃑SC-yx = a2 (β2γ + γ2β – 1) x̂, (3.98) for the illumination conditions depicted in Figure 3.16d - Figure 3.16f, respectively. The secondary captured incident optical power density contributions for SC-x and SC-y can be calculated for uniform illumination using (3.77), (3.78), (3.95) – (3.98), and the STC optical power density. A key difference between primary and secondary captured incident optical power densities is the inclusion of the incident-angle-dependent reflectance. The incident angle of light normal to any solar cell within the CC-cell can be found by taking the inverse cosine of the directional cosine normal to the solar cell. The incident angle of light found this way is defined off of the surface normal of the solar cell and can be used either with the reflectance data 97 gathered in Chapter 2 or the Fresnel equations39 for unpolarized light. The secondary captured incident optical power density contributions given the illumination case of β>α>–γ>0 are expressed as δPOP-xy(α, β, γ) = PSTCr⃑SC-xy∙A⃑⃑SC-xyA = RαPSTC (α + αγβ) (3.99) And δPOP-yx(α, β, γ) = PSTCr⃑SC-yx∙A⃑⃑SC-yxA = RβPSTC( α + 3αγ2β , β>2αα – αγ2β + 2γ – βγ2α , β<2α) . (3.100) Equations (3.99) and (3.100) use the directional cosines α and β in their reflectance calculation due to light having reflected off of SC-y onto SC-x in the case of (3.99) and due to light having reflected off of SC-x onto SC-y in the case of (3.100). The secondary captured incident optical power density contributions given the illumination case of β>–γ>α>0 are expressed as δPOP-xy(α, β, γ) = PSTCr⃑SC-xy ∙ A⃑⃑SC-xyA = RαPSTC (α + αγβ) (3.101) And δPOP-yx(α, β, γ) = PSTCr⃑SC-yx ∙ A⃑⃑SC-yxA = RβPSTC( α + 3αγ2β , β>2α–αβ2γ – αγ2β – α , β<2α) . (3.102) Equations (3.101) and (3.102) use the directional cosines α and β in their reflectance calculation due to light having reflected off of SC-x onto SC-y in the case of (3.101) and due to light having reflected off of SC-y onto SC-x in the case of (3.102). The third pair consists of the two illumination cases containing large absolute values for their γ directional cosine, –γ>α>β>0 and –γ>β>α>0. An illustration of these two illumination cases is seen in Figure 3.17. Figure 3.17a shows the areas on the CC-cell that are illuminated given the –γ>α>β>0 case. 98 Figure 3.17b shows the areas on the CC-cell that are illuminated given the –γ>β>α>0 case. The red line that is being partially covered by SC-z is representative of a single light ray incident to the CC-cell for both cases. The light ray in Figure 3.17a has a larger projection along the x-axis due to its larger α directional cosine while the light ray in Figure 3.17b has a larger projection along the y-axis due to its larger β directional cosine. Figure 3.17. Depiction of a CC-cell being illuminated by light incident from octant V given the cases –γ>α>β>0 and –γ>β>α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The light ray in (a) has a larger projection along the x-axis due to its larger α directional cosine, which causes SC-y to be illuminated more than SC-x. The light ray in (b) has a larger projection along the y-axis due to its larger β directional cosine, which causes SC-x to be illuminated more than SC-y. Light that is incident to the CC-cell interacts with either SC-x or SC-y and is then reflected outward from the CC-cell. This pair produce only a primary captured incident optical power density (in yellow) before exiting the CC-cell altogether making only (3.75) and (3.76) relevant for this analysis. The large value of γ indicates that most of the light is directed onto the CC-cell along the z-direction. The negative value of γ indicates that the light is incident onto the CC-cell from the negative z-direction. As a consequence, there is a large amount of shadowing that occurs from light hitting the back of SC-z. This means the light can barely get by the solar cell in the xy-plane and hugs the extremities of SC-x and SC-y. If α is larger, SC-y is illuminated slightly more than SC-x. If β is larger, SC-x is illuminated slightly more than SC-y. The illuminated areas on SC-x and SC-y are simply triangles and are expressed as 99 A⃑⃑SC-x=a2β2γ x̂ (3.103) and A⃑⃑SC-y=a2α2γ ŷ. (3.104) By using (3.75), (3.76), (3.103), and (3.104) with the STC optical power density, the primary captured incident optical power densities for SC-x and SC-y can be calculated given uniform illumination by finding the projection of the light rays vector onto the illumination area normal-vector, δPOP-x(α, β, γ) = PSTCr⃑SC-x ∙ A⃑⃑SC-xA = PSTC (–αβ2γ) (3.105) and δPOP-y(α, β, γ) = PSTCr⃑SC-y ∙ A⃑⃑SC-yA = PSTC (–αβ2γ) . (3.106) 100 3.2.6. Illumination Cases within Octant VI Octant VI is the octant enclosed by the negative x-axis, positive y-axis, and negative z-axis as seen in Figure 3.18. Thus, incident light can be reflected by a maximum of one surface within the CC-cell. Figure 3.18. Depiction of octant VI, which is enclosed by the negative x-axis, positive y-axis, and negative z-axis. Incident light has a chance of being reflected off of only one surface within the CC-cell. The single light vector that is used in the analysis of this octant is r⃑SC-y = + αx̂ – βŷ + γẑ. (3.107) Equation (3.107) can be thought of as the light vector directly incident onto SC-y. Once light is reflected off of SC-y it exits the CC-cell resulting in a single chance for light to be captured within the CC-cell. This single surface interaction can be predicted due to the two negative directional cosines α and γ, which indicate that light is traveling away from both SC-x and SC-z. Only the positive β directional cosine yields a surface interaction due to light traveling toward SC-y. The analysis within this octant only requires investigation into a single pair of illumination cases, β>–α>–γ>0 and β>–γ>–α>0. This can be further seen through the analysis of the square illumination area in Figure 3.19. For –α>β>–γ>0 and –α>–γ>β>0, the value of the inner most corner of the square illumination area always has its x-component larger than the side length of SC-y. For γ>–α>–β>0 and γ>–α>–β>0, the value of the inner most corner of the square illumination area always has its z-component larger than the 101 side length of SC-y. For –β>–α>γ>0 and –β>γ>–α>0, the value of the inner most corner of the square illumination area always has its x- and z-components smaller than the side length of SC-y. Figure 3.19. Depiction of a CC-cell being illuminated by light incident from octant VI given the illumination cases β>–α>–γ>0 and β>–γ>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The square illuminated area created on SC-y is expressed as A⃑⃑SC-y=a2 (1 – γβ – αβ + αγβ2) ŷ. (3.108) Combining (3.107) and (3.108) with the STC optical power density, the captured incident optical power density (in yellow) for SC-y is calculated as δPOP-y(α, β, γ) = PSTCr⃑SC-y ∙ A⃑⃑SC-yA=PSTC (β + γ + α + αγβ) . (3.109) 102 3.2.7. Illumination Cases within Octant VII Octant VII has is the octant enclosed by the negative x-axis, negative y-axis, and negative z-axis as seen in Figure 3.20. Thus, incident light cannot enter the CC-cell and cannot be reflected from a surface within the CC-cell. Its captured incident optical power density for all solar cells is therefore zero. Figure 3.20. Depiction of octant VII, which is enclosed by the negative x-axis, negative y-axis, and negative z-axis. 103 3.2.8. Illumination Cases within Octant VIII Octant VIII is the octant enclosed by the positive x-axis, negative y-axis, and negative z-axis as seen in Figure 3.21. Thus, incident light can be reflected by a maximum of one surface within the CC-cell. Figure 3.21. Depiction of octant VIII, which is enclosed by the positive x-axis, negative y-axis, and negative z-axis. The single light vector that is used in the analysis of this octant is r⃑SC-x = – αx̂ + βŷ + γẑ. (3.110) Equation (3.110) can be thought of as the light vector directly incident onto SC-x. Once light is reflected off of SC-x it exits the CC-cell resulting in a single chance for light to be captured within the CC-cell. This single surface interaction can be predicted due to the two negative directional cosines β and γ, which indicate that light is traveling away from both SC-y and SC-z. Only the positive α directional cosine yields a surface interaction due to light traveling toward SC-x. The analysis within this octant only requires investigation into a single pair of illumination cases, α>–β>–γ>0 and α>–γ>–β>0. This can be further seen through the analysis of the square illumination area in Figure 3.22. For –β>α>–γ>0 and –β>–γ>α>0, the value of the inner most corner of the square illumination area always has its y-component larger than the side length of SC-x. For –γ>α>–β>0 and –γ>–β>α>0, the value of the inner most corner of the square illumination area always has its z-component larger than the 104 side length of SC-x. For α>–β>–γ>0 and α>–γ>–β>0, the value of the inner most corner of the square illumination area always has its y- and z-components smaller than the side length of SC-x. Figure 3.22. Depiction of a CC-cell being illuminated by light incident from octant VIII given the illumination cases γ>–α>–β>0 and γ>–β>–α>0. A light ray (red arrow) is seen to aid in visualizing the illumination cases. The square illuminated area created on SC-x is expressed as A⃑⃑SC-x=a2 (1 – βα – γα + βγα2) x̂. (3.111) Combining (3.110) and (3.111) with the STC optical power density, the captured incident optical power density (in yellow) for SC-x is calculated as δPOP-x(α, β, γ) = PSTCr⃑SC-x∙A⃑⃑SC-xA=PSTC (α + β + γ + βγα) . (3.112) 105 3.2.9. Summary of Illumination Cases In this section, the preceding analysis on the optical performance characteristics of the CC-cell, given uniform incident illumination from each octant, is summarized for the three solar cell families. Similarities between illumination cases are highlighted and all illumination cases from all octants are combined into twelve equations that can define the optical performance characteristics of the CC-cell. The (benchmark) optical performance characteristics of the flat-cell are then presented along with simulations of the optical performance characteristics of the CC-cell using the aforementioned summary of equations and experimental reflectance data. The optical performance characteristics of the CC-cell are contrasted with the optical performance characteristics of the flat-cell. The prior analysis can be summarized by applying (3.5), (3.6), and (3.7) to find POP-x(α, β, γ), POP-y(α, β, γ), and POP-z(α, β, γ), respectively. Table 3.2 summarizes the captured incident optical power densities for light incident from each octant onto SC-x. For light incident onto SC-x from octant I, the overall captured incident optical power density has primary, secondary, and tertiary captured incident optical power density contributions. This is in contrast to the other seven octants, which have at most primary and secondary captured incident optical power density contributions. By examining the incident optical power density equations for SC-x in Table 3.2, we see that light incident from octants IV and V produce primary and secondary incident optical power density contributions and light incident from octant VIII produces only a primary incident optical power density contribution. Similar tables can be compiled for SC-y and SC-z and similar observations can be made in that light incident from octant I produces primary, secondary, and tertiary contributions with only slight differences for octants that produce only primary and secondary contributions. One key observation from Table 3.2 is that every illumination case with the α directional cosine projected along the negative x-axis results in a captured incident optical power density of zero. This is consistent with the prior derivation and is representative of light incident into the CC-cell simply hitting the back of SC-x as it passes through the y'z'-plane from the negative x-direction. For SC-y, light incident from octants II and V produces primary and secondary incident optical power 106 density contributions and light incident from octant VI