Abstract

Optical concentration can improve the efficiency and reduce the cost of photovoltaic power but has traditionally been too bulky, massive, and unreliable for use in space. Here, we explore a new ultra-compact and low-mass microcell concentrating photovoltaic (µCPV) paradigm for space based on the monolithic integration of transfer-printed microscale solar cells and molded microconcentrator optics. We derive basic bounds on the compactness as a function of geometric concentration ratio and angular acceptance, and show that a simple reflective parabolic concentrator provides the best combination of specific power, angular acceptance, and overall fabrication simplicity. This architecture is simulated in detail and validated experimentally with a µCPV prototype that is less than 1.7 mm thick and operates with six, 650 µm square triple-junction microcells at a geometric concentration ratio of 18.4$\times$. In outdoor testing, the system achieves a terrestrial power conversion efficiency of 25.8 $\pm$ 0.2% over a $\pm$9.5° angular range, resulting in a specific power of approximately 111 W/kg. These results lay the groundwork for future space µCPV systems and establish a realistic path to exceed 350 W/kg specific power at >$33$% power conversion efficiency by scaling down to even smaller microcells.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

J. E. Moore, K. J. Schmieder, W. Wagner, and M. P. Lumb, “Strategies for defect-tolerant microconcentrator photovoltaic modules,” J. Photonics Energy 9(01), 014501 (2019).
[Crossref]

2018 (2)

B. R. Clapp, H. J. Weigl, N. E. Goodzeit, D. R. Carter, and T. J. Rood, “GOES-R active vibration damping controller design, implementation, and on-orbit performance,” CEAS Space J. 10(4), 501–517 (2018).
[Crossref]

Matt Lumb, “Sending CPV into Space,” Compd. Semicond. 24(8), 36–40 (2018).

2017 (2)

J. S. Price, A. J. Grede, B. Wang, M. V. Lipski, B. Fisher, K.-T. Lee, J. He, G. S. Brulo, X. Ma, S. Burroughs, C. D. Rahn, R. Nuzzo, J. A. Rogers, and N. C. Giebink, “High-concentration planar microtracking photovoltaic system exceeding 30% efficiency,” Nat. Energy 2(8), 17113 (2017).
[Crossref]

M. P. Lumb, S. Mack, K. J. Schmieder, M. González, M. F. Bennett, D. Scheiman, M. Meitl, B. Fisher, S. Burroughs, K.-T. Lee, J. A. Rogers, and R. J. Walters, “GaSb-Based solar cells for full solar spectrum energy harvesting,” Adv. Energy Mater. 7(20), 1700345 (2017).
[Crossref]

2014 (4)

I. D. Sarcina, M. L. Grilli, F. Menchini, A. Piegari, S. Scaglione, A. Sytchkova, and D. Zola, “Behavior of optical thin-film materials and coatings under proton and gamma irradiation,” Appl. Opt. 53(4), A314–A320 (2014).
[Crossref]

X. Sheng, C. A. Bower, S. Bonafede, J. W. Wilson, B. Fisher, M. Meitl, H. Yuen, S. Wang, L. Shen, A. R. Banks, C. J. Corcoran, R. G. Nuzzo, S. Burroughs, and J. A. Rogers, “Printing-based assembly of quadruple-junction four-terminal microscale solar cells and their use in high-efficiency modules,” Nat. Mater. 13(6), 593–598 (2014).
[Crossref]

A. L. Lentine, G. N. Nielson, M. Okandan, J. Cruz-Campa, and A. Tauke-Pedretti, “Voltage matching and optimal cell compositions for microsystem-enabled photovoltaic modules,” IEEE J. Photovolt. 4(6), 1593–1602 (2014).
[Crossref]

S. F. Pellicori, C. L. Martinez, P. Hausgen, and D. Wilt, “Development and testing of coatings for orbital space radiation environments,” Appl. Opt. 53(4), A339–A350 (2014).
[Crossref]

2012 (1)

H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renewable Sustainable Energy Rev. 16(8), 5890–5909 (2012).
[Crossref]

2011 (2)

A. Goldstein, D. Feuermann, G. D. Conley, and J. M. Gordon, “Nested aplanats for practical maximum-performance solar concentration,” Opt. Lett. 36(15), 2836–2838 (2011).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

