1. Theory

Definition: Concentrator photovoltaics (CPV) focus direct sunlight with lenses or mirrors onto small, high-efficiency cells, usually III-V multi-junction cells, normally on a tracker and sometimes with cooling (Wikipedia). The concentration is counted in suns, which Wikipedia defines as the square of the magnification; in the efficiency tables one sun is a direct irradiance of 1000 W/m $^{2}$ and concentrator cells are measured with the ASTM G-173-03 direct-beam AM1.5 spectrum at 25 °C (Green et al.).
Concentration levels (Wikipedia): Low concentration is 2 to 100 suns, mostly with silicon cells and often without cooling or tracking; medium is 100 to 300 suns with two-axis tracking and cooling; high concentration is “1000 suns or more”, although Amonix systems are rated near 500 suns and cells are reliable at 500 to 1000 suns (50 to 100 W/cm $^{2}$ ). The current density is about 8 A/cm $^{2}$ at 500 suns, the cell resistance below 1 mΩ and the cell size typically below 1 cm $^{2}$ .
Optics and tracking: Fresnel lenses (point-focus acrylic or silicone on glass), dish and trough mirrors and heliostat arrays are used; nonimaging optics are called the most suitable; the optical efficiency is 75 to 90 %; modern high-concentration trackers must stay within 0.1 to 0.3 degrees (Wikipedia). The thermodynamic limit of concentration is 46 223 (computed with a solar half-angle of 0.2665 degrees, background knowledge).
Cell records (Table 5 of the efficiency tables, version 64): A six-junction inverted metamorphic cell (AlGaInP/AlGaAs/GaAs/GaInAs×3, 2.15/1.72/1.41/1.17/0.96/0.70 eV) reached 47.1 ± 2.6 % at 143 suns (0.099 cm $^{2}$ , NREL, March 2019); a four-junction bonded cell (GaInP/GaInAs and GaInAsP/GaInAs) 47.6 ± 2.6 % at 665 suns (0.0452 cm $^{2}$ , Fraunhofer ISE, May 2022); a four-junction monolithic cell 45.7 ± 2.3 % at 234 suns (0.09709 cm $^{2}$ , NREL, September 2014); Sharp’s inverted metamorphic triple 44.4 ± 2.6 % at 302 suns (0.1652 cm $^{2}$ , Fraunhofer ISE, April 2013); a GaInAsP/GaInAs double 35.5 ± 1.2 % at 38 suns (0.10031 cm $^{2}$ , NREL, October 2017); a GaAs cell 30.8 ± 1.9 % at 61 suns (0.0990 cm $^{2}$ , NREL, January 2022); an Amonix silicon back-contact cell 27.6 ± 1.2 % at 92 suns (1.00 cm $^{2}$ , Fraunhofer ISE, November 2004); and a CIGS cell 23.3 ± 1.2 % at 15 suns (0.09902 cm $^{2}$ , NREL, March 2014).
Module records (same table): A Fraunhofer ISE four-junction minimodule with lenses reached 43.4 ± 2.4 % (18.2 cm $^{2}$ , 340 suns, July 2015), another with 10 cells 41.4 ± 2.6 % (121.8 cm $^{2}$ , 230 suns, September 2018), a UNSW spectrum-splitting submodule (GaInP/GaInAs/Ge and silicon) 40.6 ± 2.0 % (287 cm $^{2}$ , 365 suns, NREL April 2016), a Soitec four-junction module 38.9 ± 2.5 % (812.3 cm $^{2}$ , 333 suns, Fraunhofer ISE April 2015) and an Amonix triple-junction module 35.9 ± 1.8 % (1092 cm $^{2}$ , NREL August 2013, referenced to 1000 W/m $^{2}$ direct irradiance and 25 °C). A hybrid four-terminal module of four-junction CPV and bifacial silicon (1088 cm $^{2}$ , Fraunhofer ISE, September 2019) reached 34.2 ± 1.9 % as a thermodynamic efficiency, and a luminescent minimodule with GaAs cells 7.1 ± 0.2 % at a geometric concentration of 2.5 (ECN Petten, ESTI 2008).
History: Wikipedia puts the first plant above 1 MW in Spain in 2006, a research efficiency rising from 34 % (three junctions) to 47.6 % (four junctions) and the Raygen heliostat receivers at 40.4 % in 2017. Fraunhofer ISE reported 46 % at 508 suns (50.8 W/cm $^{2}$ ) on 1 December 2014 for a four-junction cell in which each subcell converts a quarter of the photons between 300 and 1750 nm (Soitec, CEA-Leti, Fraunhofer ISE; confirmed by AIST), a module of 41.4 % on 22 November 2018 (the EU project CPVMatch), and the 47.6 % cell in May 2022, for which the previous record was 46.1 % (Soitec, 2016).
Test conditions: The IEC 62670 standard test conditions are a direct irradiance of 1000 W/m $^{2}$ , AM1.5D and a cell at 25 °C; the operating conditions are 900 W/m $^{2}$ , 20 °C ambient and 2 m/s wind (Wikipedia).
Industry: Wikipedia lists plants such as Golmud (137.8 MWp, 2012-2013), Touwsrivier (44.2 MWp, 2014) and Alamosa (35.3 MWp, 2012), a cumulative 350 MW (about 0.2 % of the 230 000 MW of global capacity), a peak of installations in 2012, almost none since 2018 and the closure of Suncore, Soitec, Amonix and SolFocus. The resource needed is roughly 2000 kWh/m $^{2}$ of direct irradiation per year.