produces only a primary incident optical power density contribution. For SC-z, light incident from octants II and IV produces primary and secondary incident optical power density contributions and light incident from octant III produces only a primary incident optical power density contribution. Table 3.2. Summary of the captured incident optical power density of SC-x for each illumination case within octants I to VIII. Octant Illumination Case POP-x(α, β, γ) I α>β>γ>0 δPOP-x + δPOP-yx + δPOP-zx + δPOP-yzx + δPOP-zyx = PSTC(1–Rα)(α + Rγ (γ –βγ2α)+Rβ (β–βγ2α)+RβRγ (βγα)) α>γ>β>0 δPOP-x + δPOP-yx + δPOP-zx + δPOP-yzx + δPOP-zyx = PSTC(1–Rα)(α + Rβ (β–βγ2α)+Rγ (γ –βγ2α)+RβRγ (βγα)) β>α>γ>0 δPOP-x + δPOP-yx + δPOP-zx + δPOP-yzx + δPOP-zyx = PSTC(1–Rα)( α+Rγ (αγ2β)+Rβ (α–αγ2β)+RβRγ{ 2γ –αγ2β–βγ2α, β<2α3αγ2β, β≥2α) β>γ>α>0 δPOP-x + δPOP-yx + δPOP-zx + δPOP-yzx + δPOP-zyx = PSTC(1–Rα)( α+Rγ (αγ2β)+Rβ (α–αγ2β)+RβRγ{ 2α –αγ2β–βα2γ, β<2γ3αγ2β, β≥2γ) γ>α>β>0 δPOP-x + δPOP-yx + δPOP-zx + δPOP-yzx + δPOP-zyx = PSTC(1–Rα)( α+Rβ (αβ2γ)+Rγ (α–αβ2γ)+RβRγ{ 2β –αβ2γ–βγ2α, γ<2α3αβ2γ, γ≥2α) γ>β>α>0 δPOP-x + δPOP-yx + δPOP-zx + δPOP-yzx + δPOP-zyx = PSTC(1–Rα)( α+Rβ (αβ2γ)+Rγ (α–αβ2γ)+RβRγ{ 2α –αβ2γ–αγ2β, γ<2β3αβ2γ, γ≥2β) II -α>β>γ>0 δPOP-x = 0 -α>γ>β>0 β>-α>γ>0 β>γ>-α>0 γ>-α>β>0 γ>β>-α>0 III -α>-β>γ>0 δPOP-x = 0 -α>γ>-β>0 -β>-α>γ>0 -β>γ>-α>0 γ>-α>-β>0 γ>-β>-α>0 IV α>-β>γ>0 δPOP-x(α, β, γ)+δPOP-zx(α, β, γ) = PSTC(1–Rα)(α+β–βγ2α+𝑅γ (γ+βγα)) α>γ>-β>0 -β>α>γ>0 δPOP-x(α, β, γ) = PSTC(1–Rα) (–αγ2β) -β>γ>α>0 107 γ>α>-β>0 δPOP-x(α, β, γ)+δPOP-zx(α, β, γ) = PSTC(1–Rα)( α+αβ2γ+𝑅γ{ (α-αβ2γ–βγ2α+2β) , γ<2α(α+3αβ2γ) , γ≥2α) γ>-β>α>0 δPOP-x(α, β, γ)+δPOP-zx(α, β, γ) = PSTC(1–Rα)( α+αβ2γ+𝑅γ{ (–αγ2β–αβ2γ–α) , γ<2α(α+3αβ2γ) , γ≥2α) V α>β>-γ>0 δPOP-x(α, β, γ)+δPOP-yx(α, β, γ) = PSTC(1–Rα)(α+γ–βγ2α+𝑅β (β+βγα)) α>-γ>β>0 β>α>-γ>0 δPOP-x(α, β, γ)+δPOP-yx(α, β, γ) = PSTC(1–Rα)( α+αγ2β+𝑅β{ (α–αγ2β–βγ2α+2γ) , β<2α(α+3αγ2β) , β≥2α) β>-γ>α>0 δPOP-x(α, β, γ)+δPOP-yx(α, β, γ) = PSTC(1–Rα)( α+αγ2β+𝑅β{ (–αβ2γ–αγ2β–α) , β<2α(α+3αγ2β) , β≥2α) -γ>α>β>0 δPOP-x(α, β, γ) = PSTC(1–Rα) (–αβ2γ) -γ>β>α>0 VI -α>β>-γ>0 δPOP-x = 0 -α>-γ>β>0 β>-α>-γ>0 β>-γ>-α>0 -γ>-α>β>0 -γ>β>-α>0 VII -α>-β>-γ>0 δPOP-x = 0 -α>-γ>-β>0 -β>-α>-γ>0 -β>-γ>-α>0 -γ>-α>-β>0 -γ>-β>-α>0 VIII α>-β>-γ>0 δPOP-x(α, β, γ) = PSTC(1–Rα) (α+β+γ+βγα) α>-γ>-β>0 δPOP-x(α, β, γ) = PSTC(1–Rα) (α+β+γ+βγα) -β>α>-γ>0 δPOP-x = 0 -β>-γ>α>0 -γ>α>-β>0 -γ>-β>α>0 The captured incident optical power density contributions of the CC-cell as a whole are simply a combination of all contributions from each solar cell family. The captured incident optical power density contributions from each octant for the CC-cell, much like that for each of the solar cell families, can be grouped into octants that produce differing contributions. Light incident onto the CC-cell from octant I produces primary, secondary, and tertiary incident optical power density contributions. Light incident onto the CC-cell from this octant is subject to more surface interactions than light incident onto the CC-cell from 108 any other octant. This is because light from octant I is initially described by three positive directional cosines. This is indicative of light initially traveling toward all three of the solar cell families, which allows the maximum number of surface interactions to be predicted as three. Light incident onto the CC-cell from octants II, IV, and V produce primary and secondary incident optical power density contributions. Light incident onto the CC-cell from these three octants has a similarity in that it hits the back of one solar cell family. Likewise, light incident from octants II, VI, and V onto the CC-cell is initially described in part by only one negative directional cosine. This is indicative of light traveling toward two of the three solar cell families, which allows the maximum number of surface interactions to be predicted as two. Light incident onto the CC-cell from octants III, VI, and VIII produces only a primary incident optical power density contribution. Light incident onto the CC-cell from these three octants has a similarity in that it hits the backs of two solar cell families. As such, light incident from octants III, VI, and VIII onto the CC-cell is initially described in part by two negative directional cosines. Light incident onto the CC-cell from octant VII produces no incident optical power density contribution. Light incident onto the CC-cell from this octant only illuminates the inactive backs of the solar cell families. Once the light passes the x'y'-, y'z'-, and x'z'-planes, corresponding to SC-x, SC-y, and SC-z, respectively, it cannot hit an active region of any of the solar cell families within the CC-cell. The initial qualitative assessment of the optical performance characteristics of the CC-cell indicates that most benefit is seen while light is incident from octant I. Fortunately, for a static implementation, this is easily exploited by orienting the CC-cell with its central axis of symmetry parallel to the z-axis of the Global Coordinate Frame (facing straight up toward the sky). For the remaining seven octants, three produce primary and secondary captured incident optical power density contributions, three produce only primary captured incident optical power density contributions, and one produces no captured incident optical power density contributions. The qualitative description of each octant can be seen in more detail in Table 3.2. Table 3.3 gives a full summary of all possible captured incident optical power densities generalized for any solar cell family given the specific illumination cases. 109 Table 3.3. Summary of all possible captured incident optical power densities generalized for any solar cell family given the specific illumination cases. Each captured incident optical power density is a function of a specific group of illumination case conditions as well as the solar cell family. The captured incident optical power densities in column four are a function of the normal(,,), min(,,), mid(,,), and max(,,) functions. These functions find the normal, minimum-valued, middle-valued, and maximum-valued directional cosine components associated with the illumination case condition, respectively. POP-x Illumination Case POP-y Illumination Case POP-z Illumination Case POP-i(α,β,γ), i = x, y, or z α>β>γ>0 α>γ>β>0 β>α>γ>0 β>γ>α>0 γ>α>β>0 γ>β>α>0 =PSTC(1–Rnormal(α,β,γ))(normal(α,β,γ)+Rmin(α,β,γ) (min(α,β,γ)–min(α,β,γ)mid(α,β,γ)2max(α,β,γ))+Rmid(α,β,γ) (mid(α,β,γ)–min(α,β,γ)mid(α,β,γ)2max(α,β,γ))+Rmin(α,β,γ)Rmid(α,β,γ) (min(α,β,γ)mid(α,β,γ)max(α,β,γ))) 2β>γ>β>α>0 2γ>β>γ>α>0 2α>γ>α>β>0 2γ>α>γ>β>0 2α>β>α>γ>0 2β>α>β>γ>0 =PSTC(1–Rnormal(α,β,γ)) (normal(α,β,γ)+Rmid(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmax(α,β,γ) (min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmid(α,β,γ)Rmax(α,β,γ) (2 min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)–min(α,β,γ)max(α,β,γ)2mid(α,β,γ))) 2α>γ>α>β>0 2α>β>α>γ>0 2β>γ>β>α>0 2β>α>β>γ>0 2γ>β>γ>α>0 2γ>α>γ>β>0 =PSTC(1–Rnormal(α,β,γ)) (normal(α,β,γ)+Rmin(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmax(α,β,γ) (mid(α,β,γ)–min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmin(α,β,γ)Rmax(α,β,γ) (2 min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ)–min(α,β,γ)max(α,β,γ)2mid(α,β,γ))) γ>2α>α>β>0 β>2α>α>γ>0 γ>2β>β>α>0 α>2β>β>γ>0 β>2γ>γ>α>0 α>2γ>γ>β>0 =PSTC(1–Rnormal(α,β,γ)) (normal(α,β,γ)+Rmin(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmax(α,β,γ) (mid(α,β,γ)–min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmin(α,β,γ)Rmax(α,β,γ) (3min(α,β,γ)mid(α,β,γ)2max(α,β,γ))) γ>2β>β>α>0 β>2γ>γ>α>0 γ>2α>α>β>0 α>2γ>γ>β>0 β>2α>α>γ>0 α>2β>β>γ>0 =PSTC(1–Rnormal(α,β,γ)) (normal(α,β,γ)+Rmid(α,β,γ) (min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmax(α,β,γ) (min(α,β,γ) –min(α,β,γ)mid(α,β,γ)2 max(α,β,γ))+Rmid(α,β,γ)Rmax(α,β,γ) (3min(α,β,γ)mid(α,β,γ)2max(α,β,γ))) α>-β>γ>0 α>γ>-β>0 α>β>-γ>0 α>-γ>β>0 β>-α>γ>0 β>γ>-α>0 β>α>-γ>0 β>-γ>α>0 γ>-α>β>0 γ>β>-α>0 γ>α>-β>0 γ>-β>α>0 =PSTC(1–Rnormal(α,β,γ))(max(α,β,γ)+min(α,β,γ)–mid(α,β,γ)min(α,β,γ)2max(α,β,γ)+Rmid(α,β,γ) (mid(α,β,γ)+mid(α,β,γ)min(α,β,γ)max(α,β,γ))) -β>α>γ>0 -β>γ>α>0 -γ>α>β>0 -γ>β>α>0 -α>β>γ>0 -α>γ>β>0 -γ>α>β>0 -γ>β>α>0 -α>β>γ>0 -α>γ>β>0 -β>α>γ>0 -β>γ>α>0 =PSTC(1–Rnormal(α,β,γ))(–mid(α,β,γ)max(α,β,γ)2min(α,β,γ)) 110 2α>γ>α>-β>0 2α>β>α>-γ>0 2β>γ>β>-α>0 2β>α>β>-γ>0 2γ>β>γ>-α>0 2γ>α>γ>-β>0 =PSTC(1–Rnormal(α,β,γ))(mid(α,β,γ)(1+min(α,β,γ)2 max(α,β,γ)+Rmax(α,β,γ) (1–min(α,β,γ)2 max(α,β,γ)–min(α,β,γ)max(α,β,γ)2(mid(α,β,γ))2+2min(α,β,γ)mid(α,β,γ)))) 2α>γ>-β>α>0 2α>β>-γ>α>0 2β>γ>-α>β>0 2β>α>-γ>β>0 2γ>β>-α>γ>0 2γ>α>-β>γ>0 =PSTC(1–Rnormal(α,β,γ))(mid(α,β,γ)(1+min(α,β,γ)2 max(α,β,γ)+Rmax(α,β,γ) (–max(α,β,γ)2 min(α,β,γ)–min(α,β,γ)2 max(α,β,γ)–1))) γ>2α>α>-β>0 γ>2α>-β>α>0 γ>-β>2α>α>0 β>2α>α>-γ>0 β>2α>-γ>α>0 β>-γ>2α>α>0 γ>2β>β>-α>0 γ>2β>-α>β>0 γ>-α>2β>β>0 α>2β>β>-γ>0 α>2β>-γ>β>0 α>-γ>2β>β>0 β>2γ>γ>-α>0 β>2γ>-α>γ>0 β>-α>2γ>γ>0 α>2γ>γ>-β>0 α>2γ>-β>γ>0 α>-β>2γ>γ>0 =PSTC(1–Rnormal(α,β,γ)) (mid(α,β,γ)(1+min(α,β,γ)2 max(α,β,γ)+Rmax(α,β,γ) (1+3 min(α,β,γ)2 max(α,β,γ)))) α>-β>-γ>0 α>-γ>-β>0 β>-α>-γ>0 β>-γ>-α>0 γ>-α>-β>0 γ>-β>-α>0 =PSTC(1–Rnormal(α,β,γ)) (max(α,β,γ)+mid(α,β,γ)+min(α,β,γ)+mid(α,β,γ)min(α,β,γ)max(α,β,γ)) -β>α>-γ>0 -β>-γ>α>0 -γ>α>-β>0 -γ>-β>α>0 α<0 -α>β>-γ>0 -α>-γ>β>0 -γ>-α>β>0 -γ>β>-α>0 β<0 -α>-β>γ>0 -α>γ>-β>0 -β>-α>γ>0 -β>γ>-α>0 γ<0 =0 The captured incident optical power densities in Table 3.3 are used to calculate the captured incident optical power density of the CC-cell, which is defined as the total optical power transmitted into the surfaces of SC-x, SC-y, and SC-z, divided by the deployment area of the CC-cell, √3A, being the area of the CC-cell projected in the x'y'-plane. This is done by converting the per unit solar cell area to a per unit projected CC-cell area by multiplying POP-x(α,β,γ) + POP-y(α,β,γ) + POP-z(α,β,γ) by A √3A⁄ . Thus, the captured incident optical power density for the CC-cell (and its array) is POP-CC = POP-x(α,β,γ)+POP-y(α,β,γ)+POP-z(α,β,γ)√3. (3.113) The captured incident optical power densities of the CC-cell and flat-cell can be compared. This comparison is valid as both captured incident optical power densities are functions of the optical power transmitted through all active regions on each device, divided by their respective areas projected in the x'y'-plane. Figure 3.23 shows the percent improvement in captured incident optical power density of the CC-cell over that of the flat-cell. This comparison uses the reflectance data gathered in Chapter 2 for the 111 protected-and-textured Si solar cell, the equations in Table 3.3, and the solar zenith and solar azimuth angles varied over (most) possible angles, i.e., 0° < θ < 85° and -180°< ϕ <180°. The reason for choosing an upper limit of 85° is discussed in the coming analysis. Within this region of analysis, the results suggest that the CC-cell yields far greater captured incident optical power density, in comparison to that of the flat-cell, when the solar zenith angle is large and the solar azimuth angle is an integer multiple of 60°. Specifically, for solar zenith angles of  = 60° to  = 85° and solar azimuth angles of -180°, -120°, -60°, 0°, 60°, 120°, and 180°, the captured incident optical power density of the CC-cell outweighs that of the flat-cell by more than 10%. The improved performance of the CC-cell over the flat-cell is attributed to the increasing reflectance of the flat-cell surface for an increasing incident angle (i.e., the solar zenith angle), as seen in Figure 2.15, and the smaller normal component of light incident to the flat-cell. The CC-cell, on the other hand, has its solar cell families nearly normal to the light incident at large solar zenith angles, being associated with octants III, VI, and VIII. The threefold symmetry seen in these results is attributed simply to the threefold symmetry of the CC-cell's structure. It should be noted, however, that the strong performance improvements seen for the CC-cell while the solar zenith angle is large do not occur as regularly throughout the day as the less significant performance improvements while the solar zenith angle is small. Due to this infrequency, the median performance improvement across all the given angles is roughly 3.88%. In reality, the performance improvement of the CC-cell over the flat-cell would also be greater than this worst-case-scenario simply because the CC-cell would be oriented with consideration to the latitude of its deployment and not with its central axis of symmetry aligned with the z'-axis of the Global Coordinate Frame. The next chapter looks at the dependence of the CC-cell performance on the deployment location as well as the implementation of its electrical power transfer to the load via one or more MPPT units. 