2007 (1)

O. Korech, B. Hirsch, E. A. Katz, and J. M. Gordon, “High-flux characterization of ultrasmall multijunction concentrator solar cells,” Appl. Phys. Lett. 91(6), 064101 (2007).
[Crossref]

2005 (2)

2002 (1)

2000 (1)

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
[Crossref]

1996 (1)

R. P. Friedman, J. M. Gordon, and H. Ries, “Compact high-flux two-stage solar collectors based on tailored edge-ray concentrators,” Sol. Energy 56(6), 607–615 (1996).
[Crossref]

1995 (1)

1986 (1)

R. D. Aines and G. R. Rossman, “Relationships between radiation damage and trace water in zircon, quartz, and topaz,” Am. Mineral. 71(9-1), 1186–1193 (1986).

1970 (1)

R. Schnadt and J. Schneider, “The electronic structure of the trapped-hole center in smoky quartz,” Eur. Phys. J. B 11(1), 19–42 (1970).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Aines, R. D.

R. D. Aines and G. R. Rossman, “Relationships between radiation damage and trace water in zircon, quartz, and topaz,” Am. Mineral. 71(9-1), 1186–1193 (1986).

Alea, P.

M. Perry, P. Alea, M. Cully, M. McCullough, P. Sanneman, N. Teti, and B. Zink, “Earth Observing-1 Spacecraft Bus,” in Proceedings of the 15th Annual AIAA/USU Conference on Small Satellites (IEEE, 2001), SSC01-V-6.

Allen, D. M.

D. M. Allen, P. J. Jones, D. M. Murphy, and M. F. Piszczor, “The SCARLET light concentrating solar array,” in Conference Record of the Twenty Fifth IEEE Photovoltaic Specialists Conference - 1996 (IEEE, 1996), pp. 353–356.

D. M. Murphy and D. M. Allen, “SCARLET development, fabrication, and testing for the Deep Space 1 spacecraft,” in IECEC-97 Proceedings of the Thirty-Second Intersociety Energy Conversion Engineering Conference (Cat. No.97CH6203) (American Institute of Chemical Engineers, 1997), pp. 2237–2245.

Baig, H.

H. Baig, K. C. Heasman, and T. K. Mallick, “Non-uniform illumination in concentrating solar cells,” Renewable Sustainable Energy Rev. 16(8), 5890–5909 (2012).
[Crossref]

Banks, A. R.

X. Sheng, C. A. Bower, S. Bonafede, J. W. Wilson, B. Fisher, M. Meitl, H. Yuen, S. Wang, L. Shen, A. R. Banks, C. J. Corcoran, R. G. Nuzzo, S. Burroughs, and J. A. Rogers, “Printing-based assembly of quadruple-junction four-terminal microscale solar cells and their use in high-efficiency modules,” Nat. Mater. 13(6), 593–598 (2014).
[Crossref]

Barrie, J. D.

Benítez, P.

J. C. Miñano, P. Benítez, and J. C. González, “RX: a nonimaging concentrator,” Appl. Opt. 34(13), 2226–2235 (1995).
[Crossref]

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier, 2005).

Bennett, M. F.

M. P. Lumb, S. Mack, K. J. Schmieder, M. González, M. F. Bennett, D. Scheiman, M. Meitl, B. Fisher, S. Burroughs, K.-T. Lee, J. A. Rogers, and R. J. Walters, “GaSb-Based solar cells for full solar spectrum energy harvesting,” Adv. Energy Mater. 7(20), 1700345 (2017).
[Crossref]

Bett, A. W.

R. Hoheisel, A. W. Bett, J. H. Warner, R. J. Walters, and P. P. Jenkins, “Low temperature low intensity effects in III-V photovoltaic devices for deep space missions,” in 2018 IEEE 7th World Conference on Photovoltaic Energy Conversion (WCPEC) (A Joint Conference of 45th IEEE PVSC, 28th PVSEC 34th EU PVSEC) (IEEE, 2018), pp. 3763–3767.