2. Working Principle

Optical concentration: The lens or mirror collects the direct beam over the aperture and focuses it on the cell, so a cell of 0.0452 cm $^{2}$ at 665 suns receives 3.01 W and must lose 1.57 W as heat, 34.8 W/cm $^{2}$ (computed, direct-beam basis, reflection neglected). Diffuse light is not collected, which is why the direct-beam efficiency must be multiplied by a factor of 0.8746 to give a thermodynamic one (Green et al.).
Voltage and efficiency: The current grows in proportion to the concentration and the voltage by $N (k_{B} T / q) ln X$ , 0.765 V for six junctions at 143 suns and 0.668 V for four junctions at 665 suns (computed). For an ideal six-junction cell this means 54.44 % at one sun, 61.58 % at 143 suns and 64.38 % at 1000 suns (computed). The relative gain is $Δ V / V_{oc}$ , as in the table of Wikipedia (computed).
Why multi-junction cells: The current per junction is small (7.0 A/cm $^{2}$ at 508 suns for an equal split of the photons, computed), so the series-resistance drop is small compared with a stack voltage of several volts; for the six-junction cell 6.0 mV out of 5.5 V at 143 suns with 5 mΩ cm $^{2}$ (computed, assumed). A GaAs cell would lose 6.6 % at 500 suns with the same resistance (computed, assumed).
Spectrum sensitivity: Series-connected junctions need a matched current, so the direct spectrum, haze and diffuse light change the output (Wikipedia). For the six-junction gaps of Table 5 my model gives 54.44 % for the direct and 52.17 % for the global spectrum at one sun (computed).
Temperature: The heat flux of 4 to 35 W/cm $^{2}$ in the records is removed by passive heat sinks (Wikipedia); a cell limit near 110 °C and a receiver limit near 80 °C apply (Wikipedia).
Module: The module efficiency is the cell efficiency times the optical and other factors: Wikipedia gives 44 % times 75 to 90 % for about 36 %, and the records are 0.84 to 0.94 of the best cells (computed).

3. Applications

Utility plants in high-DNI regions: Golmud, Touwsrivier and Alamosa (Wikipedia); Fraunhofer ISE names regions with high direct normal irradiance as the market.
Concentrated PV with heat and storage: Wikipedia lists heliostat receivers (Raygen, 250 kW, 40.4 %), concentrated PV and thermal systems with over 40 % thermal and up to 35 % electrical output, and an Australian system (2023) with 4 MW and 51 MWh storage.
Hybrids and splitting: The four-terminal hybrid of CPV and bifacial silicon (34.2 % thermodynamic) and the spectrum-splitting submodule (40.6 %) use the diffuse light or the infrared that the III-V cells do not (Green et al.; my reading).
Low concentration: Silicon cells at 11 to 92 suns reached 21.7 % (UNSW, 1990) and 27.6 % (Amonix, 2004), and a luminescent concentrator minimodule of 7.1 % is the thin-film end of the range; luminescent concentrators are an optional later device of the plan.