112 Figure 3.23. The percent improvement in captured incident optical power density of the CC-cell compared to that of the flat-cell. This comparison uses the reflectance data gathered in Chapter 2 for a protected-and-textured Si solar cell, the equations in Table 3.3, and the solar zenith and solar azimuth angles varied over (most) possible angles, i.e., 0°< θ <85° and -180°< ϕ <180°. The results suggest a median improvement in captured incident optical power density of roughly 3.88%, across all the displayed angles, for the CC-cell in comparison to the flat-cell over all chosen angles. 113 3.3. Electrical Performance Characteristics of a CC-cell The electrical performance characteristics of a given solar technology can be optimized by maximizing the electrical power delivered from the solar cells to the load. This is achieved by an appropriate MPPT implementation, which matches the impedance of the solar cells to the impedance of the load and thereby maximizes the generated electrical power density. For the flat-cell, a traditional mono-channel-MPPT (mono-MPPT) implementation is considered. The mono-MPPT implementation is also typical of flat-cell arrays. This is because most modules within arrays are made of flat-cells with nearly identical electrical performance characteristics. Thus, while uniformly illuminated, they have nearly identical impedances. The mono-MPPT flat-cell is used as a benchmark in the forthcoming analyses. For the CC-cell, the MPPT implementation must be more carefully considered. With uniform illumination, the optical powers on the solar cells will typically be equal within, but differ between, each of the SC-x, SC-y, and SC-z families. Thus, one MPPT channel cannot match the three distinct impedances of the solar cells to the one load impedance. With this in mind, a multi-channel-MPPT (multi-MPPT) implementation is introduced here. It uses three independent MPPT channels to maximize the electrical power delivered from the three solar cell families in the CC-cell to the load. The electrical performance characteristics of the multi-MPPT CC-cell are compared to those of a mono-MPPT CC-cell over a broad range of solar zenith and azimuth angles. The (benchmark) mono-MPPT flat-cell is considered first. Its generated electrical power density is the product of its voltage, Vpv, and current density, according to (2.11). The generated electrical power density for the flat-cell is maximized by differentiating (2.11) with respect to Vpv and setting the result to zero. Such a process gives dPEL-flat(θ)dVpv = Jsc(1 – R(θ)) cos(θ) + Jsat (1 – eVpv Vt⁄ – VpvVteVpv Vt⁄) = 0. (3.114) 114 The solution to this equation is the maximum power point voltage, Vpv, which yields the greatest generated electrical power density for the mono-MPPT flat-cell. The mono-MPPT and multi-MPPT CC-cell are considered next. The inherent generated electrical power densities of the SC-x, SC-y, and SC-z families are the products of the respective voltages, Vx, Vy, and Vz, and current densities, according to PEL-x(α,β,γ) = Vx (Jsc∙POP-x(α,β,γ)PSTC – Jsat(eVx Vt⁄ – 1)) , (3.115) PEL-y(α,β,γ) = Vy (Jsc∙POP-y(α,β,γ)PSTC – Jsat(eVy Vt⁄ – 1)) , (3.116) and PEL-z(α,β,γ) = Vz (Jsc∙POP-z(α,β,γ)PSTC – Jsat(eVz Vt⁄ – 1)) . (3.117) In these expressions, POP-x(,,), POP-y(,,), and POP-z(,,) are the captured incident optical power densities for the SC-x, SC-y, and SC-z families, respectively. The generated electrical power density for the CC-cell is then given by the sum of the contributions from the three solar cells, according to PEL-CC(α,β,γ) = PEL-x(α,β,γ) + PEL-y(α,β,γ) + PEL-z(α,β,γ)√3. (3.118) For the mono-MPPT CC-cell, the generated electrical power density is maximized with only one MPPT channel—and so the voltages on the three solar cells are equal. Thus, we set Vx = Vy = Vz = Vxyz. The generated electrical power density for the CC-cell is then maximized by differentiating PEL-CC(,,) in (3.118) with respect to Vxyz and setting the result to zero. This gives dPEL-CC(α, β, γ)dVxyz = PEL-x(α, β, γ)dVxyz + PEL-y(α, β, γ)dVxyz + PEL-z(α, β, γ)dVxyz = 0. (3.119) The solution to this equation is the maximum power point voltage, Vxyz, which yields the greatest generated electrical power density for the mono-MPPT CC-cell array. 115 For the multi-MPPT CC-cell, the generated electrical power density is maximized with three independent MPPT channels—and so the voltages on the three solar cells can differ. The generated electrical power density for the CC-cell is then maximized by differentiating PEL-x(,,), PEL-y(,,), and PEL-z(,,) in (3.115), (3.116), and (3.117) with respect to Vx, Vy, and Vz, respectively, and setting the individual results to zero. This gives dPEL-CC(α, β, γ)dVSC = PEL-x(α, β, γ)dVx = PEL-y(α, β, γ)dVy = PEL-z(α, β, γ)dVz = 0. (3.120) The solar cell voltage, VSC, in (3.120) is simply representative of the individual solar cells that make up each of the solar cell families within the CC-cell. The solutions to these equations are the maximum power point voltages Vx, Vy, and Vz, which yield the greatest generated electrical power density for the multi-MPPT CC-cell. This is equivalent to the solar cell families being connected in parallel. We note that the expressions in (3.119) and (3.120), for maximizing the generated electrical power densities of the mono-MPPT and multi-MPPT CC-cell, respectively, have terms that are similar but operations that differ. The first case sets the derivative of PEL-CC(,,) to zero by summing the individual generated electrical power densities and then differentiating and zeroing this sum. The second case sets the derivative of PEL-CC(,,) to zero by differentiating and zeroing the individual generated electrical power densities. This distinction can yield dissimilar maximum power point voltages and generated electrical power densities if the optical powers on the SC-x, SC-y, and SC-z families differ greatly. With this in mind, the electrical performance characteristics of the (benchmark) mono-MPPT flat-cell, mono-MPPT CC-cell, and multi-MPPT CC-cell are analysed for two deployments. The first deployment illuminates the CC-cell with a high level of symmetry—in that the central axis of symmetry of the CC-cell bisects the solar trajectory at solar noon. Such a condition occurs on the summer solstice, June 21, at a latitude of 23.45°N. This latitude aligns roughly with the geographical location of Dubai, United Arab Emirates. Three representative views of the deployed CC-cell in an arrayed implementation for this deployment are shown in Figure 3.24a, Figure 3.24b, and Figure 3.24c as viewed 116 along incident light rays at 06:22 ( = 76.00°,  = 109.80°), 12:00 ( = 0.00°,  = 0.00°), and 14:22 ( = 32.48°,  = -97.25°), respectively. Figure 3.24. Three representative views of the CC-cell in an arrayed implementation for the first deployment, which illuminates the CC-cell with a high level of symmetry. The CC-cell array is viewed along the direction of illumination for (a) a view of the CC-cell array at t1 = 06:22, corresponding to illumination along  = 76.00° and  = 109.80°, (b) a view of the CC-cell array at t2 = 12:00, corresponding to illumination along  = 0.00° and  = 0.00°, and (c) a view of the CC-cell array at t3 = 14:22, corresponding to illumination along  = 32.48° and  = -97.25°. The generated electrical power densities for the flat-cell and the SC-x, SC-y, and SC-z families in the CC-cell are shown in Figure 3.25a for the first deployment. The responses are shown as a function of time of day. The response of the mono-MPPT flat-cell is shown as a solid black curve. The responses of the SC-x family in the mono-MPPT and multi-MPPT CC-cell are labelled PEL-x and are shown as dashed and solid green curves, respectively. The responses of the SC-y family in the mono-MPPT and multi-MPPT CC-cell are labelled PEL-y and are shown as dashed and solid blue curves, respectively. The responses of the SC-z family in the mono-MPPT and multi-MPPT CC-cell are labelled PEL-z and are shown as dashed and solid red curves, respectively. Three conclusions can be made from the displayed responses for the first deployment. First, in terms of amplitudes, the PEL-flat response is lower than the PEL-x and PEL-y responses for the SC-x and SC-y families in the CC-cell for most of the day. This is due to the ability of the CC-cell to capture internal 117 reflections. In this deployment, the SC-x, SC-y, and SC-z families all receive primary, secondary, and tertiary incident optical power throughout most of the day, so their contributions outperform that of the flat-cell. Second, in terms of symmetry, the PEL-flat response shows simple symmetry with respect to its maximum, around 12:00 ( = 0.00°,  = 0.00°), while the PEL-x, PEL-y and PEL-z responses are complementary. The PEL-x response peaks later in the day, around 14:10 ( = 29.17°,  = -96.47°), the PEL-y response peaks early in the day, around 09:50 ( = 29.17°,  = 96.47°), and the PEL-z response peaks at the midpoint of the day, 12:00 ( = 0.00°,  = 0.00°). Third, in terms of electrical power tracking, the responses of the mono-MPPT and multi-MPPT CC-cell, shown as the respective dashed and solid curves, are very similar. They differ on average by approximately 1.42 W/m2 (1.60%) throughout the day. This similarity is due to the high level of symmetry for illumination in the first deployment. The symmetry yields comparable incident optical powers and impedances for the SC-x, SC-y, and SC-z families, which forms similar generated electrical power densities for the mono-MPPT and multi-MPPT CC-cell. The generated electrical power density (on the left axis) and generated electrical energy density (on the right axis) for the flat-cell and CC-cell are shown in Figure 3.25b for the first deployment. The responses are a function of time of day. The response of the mono-MPPT flat-cell is labelled PEL-flat and is shown as a solid black curve. The responses of the mono-MPPT and multi-MPPT CC-cell are labelled PEL-CC and are shown as solid orange and purple curves, respectively. It is immediately apparent that the generated electrical power density for the CC-cell is greater than that of the flat-cell. At 12:00 ( = 0.00°,  = 0.00°), the PEL-CC response for the multi-MPPT CC-cell implementation is 2.88 W/m2 (1.59%) higher than that of the PEL-flat response. The larger generated electrical power density for the CC-cell leads to a corresponding predominance of the generated electrical energy density—which is the summed power over the day. By the end of the day, the CC-cell has a generated electrical energy density that is 80.5 W·hr/m2 (5.53%) higher than that of the flat-cell. Clearly, the ability of the CC-cell to capture internal reflections, and mitigate the reflective losses, leads to increased generation of electrical energy. It is also worth noting that the results of Figure 3.25b, like those of Figure 3.25a, show a strong similarity between the mono-MPPT and multi-118 MPPT CC-cell. Again, this is due to the first deployment and how it illuminates the CC-cell with a high level of symmetry. By the end of the day, the generated electrical energy density of the multi-MPPT CC-cell is negligibly higher than that of the mono-MPPT CC-cell. Figure 3.25. Generated electrical power density and generated electrical energy density of the flat-cell and CC-cell versus time of day for the first deployment, which illuminates the CC-cell with a high level of symmetry. Results are shown for (a) the generated electrical power density of the flat-cell and the SC-x, SC-y, and SC-z families in