R. Hoheisel, R. J. Walters, and A. W. Bett, “Low temperature effects in photovoltaic devices for deep space missions,” in 2015 IEEE 42nd Photovoltaic Specialist Conference (PVSC) (IEEE, 2015), pp. 1–5.

Bonafede, S.

X. Sheng, C. A. Bower, S. Bonafede, J. W. Wilson, B. Fisher, M. Meitl, H. Yuen, S. Wang, L. Shen, A. R. Banks, C. J. Corcoran, R. G. Nuzzo, S. Burroughs, and J. A. Rogers, “Printing-based assembly of quadruple-junction four-terminal microscale solar cells and their use in high-efficiency modules,” Nat. Mater. 13(6), 593–598 (2014).
[Crossref]

Bortz, J. C.

R. Winston, J. C. Miñano, P. Benítez, N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier, 2005).

Bower, C. A.

X. Sheng, C. A. Bower, S. Bonafede, J. W. Wilson, B. Fisher, M. Meitl, H. Yuen, S. Wang, L. Shen, A. R. Banks, C. J. Corcoran, R. G. Nuzzo, S. Burroughs, and J. A. Rogers, “Printing-based assembly of quadruple-junction four-terminal microscale solar cells and their use in high-efficiency modules,” Nat. Mater. 13(6), 593–598 (2014).
[Crossref]

Brandhorst, H.

M. O’Neill, A. J. McDanal, H. Brandhorst, K. Schmid, P. LaCorte, M. Piszczor, and M. Myers, “Recent space PV concentrator advances: More robust, lighter, and easier to track,” in 2015 IEEE 42nd Photovoltaic Specialist Conference (PVSC) (IEEE, 2015), pp. 1–6.

Brulo, G. S.

J. S. Price, A. J. Grede, B. Wang, M. V. Lipski, B. Fisher, K.-T. Lee, J. He, G. S. Brulo, X. Ma, S. Burroughs, C. D. Rahn, R. Nuzzo, J. A. Rogers, and N. C. Giebink, “High-concentration planar microtracking photovoltaic system exceeding 30% efficiency,” Nat. Energy 2(8), 17113 (2017).
[Crossref]

Burroughs, S.

J. S. Price, A. J. Grede, B. Wang, M. V. Lipski, B. Fisher, K.-T. Lee, J. He, G. S. Brulo, X. Ma, S. Burroughs, C. D. Rahn, R. Nuzzo, J. A. Rogers, and N. C. Giebink, “High-concentration planar microtracking photovoltaic system exceeding 30% efficiency,” Nat. Energy 2(8), 17113 (2017).
[Crossref]

M. P. Lumb, S. Mack, K. J. Schmieder, M. González, M. F. Bennett, D. Scheiman, M. Meitl, B. Fisher, S. Burroughs, K.-T. Lee, J. A. Rogers, and R. J. Walters, “GaSb-Based solar cells for full solar spectrum energy harvesting,” Adv. Energy Mater. 7(20), 1700345 (2017).
[Crossref]

X. Sheng, C. A. Bower, S. Bonafede, J. W. Wilson, B. Fisher, M. Meitl, H. Yuen, S. Wang, L. Shen, A. R. Banks, C. J. Corcoran, R. G. Nuzzo, S. Burroughs, and J. A. Rogers, “Printing-based assembly of quadruple-junction four-terminal microscale solar cells and their use in high-efficiency modules,” Nat. Mater. 13(6), 593–598 (2014).
[Crossref]

E. Menard, W. Wagner, B. Furman, K. Ghosal, J. Gabriel, M. Meitl, and S. Burroughs, “Multi-physics circuit network performance model for CPV modules/systems,” in 2011 37th IEEE Photovoltaic Specialists Conference (IEEE, 2011), pp. 002268–002272.

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref]

Carter, D. R.

B. R. Clapp, H. J. Weigl, N. E. Goodzeit, D. R. Carter, and T. J. Rood, “GOES-R active vibration damping controller design, implementation, and on-orbit performance,” CEAS Space J. 10(4), 501–517 (2018).
[Crossref]

Chaves, J.

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
[Crossref]

Clapp, B. R.