Concentration, Optics and Heat

Name	Formula	Related Formulae	Variables/Constants
Concentration in suns	$X = η_{o pt} C_{g} \frac{G _{D N I}}{G _{r e f}}$	$C_{g} = \frac{A _{a p}}{A _{ce l l}}, X = \frac{G _{ce l l}}{G _{r e f}}$	$X : co n ce n t r a t i o n in s u n s$ $η_{o pt} : o pt i c a l e f f i c i e n cy$ $C_{g} : g eo m e t r i c co n ce n t r a t i o n$ $G_{D N I} : d i r ec t n or ma l i r r a d ian ce$ $G_{r e f} : 1000 W / m^{2} (o n e s u n i s a d i r ec t i r r a d ian ce o f 1000 W / m^{2} in G r ee n e t a l .)$ $A_{a p}, A_{ce l l} : a p er t u r e an d ce l l a r e a s$ $G_{ce l l} : i r r a d ian ce o n t h e ce l l, W ik i p e d ia g i v es o pt i c a l e f f i c i e n c i es o f 75 t o 90 %, a co n ce n t r a t i o n m e a s u r e d in s u n s a s t h e s q u a r e o f t h e ma g ni f i c a t i o n, 1000 W / m^{2} f or t h e I E C 62670 r e f er e n ce co n d i t i o n s an d 900 W / m^{2} f or t h e o p er a t in g co n d i t i o n s$ $(co m p u t e d w i t h a ss u m e d v a l u es C_{g} = 700, η_{o pt} = 0.85 an d G_{D N I} = 900 W / m^{2} : X = 535.5; a ce l l o f 0.0452 c m^{2} a t 665 s u n s r ece i v es 665 \times 0.1 \times 0.0452 = 3.01 W; 500 s u n s a r e 50 W / c m^{2} an d 1000 s u n s 100 W / c m^{2}; t h e 8 A / c m^{2} t ha t W ik i p e d ia g i v es a t 500 s u n s m e an s 16 m A / c m^{2} p er s u n, an d t h e e q u a l s pl i t ce l l o f t h e r o w s b e l o w g i v es 7.0 A / c m^{2} a t 508 s u n s)$
Maximum concentration and acceptance angle	$C_{ma x} = \frac{n ^{2}}{s i n ^{2} θ _{s}}$	$θ_{a} = arcsin \frac{n}{X}$	$C_{ma x} : t h er m o d y nami c l imi t$ $n : r e f r a c t i v e in d e x a r o u n d t h e ce l l$ $θ_{s} : ha l f - an g l e o f t h e s u n (0.2665 d e g r ees, ba c k g r o u n d k n o w l e d g e)$ $θ_{a} : l a r g es t a cce pt an ce ha l f - an g l e o f an i d e a l co n ce n t r a t or f or t h e co n ce n t r a t i o n X, W ik i p e d ia g i v es n o n u m er i c a cce pt an ce an g l e an d n o l imi t, an d r e q u i r es t r a c k er s w i t hin 0.1 t o 0.3 d e g r ees an d tw o - a x i s t r a c k in g f r o m ab o u t 100 s u n s$ $(co m p u t e d : C_{ma x} = 46223 f or n = 1 an d ab o u t 104000 f or n = 1.5; θ_{a} = 7.36 d e g r ees a t 61 s u n s, 4.80 a t 143, 3.75 a t 234, 3.30 a t 302, 2.56 a t 500, 2.22 a t 665, 1.81 a t 1000 an d 0.27 a t 46200; 0.3 d e g r ees i s 0.135 o f t h e an g l e a t 665 s u n s)$