the CC-cell, and (b) the generated electrical power density and generated electrical energy density of the flat-cell and CC-cell. In (a), the results for the flat-cell are denoted by the solid black curve, while the results for the CC-cell array are denoted by the green dashed/solid curve for SC-x labeled PEL-x, the blue dashed/solid curve for SC-y labeled PEL-y, and the red dashed/solid curve for SC-z labeled PEL-z. In (b), the results for the generated electrical power density (labeled PEL-flat) and generated electrical energy density (rising 119 to the right) of the flat-cell are denoted by solid/dashed black curves, respectively. The results for the generated electrical power density (labeled PEL-CC) and generated electrical energy density (rising to the right) of the CC-cell for the multi- and mono-MPPT implementations are denoted by solid/dashed orange and purple curves, respectively. Vertical delineations are shown at times of t1 = 06:22, t2 = 12:00, and t3 = 14:22. The second deployment is chosen to illuminate the CC-cell with a low level of symmetry—in that the SC-z family in the CC-cell has minimal illumination at noon. Such a condition occurs on September 22, at a latitude of 35.00°N. This latitude aligns roughly with the geographical location of Santa Fe, New Mexico, USA. Three representative views of the deployed CC-cell for this deployment are shown in Figure 3.26a, Figure 3.26b, and Figure 3.26c, as viewed along incident light rays at 09:08 ( = 53.63°,  = 57.88°), 12:00 ( = 35.61°,  = 0.00°), and 17:07 ( = 79.47°,  = -81.77°), respectively. The generated electrical power densities for the flat-cell and the SC-x, SC-y, and SC-z families in the CC-cell are shown in Figure 3.27a for the second deployment. The responses are a function of time of day. The response of the mono-MPPT flat-cell is shown as a solid black curve. The responses of the SC-x family in the mono-MPPT and multi-MPPT CC-cell are labelled PEL-x and are shown as dashed and solid green curves, respectively. The responses of the SC-y family in the mono-MPPT and multi-MPPT CC-cell are labelled PEL-y and are shown as dashed and solid blue curves, respectively. The responses of the SC-z family in the mono-MPPT and multi-MPPT CC-cell are labelled PEL-z and are shown as dashed and solid red curves, respectively. Three conclusions can be made from the displayed responses for the second deployment. First, in terms of amplitudes, the PEL-flat response is lower than the PEL-x and PEL-y responses for the SC-x and SC-y families in the CC-cell. However, the PEL-flat response is higher than the PEL-z response for the SC-z family in the CC-cell. The PEL-z response is lower here because the SC-z family is almost completely shadowed in this deployment. Additionally, the SC-z family receives no secondary reflections from SC-x or SC-y. 120 Figure 3.26. Three representative views of the CC-cell array for the second deployment, which has a low level of symmetry for illumination. The CC-cell array is viewed along the direction of illumination for (a) a view of the CC-cell array at t1 = 09:08, corresponding to illumination along  = 53.63° and  = 57.88°, (b) a view of the CC-cell array at t2 = 12:00, corresponding to illumination along  = 35.61° and  = 0.00°, and (c) a view of the CC-cell array at t3 = 17:07, corresponding to illumination along  = 79.47° and = -81.77°. However, the SC-x and SC-y families are nearly normal to the sun in this deployment and receive strong secondary reflections from each other, thus they are able to capture substantially more optical power than the flat-cell. Second, in terms of symmetry, the PEL-flat response shows simple symmetry with respect to its maximum, at 12:00 ( = 35.61°,  = 0.00°), while the PEL-x, PEL-y and PEL-z responses are complementary. The PEL-x response peaks later in the day, at 14:15 ( = 47.06°,  = -48.00°), the PEL-y response peaks early in the day, at 09:50 ( = 47.06°,  = 48.00°), and the PEL-z response stays relatively flat throughout most of the day. Third, in terms of electrical power tracking, the responses of the mono-MPPT and multi-MPPT CC-cell, shown as the respective dashed and solid curves, show some differences. They differ on average by 3.36 W/m2 (4.45%) throughout the day. This relatively large difference is in contrast to that seen for the first deployment and is due to the low level of symmetry for illumination with this second deployment. The symmetry yields a lower incident optical power on the SC-z family, with respect to the SC-x and SC-y families, which ultimately forms dissimilar generated electrical power densities for the mono-MPPT and multi-MPPT CC-cell. 121 Figure 3.27. Generated electrical power density and generated electrical energy density of the flat-cell and CC-cell versus time of day for the second deployment, which illuminated the CC-cell with a low level of symmetry. Results are shown for (a) the generated electrical power density of the flat-cell and the SC-x, SC-y, and SC-z families in the CC-cell, and (b) the generated electrical power density and generated electrical energy density of the flat-cell and CC-cell. In (a), the results for the flat-cell are denoted by the solid black curve, while the results for the CC-cell are denoted by the green dashed/solid curve for SC-x labeled PEL-x, the blue dashed/solid curve for SC-y labeled PEL-y, and the red dashed/solid curve for SC-z labeled PEL-z. In (b), the results for the generated electrical power density (labeled PEL-flat) and generated electrical energy density (rising to the right) of the flat-cell are denoted by solid/dashed black curves, respectively. The results for the generated electrical power density (labeled PEL-CC) and generated electrical energy density (rising to the right) of the CC-cell for 122 the multi- and mono-MPPT implementations are denoted by solid/dashed orange and purple curves, respectively. Vertical delineations are shown at times of t1 = 09:08, t2 = 12:00, and t3 = 17:07. The generated electrical power density (on the left axis) and generated electrical energy density (on the right axis) for the flat-cell and CC-cell are shown in Figure 3.27b for the second deployment, as a function of time of day. The responses of the mono-MPPT and multi-MPPT CC-cell arrays are labelled PEL-CC and are shown as solid orange and purple curves, respectively. It is immediately apparent that the generated electrical power density for the CC-cell is greater than that of the flat-cell. At 12:00 ( = 35.61°,  = 0.00°), the PEL-CC response for the multi-MPPT CC-cell implementation is 5.50 W/m2 (3.80%) higher than the PEL-flat response. The larger generated electrical power density for the CC-cell leads to a corresponding predominance of the generated electrical energy density—which is the summed power over the day. By the end of the day, the CC-cell has a generated electrical energy density that is 85.7 W·hr/m2 (8.29%) higher than that of the flat-cell. Clearly, the ability of the CC-cell to capture internal reflections, and mitigate the losses, leads to increased generation of electrical energy. It is also worth noting that the results of Figure 3.27b, like those of Figure 3.27a, show a visible difference between the mono-MPPT and multi-MPPT CC-cell. This is due to the second deployment and how it illuminates the CC-cell with a low level of symmetry. By the end of the day, the generated electrical energy density of the multi-MPPT CC-cell is 40.2 W·hr/m2 (3.72%) higher than that of the mono-MPPT CC-cell. The deployments discussed above are illustrative of possible illumination extremes. However, in general, an effective solar technology must yield benefits at many latitudes over the entire year. To this end, Figure 3.28, Figure 3.29, and Figure 3.30 present full-scale simulations of the generated electrical energy densities of the mono-MPPT flat-cell, mono-MPPT CC-cell, and multi-MPPT CC-cell, respectively, for every day of the year at latitudes spanning from 0° to 66° in the northern hemisphere. Figure 3.28 shows the generated electrical energy density of the mono-MPPT flat-cell as a function of the day of the year and the latitude. The mono-MPPT flat-cell has an annual average generated electrical energy density of 123 approximately 993 W·hr/m2, while considering all latitudes, with the highest performance culminating near the summer solstice at latitudes in the range of 30°-40°. Figure 3.28. The generated electrical energy density of the mono-MPPT flat-cell as a function of day of the year and latitude. The mono-MPPT flat-cell has an annual average generated electrical energy density of approximately 993 W·hr/m2, while considering all latitudes, with the highest performance culminating near the summer solstice at latitudes in the range of 30°-40°. Figure 3.29 shows the generated electrical energy density of the mono-MPPT CC-cell as a function of the day of the year and the latitude. The mono-MPPT CC-cell has an annual average generated electrical energy density of about 1,061 W·hr/m2, while considering all latitudes, with the highest performance culminating around the summer solstice at latitudes around 22°-60°. The greatest difference between the mono-MPPT flat-cell and mono-MPPT CC-cell is the higher overall performance that the mono-MPPT CC-cell offers in comparison to the mono-MPPT flat-cell. Here, the benefits of the internal reflections of the CC-cell structure allow solar technology to be used throughout most of the spring and summer across 124 most latitudes with notable benefits. The mono-MPPT CC-cell offers an annual average improvement of roughly 6.85% when compared to the mono-MPPT flat-cell. Figure 3.29. The generated electrical energy density of the mono-MPPT CC-cell as a function of day of the year and latitude. The mono-MPPT CC-cell has an annual average generated electrical energy density of about 1,061 W·hr/m2, while considering all latitudes, with the highest performance culminating around the summer solstice at latitudes around 22°-60°. Figure 3.30 shows the generated electrical energy density of the multi-MPPT CC-cell as a function of the day of the year and the latitude. The multi-MPPT CC-cell has an annual average generated electrical energy density of about 1,088 W·hr/m2, while considering all latitudes, with the highest performance culminating around the summer solstice at latitudes around 20°-66°. The greatest difference between the mono-MPPT flat-cell and multi-MPPT CC-cell is the higher overall performance that the multi-MPPT CC-cell offers in comparison to the mono-MPPT flat-cell. Here, the benefits of the internal reflections of the CC-cell structure and the multi-MPPT implementation allow solar technology to be used throughout the entire spring and summer across almost all latitudes with significant benefits. The multi-MPPT CC-cell 125 offers an annual average improvement of roughly 9.57% when compared to the mono-MPPT flat-cell. The improvement seen by implementing the multi-MPPT CC-cell over the mono-MPPT CC-cell, in terms of this generated electrical energy density, is approximately 2.54%. Figure 3.30. The generated electrical energy density of the multi-MPPT CC-cell as a function of day of the year and latitude. The multi-MPPT CC-cell has an annual average generated electrical energy density of about 1,088 W·hr/m2, while considering all latitudes, with the highest performance culminating around the summer solstice at latitudes around 20°-66°. The analyses of Figure 3.28, Figure 3.29, and Figure 3.30 bring about two observations. The first observation is that highest performance culminates around the summer solstice, which is not at all surprising. The second observation is that the latitudes for the highest performance are relatively far from the equator, which is much more surprising. This happens for two reasons. First, the length of a day varies more dramatically the further north you travel. For example, the summer solstice at a latitude of 23° yields a 13-hour day while at a latitude of 66° it is a 23-hour day. This results in solar devices simply being exposed to more sunlight. Figure 3.31 illustrates this concept by showing