B. R. Clapp, H. J. Weigl, N. E. Goodzeit, D. R. Carter, and T. J. Rood, “GOES-R active vibration damping controller design, implementation, and on-orbit performance,” CEAS Space J. 10(4), 501–517 (2018).
[Crossref]

Collares-Pereira, M.

J. Chaves and M. Collares-Pereira, “Ultra flat ideal concentrators of high concentration,” Sol. Energy 69(4), 269–281 (2000).
[Crossref]

Conley, G. D.

Corcoran, C. J.

X. Sheng, C. A. Bower, S. Bonafede, J. W. Wilson, B. Fisher, M. Meitl, H. Yuen, S. Wang, L. Shen, A. R. Banks, C. J. Corcoran, R. G. Nuzzo, S. Burroughs, and J. A. Rogers, “Printing-based assembly of quadruple-junction four-terminal microscale solar cells and their use in high-efficiency modules,” Nat. Mater. 13(6), 593–598 (2014).
[Crossref]

Crassidis, J. L.

F. L. Markley and J. L. Crassidis, Fundamentals of spacecraft attitude determination and control (Springer, 2014), Chap. 7.

Cruz-Campa, J.

A. L. Lentine, G. N. Nielson, M. Okandan, J. Cruz-Campa, and A. Tauke-Pedretti, “Voltage matching and optimal cell compositions for microsystem-enabled photovoltaic modules,” IEEE J. Photovolt. 4(6), 1593–1602 (2014).
[Crossref]

Cruz-Campa, J. L.

A. L. Lentine, G. N. Nielson, M. Okandan, W. C. Sweatt, J. L. Cruz-Campa, and V. Gupta, “Optimal cell connections for improved shading, reliability, and spectral performance of microsystem enabled photovoltaic (MEPV) modules,” in 2010 35th IEEE Photovoltaic Specialists Conference (IEEE, 2010), pp. 003048–003054.

Cully, M.

M. Perry, P. Alea, M. Cully, M. McCullough, P. Sanneman, N. Teti, and B. Zink, “Earth Observing-1 Spacecraft Bus,” in Proceedings of the 15th Annual AIAA/USU Conference on Small Satellites (IEEE, 2001), SSC01-V-6.

Dawson, S. F.

S. F. Dawson, P. Stella, W. McAlpine, and B. Smith, “JUNO Photovoltaic power at Jupiter,” in 10th International Energy Conversion Engineering Conference (American Institute of Aeronautics and Astronautics, 2012), pp. 2012–3833.

Eichgrün, K.

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X. Sheng, C. A. Bower, S. Bonafede, J. W. Wilson, B. Fisher, M. Meitl, H. Yuen, S. Wang, L. Shen, A. R. Banks, C. J. Corcoran, R. G. Nuzzo, S. Burroughs, and J. A. Rogers, “Printing-based assembly of quadruple-junction four-terminal microscale solar cells and their use in high-efficiency modules,” Nat. Mater. 13(6), 593–598 (2014).
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M. O’Neill, A. J. McDanal, M. Piszczor, T. Peshek, M. Myers, C. McPheeters, J. Steinfeldt, B. Heintz, P. Sharps, M. Puglia, and C. Kumar, “Advanced development of space photovoltaic concentrators using robust lenses, multi-junction cells, & graphene radiators,” in 2018 IEEE 7th World Conference on Photovoltaic Energy Conversion (WCPEC) (A Joint Conference of 45th IEEE PVSC, 28th PVSEC 34th EU PVSEC) (IEEE, 2018), pp. 3378–3383.

M. O’Neill, A. J. McDanal, H. Brandhorst, K. Schmid, P. LaCorte, M. Piszczor, and M. Myers, “Recent space PV concentrator advances: More robust, lighter, and easier to track,” in 2015 IEEE 42nd Photovoltaic Specialist Conference (PVSC) (IEEE, 2015), pp. 1–6.