Heat flux through the cell	$q = X G_{r e f} (1 - η)$	$P_{ce l l} = X G_{r e f} A_{ce l l} η$	$q : h e a t f l ux t o b e r e m o v e d$ $η : e f f i c i e n cy o n t h e d i r ec t - b e am ba s i s$ $P_{ce l l} : e l ec t r i c a l p o w er, r e f l ec t i o n an d p a r a s i t i c ab sor pt i o n a r e n e g l ec t e d (a ss u m e d), W ik i p e d ia g i v es ce l l r e l iabi l i t y a t 500 t o 1000 s u n s (50 t o 100 W / c m^{2}), p a ss i v e h e a t s ink s a s p r e f er r e d, a ce l l l imi t n e a r 110 C an d a r ece i v er l imi t n e a r 80 C$ $(co m p u t e d : 665 s u n s a t 47.6 % g i v e 66.5 W / c m^{2} in c i d e n t an d 34.8 W / c m^{2} o f h e a t, an d o n 0.0452 c m^{2} 3.01 W in c i d e n t, 1.43 W e l ec t r i c a l; 143 s u n s a t 47.1 % g i v e 7.56 W / c m^{2}; 61 s u n s a t 30.8 % g i v e 4.22 W / c m^{2})$
Module efficiency from cell efficiency	$η_{m o d} = η_{ce l l} η_{o pt}$	$η_{o pt} = \frac{η _{m o d}}{η _{ce l l}}$	$η_{m o d} : m o d u l e e f f i c i e n cy$ $η_{ce l l} : e f f i c i e n cy o f t h e ce l l u n d er t h e s am e co n d i t i o n s$ $η_{o pt} : p r o d u c t o f t h e o pt i c a l an d a l l o t h er m o d u l e f a c t or s, W ik i p e d ia g i v es a 44 % ce l l (850 W / m^{2}, 25 C), o pt i cs o f 75 t o 90 % an d a m o d u l e n e a r 36 %, t h e m o d u l es o f G r ee n e t a l . an d F r a u nh o f er I S E d o n o t s t a t e w hi c h ce l l s t h ey u se$ $(co m p u t e d : 44 \times 0.75 t o 0.90 = 33.0 t o 39.6 % an d 36/44 = 0.82; t h e 43.4 % minim o d u l e i s 0.912 o f 47.6 % an d 0.941 o f 46.1 %; t h e 38.9 % S o i t ec m o d u l e i s 0.844 o f 46.1 %; t h e 41.4 % minim o d u l e i s 0.870 o f 47.6 % (a ss u m e d t ha t t h e r ecor d ce l l s a r e co m p a r ab l e w i t h t h e ce l l s in t h e m o d u l es))$