the solar zenith angles during the 126 summer solstice (throughout the day) for latitudes of 23°, 35°, and 66°, as represented by solid red, blue, and green curves, respectively. It is evident that solar devices implemented further north are exposed to sunlight for longer on the summer solstice. However, this does not represent the actual amount of optical power that can be transmitted through the surface of a flat-cell. In fact, by noting Figure 2.15 of Chapter 2, we can see that a flat-cell being exposed to sunlight for 23 hours has most of the light being at a large incident angle, i.e., a large solar zenith angle. This results in large reflected optical power losses. It is for this reason the latitudes of 30°-40° represent an optimal balance of (maximal) sunlight exposure and (minimal) reflected optical loss for the flat-cell. When considering the CC-cell, this sensitivity is less pronounced due to light being nearly normal to one of the solar cell families during times when the solar zenith angle is large. Figure 3.31. Solar zenith angles at latitudes of 23°, 35°, and 66° during the summer solstice, June 21, represented as solid red, blue, and green curves, respectively. Here it is observed that the length of a day increases the further north one travels during the summer solstice. A more detailed analysis of the percent improvements between the two CC-cell implementations and the flat-cell implementation can be carried out by considering the daily generated electrical energy densities at each latitude. The low levels of generated electrical energy densities for the mono-MPPT flat-127 cell, mono-MPPT CC-cell, and multi-MPPT CC-cell skew the average percent improvement of the CC-cell over the flat-cell, regardless of the MPPT implementation. In fact, the improvement can exceed 50%. Although this supports the use of the CC-cell at the more northern latitudes, such a conclusion must be applied judiciously. First, solar technology is often implemented in mountainous regions and at latitudes that are closer to the equator. Second, when comparing the percent improvement of the CC-cell over the flat-cell at higher northern latitudes, we mask the underlying difference in generated electrical energy density, which is far less than it is at latitudes closer to the equator and/or during the spring or summer. Instead, the large percent improvements at higher northern latitudes are artefacts of the low generated electrical energy density of the flat-cell and the slightly higher generated electrical energy density of the CC-cell. The large percent improvements at lower values of generated electrical energy density skew the average and do not provide an accurate representation of the percent improvement during more useful times, when solar technology is usually implemented for electrical energy generation. For these reasons the median is used in this work to provide a more modest representation of the percent improvement of the mono-MPPT CC-cell and multi-MPPT CC-cell over the mono-MPPT flat-cell. The median percent improvement in generated electrical energy density between the mono-MPPT CC-cell and mono-MPPT flat-cell is approximately 4.22%. The corresponding improvement between the multi-MPPT CC-cell and mono-MPPT flat-cell is approximately 7.53%. The percent improvements seen at the more northern latitudes are far less extreme for comparing the generated electrical energy densities of the mono- and multi-MPPT CC-cell. However, for consistency the median is still used as opposed to the average for the aforementioned reason regarding the majority of global solar power system implementations and difference in generated electrical energy density. The median percent improvement in generated electrical energy density between the multi-MPPT CC-cell and the mono-MPPT CC-cell is approximately 3.51%. Given that the above analysis, and in particular the regions of higher performance seen in Figure 3.28, Figure 3.29, and Figure 3.30, do not necessarily translate to better overall performance, we will consider next a broader perspective on the performance of the two CC-cell implementations compared to 128 that of the mono-MPPT flat-cell. The perspective takes the annually summed generated electrical energy densities seen in Figure 3.32 and compares the performance of the two CC-cell implementations to the mono-MPPT flat-cell. Here, the dominating performance of the CC-cell over the flat-cell at all latitudes is observed. The vertical axis on the left of Figure 3.32 shows the summed annual generated electrical energy density of the mono-MPPT flat-cell, mono-MPPT CC-cell, and multi-MPPT CC-cell as solid black, orange, and purple curves, respectively. The generated electrical energy density of the multi-MPPT CC-cell over the course of a year is 242,900 W·hr/m2 as opposed to the 228,900 W·hr/m2 of the mono-MPPT flat-cell at the equator. This is the smallest annual difference between the multi-MPPT CC-cell and the mono-MPPT flat-cell. The vertical axis on the right side of Figure 3.32 shows the annual percent improvement of the mono-MPPT CC-cell over the mono-MPPT flat-cell, of the multi-MPPT CC-cell over the mono-MPPT flat-cell, and of the multi-MPPT CC-cell over the mono-MPPT CC-cell as d red, blue, and green curves, respectively. The percent improvement of the CC-cell over the flat-cell differs greatly from roughly 6.1% (being a difference of 14,000 W·hr/m2) at the equator to over 20.0% (being a difference of 21,700 W·hr/m2) at a latitude of 66°. The annual percent improvement between the multi- and mono-MPPT CC-cell stays linear with a trend of increased improvement with increasing latitude. The annual percent improvement varies from about 2.0% (being a difference of 4,600 W·hr/m2) at the equator to about 4.5% (being a difference of 4,200 W·hr/m2) at a latitude of 66°. Ultimately, this broader (annual) perspective to the analysis reveals the general superiority of the CC-cell at virtually every northern latitude. Such results can also be generalized for the southern latitudes, as there would be no difference in the flat-cell performance. The CC-cell would simply have to be rotated by 180° such that SC-z was facing directly north. 129 Figure 3.32. The summed generated electrical energy density of the mono-MPPT flat-cell, mono-MPPT CC-cell, and multi-MPPT CC-cell over the course of a year and comparisons between all three implementations for latitudes from the equator to 66°. The solid purple, orange, and black curves, linked to the left vertical axis, show the summed generated electrical energy densities of the multi-MPPT CC-cell, mono-MPPT CC-cell, and mono-MPPT flat-cell, respectively. The dashed blue, red, and green curves, linked to the right vertical axis, show the percent improvement between the multi-MPPT CC-cell over the mono-MPPT flat-cell, the mono-MPPT CC-cell over the mono-MPPT flat-cell, and the multi-MPPT CC-cell over the mono-MPPT CC-cell, respectively. The CC-cell has been theoretically shown, via experimentally-seeded simulations, to be superior to a flat-cell over all latitudes during the entire year despite its larger overall projected area. The analysis assumes that the electrical properties of each solar cell making up the flat-cell and CC-cell are identical and that the optical properties of each solar cell mimic that of the protected-and-textured Si solar cell. The next section tests these assumptions by comparing the electrical performance of the multi-MPPT CC-cell to that of the mono-MPPT flat-cell via experimental implementations. 130 3.4. Experimental Implementation of a CC-cell The experimental implementation of a CC-cell is carried out in three steps. The first step is to prepare multiple solar cells for measurement. The second step is to compare the I-V characteristics of the solar cells. The third step is to create the CC-cell and test its overall electrical performance characteristics. The first step, being to prepare multiple solar cells for measurement, requires soldering of electrical contacts and adhering of a protective layer of glass onto each solar cell. Soldering the electrical contacts to the solar cells is critical to their electrical performance characteristics, as stated in section 2.3, but it was found to be difficult to execute. Three distinct attempts to solder contacts onto the surface of the solar cells were made. First, copper wire was soldered onto the surface contact of the solar cells. The glass was adhered to the surface of the solar cells using the UV-curable adhesive NOA 60. This solution gave the needed (negligible) series resistance, but it could not accommodate the placement of glass uniformly over the surface of the solar cells. Second, a copper ribbon with adhesive was used. The copper ribbon was glued onto the surface of the glass and then pressed onto the solar cell. The glass was adhered to the solar cell using the UV-curable adhesive NOA 60 while under pressure to keep the copper ribbon firmly in contact with the surface contact of the solar cell. This solution could accommodate the placement of glass uniformly over the surface of the solar cells, but it introduced varying levels of series resistance, based on the quality of the contact to the solar cell. Third, a tin-lead ribbon, which was thicker than the copper ribbon, was soldered onto the surface of the solar cells. The glass was adhered to the solar cell using the index-matching adhesive ethyl vinyl acetate (EVA), which is the traditional means of adhering solar cells to the glass surface of solar modules in industry.66 This solution could accommodate the placement of glass uniformly over the surface of the solar cells and introduced smaller levels of varying series resistance based on the quality of the contact to the solar cell. Overall, the series resistance introduced by the contacts was found to play a pivotal role in the electrical performance characteristics of the solar cells. With the tin-lead contact identified as the optimal contact, the I-V characteristics of individual solar cells could be acquired. 131 The second step, being to compare the I-V characteristics of solar cells, requires a measurement tool that can load solar cells with a variable impedance while scanning the internal impedance of the solar cells. A circuit schematic of the measurement tool devised for this work can be seen in Figure 3.33. The tool uses metal oxide semiconductor field effect transistors (MOSFETs) and low-resistance power resistor banks to vary the resistance of the load connected to each solar cell. The MOSFETs are used as a voltage-controlled variable resistance. By varying their gate voltages, the MOSFETs would have their drain-to-source resistances varied between negligible and large resistances. A summing amplifier connected to two National Instruments Data Acquisition (NI DAQ) units is employed to control and increment the gate voltages of the MOSFETs. Each solar cell voltage, Vpv, is measured directly with the NI DAQ units and each solar cell current is measured indirectly by measuring the voltage, Vcur, across the power resistor bank. The known value of the power resistor bank enables the solar cell current, Ipv, to be calculated after the fact by dividing Vcur by the equivalent resistance of the power resistor bank, which is 0.2 Ω. An instrumentation amplifier is employed to measure Vcur. All of the operational amplifiers within the measurement tool where powered by the same DC Power Supply (BK Precision 1673). The current and voltage data collected during the scan is then combined to create an I-V curve, which is used to determine the solar cell’s overall electrical performance characteristics. To this end, the solar cells were individually tested and grouped with solar cells having similar electrical performance characteristics. Three solar cells with similar electrical performance characteristics were then used to create the CC-cell and the (benchmark) flat-cell. Using the same three solar cells for the CC-cell and flat-cell minimized electrical variances in the solar cells between the two structures. Electrical performance improvements of the CC-cell over the flat-cell, if present, could then be attributed to the CC-cell’s structure. 