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A. L. Lentine, G. N. Nielson, M. Okandan, J. Cruz-Campa, and A. Tauke-Pedretti, “Voltage matching and optimal cell compositions for microsystem-enabled photovoltaic modules,” IEEE J. Photovolt. 4(6), 1593–1602 (2014).
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M. O’Neill, A. J. McDanal, M. Piszczor, T. Peshek, M. Myers, C. McPheeters, J. Steinfeldt, B. Heintz, P. Sharps, M. Puglia, and C. Kumar, “Advanced development of space photovoltaic concentrators using robust lenses, multi-junction cells, & graphene radiators,” in 2018 IEEE 7th World Conference on Photovoltaic Energy Conversion (WCPEC) (A Joint Conference of 45th IEEE PVSC, 28th PVSEC 34th EU PVSEC) (IEEE, 2018), pp. 3378–3383.

M. O’Neill, A. J. McDanal, H. Brandhorst, K. Schmid, P. LaCorte, M. Piszczor, and M. Myers, “Recent space PV concentrator advances: More robust, lighter, and easier to track,” in 2015 IEEE 42nd Photovoltaic Specialist Conference (PVSC) (IEEE, 2015), pp. 1–6.

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J. S. Price, A. J. Grede, B. Wang, M. V. Lipski, B. Fisher, K.-T. Lee, J. He, G. S. Brulo, X. Ma, S. Burroughs, C. D. Rahn, R. Nuzzo, J. A. Rogers, and N. C. Giebink, “High-concentration planar microtracking photovoltaic system exceeding 30% efficiency,” Nat. Energy 2(8), 17113 (2017).
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J. S. Price, A. J. Grede, B. Wang, M. V. Lipski, B. Fisher, K.-T. Lee, J. He, G. S. Brulo, X. Ma, S. Burroughs, C. D. Rahn, R. Nuzzo, J. A. Rogers, and N. C. Giebink, “High-concentration planar microtracking photovoltaic system exceeding 30% efficiency,” Nat. Energy 2(8), 17113 (2017).
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J. S. Price, A. J. Grede, B. Wang, M. V. Lipski, B. Fisher, K.-T. Lee, J. He, G. S. Brulo, X. Ma, S. Burroughs, C. D. Rahn, R. Nuzzo, J. A. Rogers, and N. C. Giebink, “High-concentration planar microtracking photovoltaic system exceeding 30% efficiency,” Nat. Energy 2(8), 17113 (2017).
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N. Yu, P. Genevet, M. A. Kats, F. Aieta, J. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
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Figures (8)