Voltage, Resistance and Detailed Balance

Name	Formula	Related Formulae	Variables/Constants
Voltage gain with concentration	$Δ V = N \frac{n k _{B} T}{q} ln X$	$V_{oc} (X) = V_{oc} (1) + Δ V$	$Δ V : r i se o f t h e o p e n - c i r c u i t v o l t a g e$ $N : n u mb er o f j u n c t i o n s in ser i es$ $n : i d e a l i t y f a c t or (1 a ss u m e d), k_{B} T / q = 25.693 mV a t 298.15 K (W ik i p e d ia u ses 25.85 mV a t 300 K), t h e p h o t oc u r r e n t i s t ak e n a s p r o p or t i o na l t o X$ $k_{B} : B o l t z mann co n s t an t$ $(co m p u t e d : s i x j u n c t i o n s a t 143 s u n s 0.765 V (m y m o d e l w i t h t h e g a p s o f T ab l e 5 g i v es o p e n - c i r c u i t v o l t a g es o f 7.041 - 6.276 = 0.765 V; t h e e a r l i er 6.596 - 5.835 = 0.761 V a r e t h e ma x im u m - p o w er v o l t a g es, a u d i t n o t e), f o u r j u n c t i o n s a t 665 s u n s 0.668 V, a t 508 s u n s 0.640 V an d a t 234 s u n s 0.561 V, t h r ee j u n c t i o n s a t 302 s u n s 0.440 V, tw o a t 38 s u n s 0.187 V an d G a A s a t 61 s u n s 0.106 V)$
Relative efficiency gain from the voltage	$\frac{η _{X}}{η _{1}} \approx 1 + \frac{Δ V}{V _{oc}}$	$\frac{η _{X}}{η _{1}} \approx 1 + \frac{n k _{B} T}{q V _{oc}} ln X$	$η_{X}, η_{1} : e f f i c i e n cy a t X s u n s an d a t o n e s u n$ $V_{oc} : v o l t a g e p er j u n c t i o n (or t h e s t a c k v o l t a g e w i t h t h e N in t h e g ain), t h e f i l l f a c t or an d t h e s p ec t r u m a r e t ak e n a s co n s t an t (a ss u m e d), W ik i p e d ia t ab u l a t es t h e g ain f or 550 mV (m u l t i cr y s t a l l in e s i l i co n), 700 mV (m o n ocr y s t a l l in e s i l i co n) an d 850 mV (t r i pl e - j u n c t i o n I I I - V) a t 10, 100 an d 1000 s u n s$ $k_{B} : B o l t z mann co n s t an t$ $(co m p u t e d w i t h 25.85 mV : 10.8, 21.6 an d 32.5 % f or 550 mV, 8.5, 17.0 an d 25.5 % f or 700 mV, 7.0, 14.0 an d 21.0 % f or 850 mV, w hi c h a g r ees w i t h t h e p r in t e d t ab l e; f or t h e s i x - j u n c t i o n ce l l (5.549 V a t o n e s u n) t h e g ain a t 143 s u n s i s 13.8 %)$
Series-resistance loss	$Δ V_{R} = X J_{1} r_{s}$	$\frac{Δ P}{P} \approx \frac{X J _{1} r _{s}}{V _{m p}}$	$Δ V_{R} : v o l t a g e d r o p$ $J_{1} : c u r r e n t d e n s i t y a t o n e s u n$ $r_{s} : s p ec i f i c ser i es r es i s t an ce o f t h e ce l l (m Ω c m^{2})$ $V_{m p} : v o l t a g e a t ma x im u m p o w er, W ik i p e d ia g i v es a ser i es r es i s t an ce b e l o w 1 m Ω, ce l l s t y p i c a l l y b e l o w 1 c m^{2} an d ab o u t 8 A / c m^{2} a t 500 s u n s, t h e r es i s t an ce v a l u es b e l o w a r e a ss u m e d$ $(co m p u t e d : 8 A / c m^{2} t h r o ug h 1 m Ω c m^{2} i s 8 mV; t h e G a A s ce l l (29.78 m A / c m^{2}) a t 61 s u n s c a r r i es 1.817 A / c m^{2} an d l oses 9.1 mV (0.8 % o f 1.127 V) f or 5 m Ω c m^{2} (a ss u m e d), an d 74.5 mV (6.6 %) a t 500 s u n s; t h e s i x - j u n c t i o n ce l l (8.457 m A / c m^{2}) a t 143 s u n s c a r r i es 1.209 A / c m^{2} an d l oses 6.0 mV (0.11 % o f 5.549 V))$