132 Figure 3.33. A schematic of the measurement tool connected to one solar cell. The four sections of the measurement tool are outlined in four unique colours. The green section shows the summing amplifier. Its negative terminal is attached to the voltage output terminals on the NI DAQ units, Vout1 and Vout2. The voltage output of the summing amplifier is negative. The blue section shows the unity gain amplifier. This section is used to invert the negative voltage to incrementally vary the gate voltage of the n-channel MOSFET. The orange section shows the variable resistive load. The variable resistive load is made up of a power resistor bank with an equivalent resistance of 0.2 and a MOSFET, which has gate subjected to a varying voltage. The red section shows the instrumentation amplifier. This section is used to indirectly measure the current generated by the solar cell that is attached to the variable resistive load with the NI DAQ units. The current is inferred by taking the output voltage of the instrumentation amplifier, Vcur, and dividing it by 0.2 . The output voltage of the solar cell, Vpv, is measured directly with the NI DAQ units. The voltage rails, VCC and VEE, used to power all of the operational amplifiers are supplied by a DC Power Supply (BK Precision 1673). 133 The third step, being the creation of the CC-cell and testing of its overall electrical performance characteristics, requires a large uniform illumination source, solar cells with similar electrical performance characteristics, and the aforementioned measurement tool. The large uniform illumination source used in this work is a 500 W halogen lamp, as seen in the picture of the experimental setup in Figure 3.34a. A halogen lamp is used in this work due to its high optical power output and accessibility. The use of a halogen lamp as opposed to a light source capable of mimicking solar irradiance is a compromise between cost and spectral irradiance similarity. The normalized spectral power density of the halogen lamp, seen in Figure 3.35, is heavily weighted towards the infrared region of the electromagnetic spectrum. However, the difference between the halogen and solar spectral power densities has a negligible effect when it comes to the optical power that is reflected off of the protective layer of glass on the solar cells. The percent difference in reflectance of low iron glass given a weighted refractive index based on the solar spectral power density and the halogen spectral power density is in the range of 2-3%. This range was calculated using empirical values for the wavelength dependent refractive index of low iron glass found in literature64 and the Fresnel equations.39 This low percent difference is due to the wavelength dependent refractive index of low iron glass being nearly constant over both the halogen and solar spectral power densities. Ultimately, this allows the experimentally-seeded simulation discussed in section 3.3 to be compared to the experimental data that is collected. The last two items needed, being the solar cells with similar electrical performance characteristics and the measurement tool, are used to form the CC-cell and flat-cell, and to measure the electrical performance of these structures. The Global Coordinate Frame used for such measurements is shown in Figure 3.34b. 134 Figure 3.34. Photographs of the setup for experimental implementation of a CC-cell showing the (a) overall setup and (b) positioning of the halogen lamp. The device under test (DUT) can be the CC-cell, as is shown, or the flat-cell. The DUT collects optical power from the halogen lamp. The measurement tool, powered by the DC power supply, and NI DAQ units collect the voltage and current data of each solar cell within the DUT. The DUT is centred on the origin of the Global Coordinate Frame and the halogen lamp is placed 60 cm away at varying solar zenith () and azimuth () angles. The DUT is attached to an optical breadboard for stability. 135 Figure 3.35. Normalized spectral power density of the halogen lamp used in the experimental implementation of the CC-cell. The spectral power density of the halogen lamp is shifted to the near infrared compared to that of the solar spectral power density. For the measurements, the CC-cell or flat-cell is mounted at the origin of the Global Coordinate Frame in Figure 3.34b while the halogen lamp is swept overhead. The CC-cell is mounted on a support structure to have its central axis of symmetry aligned along the z'-axis, while the flat-cell is laid flat on the optical breadboard. After the measurements are taken of the CC-cell, its solar cells are arranged side-by-side at the origin to have them act as the flat-cell (with its measurements being averaged across the three solar cells). The solar cells used in this process are kept connected to the same measurement channels for consistency between measurements. This reduces the risk of introducing changes in series resistance that could otherwise affect the performance and prevent a fair comparison between the two structures. The halogen lamp is positioned 60 cm away from the origin with solar zenith angles of 0°, 25°, 50°, and 75° at solar azimuth angles of 90° and -90°. This combination of angles has the halogen lamp pass directly overhead of the origin from East to West. The high level of symmetry for this illumination corresponds to a solar trajectory similar to the first deployment of section 3.3 and is thus expected to show the smallest percent improvement for the generated electrical energy density of the CC-cell over the flat-cell. This is done to yield conservative conclusions regarding the performance of the CC-cell over the flat-cell. 136 Figure 3.36 shows representative results from measurements of the generated electrical power for the CC-cell and flat-cell. The generated electrical powers shown are the maximum power points found from each P-V curve produced by the measurement tool for each position over each structure. This is equivalent to the multi-MPPT implementation of the CC-cell and the mono-MPPT implementation of the flat-cell. The figure shows the generated electrical power densities, as opposed to the generated electrical energy densities, to see symmetries (if present) in the operation of the CC-cell and flat-cell and draw conclusions on their effectiveness. The generated electrical powers measured for the CC-cell are shown as circular green, blue, and red data points for the SC-x, SC-y, and SC-z solar cell families, respectively. The generated electrical power measured for the flat-cell, being an average from its three constituent solar cells, is shown as triangular black data points. To aid in visualizing the trends, the CC-cell’s data points are connected by solid curves in the respective colours and the flat-cell’s data points are connected by a dashed black curve. The solid curves show that the generated electrical powers measured for the CC-cell’s SC-x, SC-y, and SC-z solar cell families follow similar trends to the generated electrical power densities predicted from the experimentally-seeded simulations, whose results are displayed in Figure 3.25a. Specifically, the generated electrical powers of the SC-x, SC-y, and SC-z solar cell families peak late in the day (at 30°W), early in the day (at 35°E), and at midday (at 0°), respectively. Such trends can be understood by visualizing the asymmetric illumination of the solar cell families throughout the day. The dashed curve shows that the generated electrical power measured for the flat-cell follows a similar trend to the generated electrical power density from the experimentally-seeded simulation, whose results are shown in Figure 3.25b, in that the power is large at midday. Admittedly, the measured results do show a slightly larger generated electrical power later in the day, and this is attributed to the imperfect alignment of the surface normal of the solar cells with the z'-axis. Specifically, the solar cells are believed to be slightly misaligned due to the back surface contact, which acts as a pivot and causes the solar cells to face slightly eastward or westward. 137 Figure 3.36. A representative scan of the generated electrical power of the solar cell families making up the CC-cell (solid curves) and flat-cell (dashed curve). The measured values for SC-x are shown as green circles connected by a solid green curve, the measured values for SC-y are shown as blue circles connected by a solid green curve, the measured values for SC-z are shown as red circles connected by a solid red curve, and the measured average of the three solar cells making up the flat-cell is shown as black triangles connected by a dashed black curve. The halogen lamp used as the illumination source and was positioned over top of the CC-cell and flat-cell starting at an azimuth angle of 90° and a zenith angle of 75° and ending at an azimuth angle of -90° and a zenith angle of 75° (with respect to the Global Coordinate Frame). This is represented as 75°E and 75°W, respectively. Next, the measured generated electrical power of each solar cell family within the CC-cell is made into a generated electrical power density. This is done by dividing the measured generated electrical power of each solar cell within the CC-cell by A √3⁄ , being one third of the area of the CC-cell in the horizontal plane. The measured generated electrical power of the solar cells surrounding the origin for the flat-cell measurements can simply be divided by A, being the area of an individual solar cell. The performance improvement in generated electrical power density between the CC-cell and the flat-cell is then calculated and compared to the predicted performance improvement by the experimentally-seeded simulation. Figure 3.37 shows the performance improvement in measured generated electrical power density of each solar cell family and the flat-cell compared to its predicted performance improvement by the experimentally-seeded 138 simulation results. Figure 3.37a shows the experimentally-seeded simulation performance improvement (solid black curve) and measured performance improvement (purple circular data points) of SC-x. A general increase in performance improvement is seen as the illumination source sweeps from East to West. This can be expected because SC-x receives little illumination early in the day and great illumination late in the day. The overall results show relatively good agreement, with an average absolute error of 0.29% between the measured performance improvement and the experimentally-seeded simulation performance improvement. This is well within the 1% random error that was observed during measurement. Figure 3.37b shows the experimentally-seeded simulation performance improvement (solid black curve) and measured performance improvement (purple circular data points) of SC-y. A general decrease in performance improvement is seen as the illumination source sweeps from East to West. As stated above, this can be expected because SC-y receives strong illumination early in the day and little illumination late in the day. There is an average absolute error here of 0.96% between the measured performance improvement and the experimentally-seeded simulation performance improvement. This is within the 1% random error that was observed during measurement. The measurement at 75°E falls outside the 1% random error threshold and is instead due to systematic error. The CC-cell experiment is preferentially sensitive to systematic error at large solar zenith angles due to the high refractive losses off of the flat-cell compared to the near normal light to SC-y. In addition, any misalignment of SC-x, SC-y, or SC-z can cause the percent improvement in generated electrical energy density to divert slightly from the expected theoretical value for any combination of solar zenith and azimuth for any of the solar cell families. Figure 3.37c shows the experimentally-seeded simulation performance improvement (solid black curve) and measured performance improvement (purple circular data points) of SC-z. A relatively flat performance improvement is seen here as the illumination source sweeps from East to West. Again, there is a relatively low absolute error, of 0.35% seen here between the measured performance improvement and the experimentally-seeded simulation performance improvement. This is well within the 1% random error that was observed during measurement. The fact that this error and the above two errors are not closer to zero can be attributed to the imperfect collimation of the illumination source and imperfect orthogonality of the CC-cell. Nonetheless, 139 the CC-cell exhibits the expected performance improvement and validates the predictions of the experimentally-seeded simulation. 