Fig. 1.
Fig. 1. (a) Diagram showing the geometry of a reflective single surface concentrator operating at the limit of compactness. (b) Trade-off between aspect ratio and concentration ratio for different acceptance half-angles (denoted by the contour labels) according to Eqn. 1. (c) Fraction of the sine-limiting concentration that can be achieved by an ideal reflective single surface concentrator with different aspect ratios.
Fig. 2.
Fig. 2. (a) Construction of a single surface reflective concentrator with minimum aspect ratio for a given concentration ratio and acceptance angle, $\pm \theta$. The solution is a parabolic reflector with its axis tilted at $\theta ^{\prime }$ and its focus located at the near edge of the receiver; rotating the solid portion of this curve axisymmetrically about the y-axis defines the surface of the concentrator in three dimensions. (b) Comparison of simple and inverted compound parabolic concentrator designs for different acceptance half-angles. The inset illustrates the case of $CR = 25$ and $\theta =4$°, where the compound parabolic design lowers the aspect ratio to $AR=0.28$ compared with $AR=0.32$ for the normal paraboloid.
Fig. 3.
Fig. 3. (a) Ray tracing diagrams of the simple parabola, inverted compound parabolic concentrator (ICPC), and aplanatic concentrator designs. In the aplanatic system, light reflects off the parabolic primary surface to a secondary mirror which focuses the light back down onto the (upward-facing) receiver located at the vertex of the primary. Note, that the optimal position of the receiver in the simple paraboloid case is slightly below the nominal focal point as discussed in the text. (b) Simulated optical efficiency as a function of solar incidence angle for the the three concentrator designs in (a) operating at a common concentration ratio of $CR$ = 25, 4% shading loss, and aspect ratio of $AR$ = 0.39. The simulation is carried out assuming axisymmetric concentrators with dispersionless $n^{\prime }=1.5$ and no reflection or absorption losses. The horizontal red dashed line denotes 90% of the on-axis optical efficiency for the different designs.
Fig. 4.
Fig. 4. Semi-log plot of specific optical efficiency simulated for an ideal reflective parabolic concentrator (solid lines) with different edge thicknesses specified via the ratio, ER, of edge thickness to receiver diameter. The colored dots mark different acceptance angles on each curve. The dashed red line is the result for an ellipsoidal refractive concentrator described in the text. The simulations are carried out with the same assumptions as Fig. 3(a) and 3(b). The right-hand axis rescales the data to specific power assuming a fixed 30% microcell efficiency and the AM0 solar flux of 1366 W m$^{-2}$.
Fig. 5.
Fig. 5. Ideal specific power calculated for a hexagonal glass parabolic reflector array interfaced with square microcells according to Eqn. 4. In this log-log plot, $h_{\mathrm {edge}}= 200\;$µm, $w_{\mathrm {cusp}}= 75\;$µm, and the concentration ratio varies implicitly to maximize the specific power maintaining the same assumptions as in Fig. 4. In the limit of small cell size, the specific power becomes limited by the minimum thickness constraint of the optic before the cusp loss becomes dominant as indicated by the horizontal dashed lines.
Fig. 6.
Fig. 6. (a) Specific power simulated for a hexagonal µCPV array with 170 µm square microcells as shown in the inset. Acceptance half-angle (defined at 90% of the on-axis optical efficiency) is noted at the top of the plot. All optical losses and fabrication constraints are accounted for in the simulation. (b) Monte Carlo tolerance analysis expressing the spread in optical efficiency that results from random parameter variation within the tolerance ranges shown in the inset for the case of $CR = 30$.
Fig. 7.
Fig. 7. The µCPV prototype is assembled from two components, i) a thin sheet of coverglass supporting the transfer-printed microcell array and ii) the reflective optic. Viewed on-axis, the image of the microcells is magnified and their associated mirror elements appear black; they appear reflective again when viewed off-axis beyond the angular tolerance of the system as shown in the right-most photograph. The smaller images in the middle show the backside of a cell after it has been bonded to the optics (left) and the front face of a typical microcell (right). (b) Current-voltage characteristic measured for the µCPV prototype outdoors on a clear sunny day, reflecting the output of two strings of three series-connected microcells; the current density is relative to the aperture area of the concentrator. The inset shows the direct (black) and diffuse (red) components of the solar spectrum on the day of the test; the measurements reported here were taken in the blue-highlighted window. (c) Power conversion efficiency (PCE) and optical efficiency measured outdoors as a function of incidence angle (blue circles) compared with that simulated by our ray tracing model (red dashed line). The factors that contribute to the optical loss at normal incidence are broken down in the inset pie chart. More than 95% of the shading loss in this case is due to the area of the cell with the remainder due to the metal interconnect traces. The relative loss contributions do not change significantly over the acceptance angle range of the concentrator.
Fig. 8.
Fig. 8. (a) Transient change in $V_{\textrm{oc}}$ measured by suddenly exposing the µCPV prototype to direct sunlight. The data are rescaled based on the known temperature coefficient of $V_{\textrm{oc}}$ to reflect the change in average cell temperature on the right-hand axis, which agrees well with that predicted by a finite element heat transfer model indicated by the dashed red line. The inset shows a cross-section of the simulated temperature distribution at thermal equilibrium. (b) Microcell temperature predicted using the same model for a µCPV system operated in space at different concentration ratios, highlighting the passive cooling benefit of small microcell size.

Equations (4)

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tan 1 ( 1 + 1 C R 2 A R ) tan 1 ( 1 1 C R 2 A R ) = 2 sin 1 ( n n sin θ ) .
d y d x = m 1 = tan [ 1 2 ( θ + tan 1 ( r u o h edge y ) ) ] .
Δ y = u i cot [ tan 1 ( u i u i C R h edge ) + θ ] .
A M = 48 ( 3 h e d g e 2 + d 2 + 3 h e d g e ) ( 3 2 ( d 1 2 w c u s p ) 2 w c e l l 2 ) ρ [ 12 d 2 ( 3 h e d g e 2 + d 2 + 3 h e d g e ) 2 5 d 4 ] ,

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