Detailed-balance efficiency against concentration	$η (X) = max_{J} \frac{J \sum _{i} V _{i} ( J )}{X P _{d i r}}$	$V_{i} (J) = \frac{k _{B} T}{q} ln (\frac{X J _{i} - J}{J _{0, i}} + 1)$	$P_{d i r} : p o w er o f t h e A S T M G 173 d i r ec t s p ec t r u m (900.14 W / m^{2} in t e g r a t e d b y p v l ib), t h e m o d e l i s t h e o n e o f t h e p r e v i o u s p a g e (s t e p ab sor pt an ce, n o r e f l ec t i o n, e mi ss i o n in t o 2 π an d n o l u min esce n t co u pl in g, m y o w n c a l c u l a t i o n), t h e 2 π e mi ss i o n i s co n ser v a t i v e a t hi g h co n ce n t r a t i o n (ba c k g r o u n d k n o w l e d g e)$ $k_{B} : B o l t z mann co n s t an t$ $(co m p u t e d, s i x j u n c t i o n s w i t h t h e g a p s o f T ab l e 5 (2.15/1.72/1.41/1.17/0.96/0.70 e V) : 54.44 % a t 1 s u n, 57.75 % a t 10, 61.06 % a t 100, 61.58 % a t 143, 63.38 % a t 500, 64.38 % a t 1000, 67.69 % a t 10 000 an d 69.90 % a t 46200; G a A s a t 1.42 e V : 31.98 % a t 1 s u n, 35.23 % a t 61 an d 36.89 % a t 500; t h e g a p se t 2.19/1.76/1.45/1.19/0.97/0.7 e V o f t h e o n e - s u n e n t r y g i v es 65.09 % a t 143 s u n s)$
Realised fraction at concentration	$r = \frac{η _{m e a s}}{η _{i d e a l}}$	$r_{o pt} = \frac{η _{m e a s}}{η _{N, o pt} ( X )}$	$r : r a t i o o f t h e m e a s u r e d d i r ec t - b e am e f f i c i e n cy t o t h e v a l u e o f m y m o d e l f or t h e s am e g a p s an d co n ce n t r a t i o n$ $r_{o pt} : r a t i o t o t h e b es t e f f i c i e n cy o f N j u n c t i o n s a t t ha t co n ce n t r a t i o n (u se d w h e n t h e g a p s a r e n o t k n o w n), t h e m e a s u r e d v a l u es a r e f r o m T ab l e 5 o f G r ee n e t a l ., t h e 25 C ce l l t e m p er a t u r e an d t h e s p ec t r u m a r e t h e s t a t e d t es t co n d i t i o n s$ $(co m p u t e d : G a A s a t 61 s u n s 30.8/35.23 = 0.874; s i x j u n c t i o n s a t 143 s u n s 47.1/61.58 = 0.765; a g ain s t t h e o pt im u m : f o u r j u n c t i o n s a t 665 s u n s 47.6/64.02 = 0.744, f o u r j u n c t i o n s a t 234 s u n s 45.7/62.25 = 0.734, t h r ee j u n c t i o n s a t 302 s u n s 44.4/58.83 = 0.755 an d tw o j u n c t i o n s a t 38 s u n s 35.5/49.28 = 0.720; o pt im u m g a p s f or f o u r j u n c t i o n s a t 665 s u n s 1.877/1.369/0.963/0.516 e V)$
Equal photon split of a multi-junction cell	$J_{i} = \frac{J _{t o t}}{N}, J_{t o t} = q \int_{E_{g, N}}^{\infty} Φ (E) d E$	$E_{g, N} = \frac{h c}{λ _{ma x}}$	$J_{i} : p h o t oc u r r e n t o f e a c h o f N j u n c t i o n s$ $J_{t o t} : p h o t oc u r r e n t ab o v e t h e l o w es t g a p$ $λ_{ma x} : l o n g es t w a v e l e n g t h ab sor b e d, F r a u nh o f er I S E s t a t es t ha t e a c h s u b ce l l o f i t s f o u r - j u n c t i o n ce l l co n v er t s o n e q u a r t er o f t h e p h o t o n s in t h e r an g e o f 300 t o 1750 nm, t h e g a p s a r e d er i v e d b y m e f r o m t h e d i r ec t s p ec t r u m$ $(co m p u t e d : 1750 nm i s 0.708 e V, J_{t o t} = 55.24 m A / c m^{2} an d t h e f o u r g a p s a r e 1.931/1.439/1.043/0.708 e V w i t h 13.81 m A / c m^{2} e a c h; t h e m o d e l g i v es 53.44 % a t 1 s u n, 63.01 % a t 508 s u n s (7.02 A / c m^{2} p er j u n c t i o n) an d 63.42 % a t 665 s u n s; 46.0/63.01 = 0.730 an d 47.6/63.42 = 0.751)$