140 Figure 3.37. The experimentally-seeded simulation performance improvements (solid black curves) and the measured performance improvements (purple circles) in generated electrical power density of each solar cell family as compared to its flat-cell counter-part. The agreement between the measured and experimentally-141 seeded simulation results is clear. The performance improvement of SC-x, seen in (a), shows a general increase as the illumination source sweeps from East to West. The performance improvement of SC-y, seen in (b), shows a general decrease as the illumination source sweeps from East to West. The performance improvement of SC-z, seen in (c), shows a nearly constant trend as the illumination source sweeps from East to West. The measured performance improvements are seen with error bars representing the acceptable ±1% random error tolerance. The tolerance was set after taking the measurements of the system. The halogen lamp used as the illumination source was positioned over top of each structure starting with an azimuth angle of 90° and a zenith angle of 75°, i.e., 75°E, and ending with an azimuth angle of -90° and a zenith angle of 75°, i.e., 75°E, with respect to the Global Coordinate Frame. For the experimentally-seeded simulation, the solar zenith angle sweeps from 90°E to 90°W to capture the entire performance. 142 3.5. Summary In summary, Chapter 3 theoretically introduced and experimentally tested a CC-cell structure and compared it against a (benchmark) flat-cell. As stated in Chapter 1, the proposed work is motivated by the need for an effective solar technology, with effectiveness defined in terms of both low cost and high performance. The introduced CC-cell realizes this. It enables low-cost implementations due to its compatibility with conventional PV technology, which has a standardized form (square) and material (silicon) for its solar cells. At the same time, it enables high-performance operation due to its improved generated electrical power/energy densities. The manifestations of the three primary improvements are discussed in this section. First, the generated electrical energy density of the CC-cell array was found to be significantly higher than that of the flat-cell. The improvement could be as high as 8.29% for the CC-cell over that of the flat-cell, given a deployment with illumination having a low level of symmetry over the CC-cell. This improvement is due to the fundamental nature of the CC-cell's cubic geometry, which allows it to protrude above the horizontal plane and capture internal reflections over a wide range of incident angles. Second, the generated electrical energy density of the CC-cell was found to be increased by using the multi-MPPT implementation, rather than the mono-MPPT implementation. However, it should be noted that such performance improvements depend upon the deployment. A comparison of the generated electrical energy density of the multi-MPPT CC-cell array over that of the mono-MPPT CC-cell array shows a performance improvement of only 1.26% when there is illumination with a high level of symmetry over the CC-cell and a performance improvement of 3.72% when there is illumination with a low level of symmetry over the CC-cell. The basis of differing improvements can be seen by examining equations (3.119) and (3.120), which govern the electrical optimization of the mono- and multi-MPPT CC-cell, respectively. While the CC-cell is illuminated, in general, the SC-x, SC-y, and SC-z families capture different incident optical powers and are characterized by different impedances. Thus, the demands for 143 impedance matching can differ between the solar cell families. For illumination conditions with high levels of symmetry over the CC-cell, the solar cell families capture similar optical powers, so the performance metrics of the mono- and multi-MPPT CC-cell differ only slightly. For illumination conditions with low levels of symmetry over the CC-cell, however, the solar cell families may capture a wide range of optical powers, so the performance metrics of the mono- and multi-MPPT CC-cell can differ greatly. Ultimately, the decision to employ the mono- or multi-MPPT implementation should be made with consideration to both the location of the deployment on the globe and the physical orientation of the deployment. Third, the generated electrical power density of the CC-cell was found to improve upon that of the flat-cell at early and late times of day. These improvements are a consequence of the orientation of SC-x and SC-y given a symmetric due-North alignment of SC-z during sunrise and sunset. SC-y is nearly normal to the sun as it rises, yielding an increased generated electrical power density in the morning, and SC-x is nearly normal to the sun as it sets, yielding an increased generated electrical power density in the evening. This is in contrast to the flat-cell, which has the sun far off of its normal at sunrise and sunset. Moreover, the improved performance of the CC-cell over the flat-cell during early and late times of the day is seen to be further enhanced for the multi-MPPT CC-cell. This is because the multi-MPPT CC-cell is better able to function with the dissimilar optical powers on the solar cells during the early and late times of the day. The improved performance of the multi-MPPT CC-cell can be exploited, if desired, to address the peak demands for power seen at early and late times of the day. These dynamic demands are a growing concern—particularly for states such as California for which the trend in demand and electricity prices has become known as the Duck Curve.67 These three primary improvements enable the CC-cell to outperform the generated electrical energy density of the flat-cell by 6.1% at the equator on an annual basis, given the simulated conditions. The generated electrical energy density of the multi-MPPT CC-cell was also seen to outperform its mono-MPPT counterpart, in this case by 2.0% at the equator, which is impressive considering the large amount of symmetry in the solar trajectory over the CC-cell at the equator throughout the year. 144 Finally, a special case of highly symmetric illumination was tested on a CC-cell and a flat-cell in a laboratory setting. It was found that the measured values were in good agreement with the experimentally-seeded simulation values, with less than 1% in average absolute error observed. 145 Chapter 4 Conclusion In this chapter, concluding remarks are given with respect to the theoretical and experimental development of the CC-cell. The remarks include a brief summary of the work performed in section 4.1 and a look into future work in section 4.2. 146 4.1. Summary In Chapter 1 it was stated that solar cell technology is concurrently implemented at either low cost, which typically offers low performance, or at high cost, which typically offers high performance. Thus, there is an unanswered desire to implement solar technology with both low costs and high performance. Such a technology was defined as an effective solar technology—and was targeted in this work by way of the proposed CC-cell. The CC-cell can offer a low cost (due to its ability to be implemented using conventional PV technology) and high performance (due to its ability to enhance the generated electrical power/energy density). Chapter 2 laid the theoretical foundation for the optical and electrical properties of semiconductors, with an emphasis on Si. A Si photodiode was presented as a basic solar cell and its operation was explored. From there, the conversion process from Si to a conventional Si solar cell was discussed and the optical and electrical performance characteristics were summarized. Chapter 2 concluded with the optical performance limitations that are imposed on conventional PV technology due to the protective layer of glass on the surface of the solar cell. Chapter 3 introduced the CC-cell and presented a rigorous mathematical definition for its optical and electrical performance characteristics. The CC-cell's potential to offer improvements in generated electrical energy density compared to conventional PV solar technology was explored. It was seen using experimentally-seeded simulation that the CC-cell was capable of enhancing the generated electrical energy density compared to a simple flat-cell through its ability to capture internal reflections. The improvement in generated electrical energy density was seen regardless of latitude, which suggests that the CC-cell could be used as an alternative for many of the existing solar power installations, which use conventional PV technology. Chapter 3 then concluded with the experimental implementation of the CC-cell. It was found that the measured performance improvement agreed with the experimentally-seeded simulation to within 1% average absolute error. 147 4.2. Future Work The CC-cell can yield further decreases in cost and increases in performance by introducing new materials. Two specific examples are given in this section as potential for future work. First, by replacing one or two of the solar cells within the CC-cell with (less expensive) mirrors the overall cost of CC-cell can be reduced. Such a concept relies upon the fact that the CC-cell yields multiple internal reflections with multiple opportunities to absorb the incident light. Nonetheless, such a concept would also decrease the performance, so a careful cost-versus-benefit analysis would need to be performed as part of the future work. Second, by replacing all of the solar cells within the CC-cell with thin-film PV technology, with a reflective back surface, a large decrease in cost could be witnessed—potentially with strong performance. These outcomes can come about because incident optical power would pass twice through each of the three PV thin films, yielding six net passes through the thin films. 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Optical Constants of UV Transparent EVA and the Impact on the PV Module Output Power under Realistic Irradiation. in Energy Procedia 92, 523–530 (The Author(s), 2016). 67. Obi, M. & Bass, R. Trends and challenges of grid-connected photovoltaic systems - A review. Renew. Sustain. Energy Rev. 58, 1082–1094 (2016). 