Efficiency Conversions and Module Checks

Name	Formula	Related Formulae	Variables/Constants
Thermodynamic efficiency from the direct-beam efficiency	$η_{t h} = f η_{d}$	$f_{c a l c} = \frac{P _{d i r}}{P _{g l o b}}$	$η_{t h} : t h er m o d y nami c e f f i c i e n cy$ $η_{d} : e f f i c i e n cy u n d er t h e A S T M G - 173 - 03 d i r ec t - b e am s p ec t r u m$ $f : f a c t or t ha t G r ee n e t a l . es t ima t e a s 0.8746 b ec a u se t h e d i r ec t b e am n e g l ec t s t h e d i f f u se l i g h t$ $f_{c a l c} : r a t i o o f t h e in t e g r a t e d d i r ec t an d g l o ba l s p ec t r a (900.14 an d 1000.37 W / m^{2}, p v l ib), t h e m o d u l es a r e m e a s u r e d a t t h e I E C 62670 - 1 r e f er e n ce co n d i t i o n s an d t h e f a c t or i s a ppl i e d t o t h e m o n l y f or co m p a r i so n (a ss u m e d)$ $(co m p u t e d, w i t h 0.8746 (w i t h 0.8998) : 47.6 g i v es 41.63 (42.83), 47.1 g i v es 41.19 (42.38), 45.7 g i v es 39.97 (41.12), 44.4 g i v es 38.83 (39.95), 35.5 g i v es 31.05 (31.94), 43.4 g i v es 37.96 (39.05), 40.6 g i v es 35.51 (36.53), 38.9 g i v es 34.02 (35.00), 41.4 g i v es 36.21 (37.25), 30.8 g i v es 26.94 (27.71); 0.8746 \times 1000.37 = 874.9 W / m^{2})$
Measured gain from concentration	$g_{X} = \frac{η _{co n c}}{η _{1}}$	$g_{t h} = \frac{f η _{co n c}}{η _{1}}$	$g_{X} : r a t i o o f t h e co n ce n t r a t or e f f i c i e n cy (d i r ec t b e am) t o t h e o n e - s u n r ecor d (g l o ba l s p ec t r u m)$ $g_{t h} : t h e s am e w i t h t h e f a c t or f = 0.8746, t h e ce l l s a r e d i f f er e n t d e v i ces (a ss u m e d co m p a r ab l e in s t r u c t u r e), v a l u es f r o m G r ee n e t a l . (t h e o n e - s u n ce l l s a r e o n t h e p r e v i o u s p a g e an d G a A s i s o n p a g e 86)$ $(co m p u t e d : s i x j u n c t i o n s 47.1/39.2 = 1.202 (g_{t h} = 1.051); f o u r j u n c t i o n s 47.6/39.5 = 1.205 (1.054); S ha r p t r i pl e 44.4/37.9 = 1.172 (1.025); G a A s 30.8/29.1 = 1.058 (0.926); t h e m o d e l g i v es 61.58/52.17 = 1.18 f or t h e s i x - j u n c t i o n g a p s o f T ab l e 5 b e tw ee n d i r ec t 143 s u n s an d g l o ba l o n e s u n)$
Module power and cell size	$P = η A_{a p} G_{r e f}$	$A_{ce l l} = \frac{A _{a p}}{N _{c} C _{g}}$	$P : o u tp u t a t t h e r e f er e n ce co n d i t i o n s$ $A_{a p} : a p er t u r e a r e a$ $N_{c} : n u mb er o f ce l l s$ $C_{g} : g eo m e t r i c co n ce n t r a t i o n (t h e in t e n s i t y co l u mn o f G r ee n e t a l . i s t ak e n a s g eo m e t r i c h er e, w hi c h i s a ss u m e d), G_{r e f} = 100 mW / c m^{2}, t h e e n t r i es a r e f r o m G r ee n e t a l . (a p : a p er t u r e a r e a)$ $(co m p u t e d : t h e 41.4 % minim o d u l e (121.8 c m^{2}, 230 s u n s, 10 ce l l s) g i v es 5.04 W, 0.504 W p er ce l l an d 0.053 c m^{2} p er ce l l; t h e S o i t ec m o d u l e (38.9 %, 812.3 c m^{2}) 31.6 W; t h e A m o ni x m o d u l e (35.9 %, 1092 c m^{2}) 39.2 W; t h e U N S W s u bm o d u l e (40.6 %, 287 c m^{2}) 11.65 W; t h e 43.4 % minim o d u l e (18.2 c m^{2}) 0.79 W)$

Important Questions

Ques: Does the factor 0.8746 or about 0.90 convert a direct-beam efficiency into a thermodynamic one?