152 Appendices Appendix A MatLAB code for the experimentally-seeded simulation of the CC-cell. %% Corner-Cube-Cell Simulation for an Entire Year at all Latitudes (Experimental Data) % This simulation is designed to take surface reflectance data with the IV % characteristics of solar cells from literature and generate an estimated % comparison between the two structures in terms of generated electrical % energy density over the course of every day throughout the year. This % simulation also runs through every latitude on Earth for a comprehensive % look at all possible implementations of the Corner-Cube-Cell facing % straight up into the sky. % % It is recommended that surface reflectance data is formatted with the % CreateMDataSet.m file. This way the data can be properly loaded into the % program. %% Optical Performance Characteristics Section ---------------------------- % This section of the simulation first estimates solar trajectory % throughout the year for different latitudes on Earth. It then calculates % the normalized captured optical power density for both the CC-cell and % the Flat-Cell. %% Solar Trajectory Within the Global Coordinate Frame close all; clear all; clc; load('protectedcell_532nm'); data_angle=refldata(1,:); data_refl=refldata(2,:); resolution = 201; doy=1:2:365; lat=1:0.5:66; executiontime=zeros(1,length(lat)); E1MA=zeros(length(doy),length(lat)); E3MA=E1MA; EFA=E1MA; for l=1:length(lat) tic collectAST=zeros(length(doy),resolution); collectzenith=zeros(length(doy),resolution); collectazimuth=zeros(length(doy),resolution); for j=1:length(doy) dayofyear = doy(j); % [day] B = (dayofyear - 81) * (360/364) * (pi/180); % [radians] % **EQUATION OF TIME** ET = 9.87 * sin(2*B) - 7.53 * cos(B) - 1.5 * sin(B); % [minutes] 153 % **STANDARD LONGITUDE** SL = 115; % [degrees] % **LOCAL LONGITUDE** LL = 115; % [degrees] % **LOCAL LATITUDE** L = lat(l); % [degrees] L = L * (pi/180); % [radians] % **DAYLIGHT SAVINGS TIME** DS = 0; % [minutes] % **DECLINATION ANGLE** declination_d = 23.45 * sin(((360/365) * (284+dayofyear)) * (pi/180)); % [degrees] declination = declination_d * (pi/180); % [radians] % **SUNRISE & SUNSET HOURS** h_sr = -(acos(-tan(L)*tan(declination))*(180/pi)); % [degrees] h_ss = -h_sr; % [degrees] AST_sr = h_sr/0.25 + 720; AST_ss = h_ss/0.25 + 720; LST_sr = AST_sr - ET + DS; LST_ss = AST_ss - ET + DS; LST = linspace(LST_sr,LST_ss,resolution); % [minutes] **LOCAL STANDARD TIME** AST = LST + ET - (4*(SL - LL)) - DS; % [minutes] **APPARENT SOLAR TIME** % **HOUR ANGLE** (modified for minutes) h_d = (AST - 720) * 0.25; % [degrees] h = h_d * (pi/180); % [radians] % **SOLAR ELEVATION** solarelevation = asin((sin(L) * sin(declination)) +... (cos(L) * cos(declination) * cos(h))); % [radians] solarelevation_d = solarelevation * (180/pi); % [degrees] % **SOLAR ZENITH** solarzenith_d = 90 - solarelevation_d; % [degrees] for i = 1:resolution if solarzenith_d(i)>90 solarzenith_d(i)=90; end end % **SOLAR AZIMUTH** solarazimuth = zeros(1,resolution); for i = 1:resolution if cos(h(i)) > (tan(declination)/tan(L)) solarazimuth(i) = asin((cos(declination)* sin(h(i)))/(cos(solarelevation(i)))); % [radians] elseif cos(h(i)) <= (tan(declination)/tan(L)) if i < resolution/2 solarazimuth(i) = -pi + abs(asin((cos(declination)* sin(h(i)))/cos(solarelevation(i)))); elseif i > resolution/2 solarazimuth(i) = pi - abs(asin((cos(declination)* sin(h(i)))/(cos(solarelevation(i))))); end end end solarazimuth_d = solarazimuth * (180/pi); % [degrees] collectzenith(j,:)=solarzenith_d; collectazimuth(j,:)=solarazimuth_d; 154 collectAST(j,:)=AST; end %% Clear Workspace #1 clear AST_sr AST_ss B dayofyear declination declination_d DS ET h h_d... h_sr h_ss i j L LL LST LST_sr LST_ss SL AST solarzenith... solarzenith_d solarazimuth solarazimuth_d solarelevation... solarelevation_d %% Calculate Directional Cosines (Global and Body Coordinate Frame) alpha_prime = cosd(collectazimuth).*sind(collectzenith); beta_prime = sind(collectazimuth).*sind(collectzenith); gamma_prime = cosd(collectzenith); alpha = (1/sqrt(6))*alpha_prime-(1/sqrt(2))*beta_prime+... (1/sqrt(3))*gamma_prime; beta = (1/sqrt(6))*alpha_prime+(1/sqrt(2))*beta_prime+(1/sqrt(3))... *gamma_prime; gamma = -(sqrt(2)/sqrt(3))*alpha_prime+(1/sqrt(3))*gamma_prime; %% Calculate Captured Incident Optical Power on CC-Cell incident_alpha = zeros(length(doy),resolution); incident_beta = incident_alpha; incident_gamma = incident_alpha; storeR1P=zeros(length(doy),resolution); storeR2P=storeR1P; storeR3P=storeR1P; for j=1:length(doy) for i = 1:resolution R1P=refcalc3(alpha(j,i),data_angle,data_refl); R2P=refcalc3(beta(j,i),data_angle,data_refl); R3P=refcalc3(gamma(j,i),data_angle,data_refl); nmat=[1,alpha(j,i),R1P,0;2,beta(j,i),R2P,0;3,gamma(j,i),R3P,0]; nsort=sortrows(nmat,2); s_n=nsort(:,2); s_r=nsort(:,3); if (s_n(1)>=0) nsort(3,4)=s_n(3)+s_r(2)*(s_n(2)-s_n(2)*s_n(1)/(2*s_n(3)))+s_r(1)*(s_n(1)-s_n(2)*s_n(1)/(2*s_n(3)))+s_r(2)*s_r(1)*s_n(2)*s_n(1)/s_n(3); if s_n(3)>2*s_n(2) nsort(1,4)=s_n(1)+s_r(2)*s_n(1)*s_n(2)/(2*s_n(3))+s_r(3)*(s_n(1)-s_n(1)*s_n(2)/(2*s_n(3)))+s_r(2)*s_r(3)*3*s_n(1)*s_n(2)/(2*s_n(3)); nsort(2,4)=s_n(2)+s_r(1)*s_n(2)*s_n(1)/(2*s_n(3))+s_r(3)*(s_n(2)-s_n(2)*s_n(1)/(2*s_n(3)))+s_r(1)*s_r(3)*3*s_n(2)*s_n(1)/(2*s_n(3)); else nsort(1,4)=s_n(1)+s_r(2)*s_n(1)*s_n(2)/(2*s_n(3))+s_r(3)*(s_n(1)-s_n(1)*s_n(2)/(2*s_n(3)))+s_r(2)*s_r(3)*(2*s_n(1)-s_n(1)*s_n(2)/(2*s_n(3))-s_n(1)*s_n(3)/(2*s_n(2))); nsort(2,4)=s_n(2)+s_r(1)*s_n(2)*s_n(1)/(2*s_n(3))+s_r(3)*(s_n(2)-155 s_n(2)*s_n(1)/(2*s_n(3)))+s_r(1)*s_r(3)*(2*s_n(1)-s_n(1)*s_n(2)/(2*s_n(3))-s_n(1)*s_n(3)/(2*s_n(2))); end elseif s_n(2)>=0 nsort(1,4)=0; if abs(s_n(1))>s_n(3) nsort(2,4)=-s_n(2)*s_n(3)/(2*s_n(1)); nsort(3,4)=-s_n(2)*s_n(3)/(2*s_n(1)); else nsort(3,4)=s_n(3)+s_n(1)-s_n(1)*s_n(2)/(2*s_n(3))+s_r(2)*(s_n(2)+s_n(1)*s_n(2)/s_n(3)); if (abs(s_n(1))<s_n(2))&&(s_n(3)<2*s_n(2)) nsort(2,4)=s_n(2)+s_n(1)*s_n(2)/(2*s_n(3))+s_r(3)*(2*s_n(1)+s_n(2)-s_n(1)*s_n(2)/(2*s_n(3))-s_n(1)*s_n(3)/(2*s_n(2))); elseif (abs(s_n(1))>s_n(2))&&(s_n(3)<-2*s_n(1)) nsort(2,4)=s_n(2)+s_n(1)*s_n(2)/(2*s_n(3))+s_r(3)*(-s_n(2)*s_n(3)/(2*s_n(1))-s_n(1)*s_n(2)/(2*s_n(3))-s_n(2)); else nsort(2,4)=s_n(2)+s_n(1)*s_n(2)/(2*s_n(3))+s_r(3)*(s_n(2)+3*s_n(1)*s_n(2)/(2*s_n(3))); end end elseif s_n(3)>=0 nsort(1,4)=0; nsort(2,4)=0; if abs(s_n(1))<s_n(3) nsort(3,4)=s_n(1)+s_n(2)+s_n(3)+s_n(1)*s_n(2)/s_n(3); else nsort(3,4)=0; end else nsort(1,4)=0; nsort(2,4)=0; nsort(3,4)=0; end incident_alpha(j,i)=(1-R1P)*(nsort(1,4)*(nsort(1,1)==1)+nsort(2,4)*(nsort(2,1)==1)+nsort(3,4)*(nsort(3,1)==1)); incident_beta(j,i)=(1-R2P)*(nsort(1,4)*(nsort(1,1)==2)+nsort(2,4)*(nsort(2,1)==2)+nsort(3,4)*(nsort(3,1)==2)); incident_gamma(j,i)=(1-R3P)*(nsort(1,4)*(nsort(1,1)==3)+nsort(2,4)*(nsort(2,1)==3)+nsort(3,4)*(nsort(3,1)==3)); end end %% Clear Workspace #2 clear i j nmat nsort R1P R2P R3P s_n s_r %% Calculate Captured Incident Optical Power on Flat-Cell RFP=refcalc6(gamma_prime,data_angle,data_refl,resolution,length(doy)); incident_flat = (1-RFP).*cosd(collectzenith); 156 %% Clear Workspace #3 clear RFP %% Electrical Performance Characteristics Section ------------------------- % This section takes the information from the previous section and % implements the I-V characteristics of solar cells. %% Preliminaries on I-V Characteristics Claculated using Datasheets datasheet_area=245.706; %cm^2 sim_area=datasheet_area/10000; %m^2 sim_areacc=sim_area*sqrt(3); %m^2 datasheet_Jsc=36.9954; %mA/cm^2 datasheet_Voc=0.641; %V int_Isc=datasheet_Jsc*datasheet_area; %mA Isc=int_Isc/1000; %A k=1.38e-23; %Boltzmann Constant q=1.6e-19; %Elementary Charge Vt=(k/q)*298.15; %V datasheet_Vmp=0.540; %V datasheet_Imp=8.59; %A Isat=(Isc-datasheet_Imp)/... (exp(datasheet_Vmp/Vt)-1); %A Gsim = 1000; %W/m^2 Gstc = 1000; %W/m^2 %% Generate Photocurrents (Iphs) for all Solar Trajectories Iphx=Isc*incident_alpha*(Gsim/Gstc); %A Iphy=Isc*incident_beta*(Gsim/Gstc); %A Iphz=Isc*incident_gamma*(Gsim/Gstc); %A Iphf=Isc*incident_flat*(Gsim/Gstc); %A %% Clear Workspace #3 clear Gsim Gstc int_Isc k q %% Generate Maximum Power Voltages (Vpps) and Currents (Ipps) Vpp1M=zeros(length(doy),resolution); Vpp3Mx=Vpp1M; Vpp3My=Vpp1M; Vpp3Mz=Vpp1M; VppF=Vpp1M; Ipp1Mx=Vpp1M; Ipp1My=Vpp1M; Ipp1Mz=Vpp1M; Ipp3Mx=Vpp1M; Ipp3My=Vpp1M; Ipp3Mz=Vpp1M; IppF=Vpp1M; for j=1:length(doy) for i=1:resolution % Find Maximum Power Voltage and Current for CC-Cell (1MPPT) Vpp1M(j,i) = fzero(@(V)(Iphx(j,i) + Iphy(j,i) + Iphz(j,i) + ... %V 3*(Isat - (Isat*exp(V/Vt)) - (V/Vt)*(Isat*exp(V/Vt)))),0.6); 157 Ipp1Mx(j,i) = (Iphx(j,i) - Isat*(exp(Vpp1M(j,i)/Vt)-1)); %A Ipp1My(j,i) = (Iphy(j,i) - Isat*(exp(Vpp1M(j,i)/Vt)-1)); %A Ipp1Mz(j,i) = (Iphz(j,i) - Isat*(exp(Vpp1M(j,i)/Vt)-1)); %A % Find Maximum Power Voltage and Current for CC-Cell (3MPPT) Vpp3Mx(j,i) = fzero(@(V)(Iphx(j,i) + Isat - (Isat*exp(V/Vt))... - (V/Vt)*(Isat*exp(V/Vt))),0.6); %V Vpp3My(j,i) = fzero(@(V)(Iphy(j,i) + Isat - (Isat*exp(V/Vt))... - (V/Vt)*(Isat*exp(V/Vt))),0.6); %V Vpp3Mz(j,i) = fzero(@(V)(Iphz(j,i) + Isat - (Isat*exp(V/Vt))... - (V/Vt)*(Isat*exp(V/Vt))),0.6); %V Ipp3Mx(j,i) = (Iphx(j,i) - Isat*(exp(Vpp3Mx(j,i)/Vt)-1)); %A Ipp3My(j,i) = (Iphy(j,i) - Isat*(exp(Vpp3My(j,i)/Vt)-1)); %A Ipp3Mz(j,i) = (Iphz(j,i) - Isat*(exp(Vpp3Mz(j,i)/Vt)-1)); %A % Find Maximum Power Voltage and Current for Flat-Cell (1MPPT) VppF(j,i)=fzero(@(V)(Iphf(j,i)+Isat-(Isat*... exp(V/Vt))-(V/Vt)*(Isat*exp(V/Vt))),0.6); %V IppF(j,i) = (Iphf(j,i) - Isat*(exp(VppF(j,i)/Vt)-1)); %A end end %% Calculate Generated Electrical Power Density % Calculate Maximum Power Output for CC-Cell with 1MPPT Mpp1Mx=Vpp1M.*Ipp1Mx; %W Mpp1My=Vpp1M.*Ipp1My; %W Mpp1Mz=Vpp1M.*Ipp1Mz; %W P1M = Mpp1Mx+Mpp1My+Mpp1Mz; %W P1MA = P1M/sim_areacc; %W/m^2 % Calculate Maximum Power Output for CC-Cell with 3MPPT Mpp3Mx=Vpp3Mx.*Ipp3Mx; %W Mpp3My=Vpp3My.*Ipp3My; %W Mpp3Mz=Vpp3Mz.*Ipp3Mz; %W P3M=Mpp3Mx+Mpp3My+Mpp3Mz; %W P3MA=P3M/sim_areacc; %W/m^2 % Calculate Maximum Power Output for Flat-Cell with 1MPPT PF=VppF.*IppF; %W PFA=PF/sim_area; %W/m^2 %% Calculate Generated Electrical Energy Density for each Day of the Year E1MAtemp=zeros(length(doy),resolution); E3MAtemp=E1MAtemp; EFAtemp=E1MAtemp; d_t=E1MAtemp; for j=1:length(doy) for i=1:resolution-1 d_t(j,i)=abs((collectAST(j,i+1)/60)-(collectAST(j,i)/60)); %hr E1MAtemp(j,i+1)=E1MAtemp(j,i)+(P1MA(j,i)*(d_t(j,i))); %Whr/m^2 E3MAtemp(j,i+1)=E3MAtemp(j,i)+(P3MA(j,i)*(d_t(j,i))); %Whr/m^2 EFAtemp(j,i+1)=EFAtemp(j,i)+(PFA(j,i)*(d_t(j,i))); %Whr/m^2 158 end E1MA(j,l)=E1MAtemp(j,resolution); %Whr/m^2 E3MA(j,l)=E3MAtemp(j,resolution); %Whr/m^2 EFA(j,l)=EFAtemp(j,resolution); %Whr/m^2 end %% Clear Workspace #4 clear d_t E1MAtemp E3MAtemp EFAtemp i j l executiontime(1,l)=toc; executiontime(l) end %% Clear Workspace #5 clear alpha beta gamma alpha_prime beta_prime gamma_prime collectAST... collectazimuth collectzenith data_refl data_angle datasheet_area... datasheet_Jsc datasheet_Voc incident_alpha incident_beta... incident_gamma incident_flat Iphf Iphx Iphy Iphz Ipp1Mx Ipp1My... Ipp1Mz Ipp3Mx Ipp3My Ipp3Mz IppF Isat Isc l Mpp1Mx Mpp1My Mpp1Mz... P1M P1MA P3M P3MA PF refldata sim_area sim_areacc Vpp1M Vpp3Mx... Vpp3My Vpp3Mz VppF Vt Mpp3Mx Mpp3My Mpp3Mz PFA storeR1P storeR2P... storeR3P %% Simulation Ends E1MA_EFA=E1MA./EFA; E3MA_EFA=E3MA./EFA; E3MA_E1MA=E3MA./E1MA; 159 Appendix B Datasheet for the XS156B3-210R+ series solar cell by Motch Industries Inc. 160 161 Appendix C Ultra-violet transmission experiment on glass and glass with NOA 60 cured to it. Measurements were taken of the optical power transmitted through a piece of glass and a piece of glass with NOA 60 cured to it for wavelengths of 365 nm and 405 nm. The first measurement, through glass, is seen in Figure C.1 as a solid orange curve for both wavelengths. This measurement is used as the reference point for determining how much ultra-violet light is absorbed by the NOA 60. The second measurement, through glass with NOA 60 cured to it, is seen in Figure C.1 as a solid green curve for both wavelengths. It is evident that the energetic light is absorbed more readily due to the presence of NOA 60. However, the NOA 60 only absorbs about 5% of the 365 nm light and only about 1% of the 405 nm light. Most of the optical power within the ultra-violet region of solar spectral irradiance is already absorbed by O3 before it touches Earth's surface, as seen in Figure 2.9, making this minor absorption practically inert to the overall electrical performance of Si solar cells. Figure C.1. Ultra-violet transmission experiment on glass and glass with NOA 60 cured to it. Measurements were taken of the optical power transmitted through a piece of glass and a piece of glass with NOA 60 cured to it for wavelengths of 365 nm and 405 nm. The first measurement, through glass, is seen as a solid orange curve for both wavelengths. The second measurement, through glass with NOA 60 cured to it, is seen as a solid green 162 curve for both wavelengths. It is evident that as the light becomes more energetic the absorption due to the presence of NOA 60 increases. While this is problematic, most of the optical power within the ultra-violet region of solar spectral irradiance is absorbed before it touches Earth's surface by O3.
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
A novel photovoltaic device for improvements in solar energy collection
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