Green et al. say under Table 5 that direct-beam efficiencies neglect the diffuse light and must be multiplied by a factor estimated as 0.8746, so 47.6 % becomes 41.63 % (computed). My integral of the ASTM G173 direct spectrum is 900.14 $W/ m^{2}$ against 1000.37 for the global one, a ratio of 0.8998 (computed), which gives 42.83 %. The factors differ by 2.9 %, or 25 $W/ m^{2}$ (computed), and the extract does not say how 0.8746 was estimated; the 874.9 $W/ m^{2}$ it implies is not a value I can match to a spectrum. Fraunhofer ISE defines its 41.4 % module as the share of the direct irradiation converted, not as a thermodynamic value. The table’s own factor is the more likely one for comparing with those tables, and 0.90 is only my spectral ratio; the two should not be mixed (my reading). Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Fraunhofer ISE press release #25, ‘Concentrator photovoltaics shows peak performance: module efficiency reaches 41.4 percent’ (22 November 2018).

Ques: Was the previous concentrator record 46.0 % (2014) or 46.1 % (2016)?

Fraunhofer ISE announced 46 % at 508 suns (50.8 $W/c m^{2}$ ) on 1 December 2014 for a four-junction cell of Soitec, CEA-Leti and Fraunhofer ISE, confirmed by AIST, made on a 100 mm wafer of about 500 devices. Its 2022 release names 46.1 % (Soitec, 2016) as the record that the 47.6 % cell beat. The difference of 0.1 points is far inside the uncertainty of 2.6 points of the table, so the values cannot be told apart, and the dates differ by two years. They are most likely two different cells, the later one slightly better (my reading), but the extracts give no area or gap set for either. Green et al. list neither number, only 47.6, 47.1 and 45.7 %, and Wikipedia mentions only the 47.6 % cell. I take 46.1 % as the later and more likely predecessor. Reference: Fraunhofer ISE press release ‘World record: 46 percent’ (1 December 2014; Soitec, CEA-Leti, Fraunhofer ISE); Fraunhofer ISE press release on the 47.6 % concentrator cell (30 May 2022); Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Wikipedia, ‘Concentrator photovoltaics’.

Ques: Are the gaps of the six-junction cell those of the one-sun entry or of Table 5?

Green et al. print 2.19/1.76/1.45/1.19/0.97/0.7 eV for the one-sun six-junction entry (39.2 %, 0.247 $c m^{2}$ ) and 2.15/1.72/1.41/1.17/0.96/0.70 eV for the concentrator cell (47.1 % at 143 suns, 0.099 $c m^{2}$ ). The sets differ by 0 to 0.04 eV and the areas differ too, so the two entries may be separate devices or two descriptions of one structure; the extracts do not say which. My model gives 65.09 % at 143 suns for the first set and 61.58 % for the second (computed), so the realised fraction is 0.724 or 0.765. I use the Table 5 set for the concentrator cell, since it is printed with that entry, and I cannot decide which set is the more accurate. Semiconductor Today describes one cell measured at both conditions (my reading). Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Semiconductor Today, report on the six-junction inverted metamorphic cell (14 April 2020; Geisz et al., Nature Energy 2020); III-V Multi-Junction Solar Cell page of this vault.

Ques: Is high-concentration CPV 1000 suns or more, or 300 to 700 suns?

Wikipedia defines high concentration as 1000 suns or more, medium as 100 to 300 suns and says that cells are reliable at 500 to 1000 suns, yet it rates Amonix systems near 500 suns and thermal-electric systems above 700 suns. In Green et al. the cell records are at 143, 234, 302 and 665 suns and the modules at 230 to 365 suns; the Fraunhofer 2014 cell was measured at 508 suns. Nothing in the tables reaches 1000 suns, and 665 suns is the highest record (computed from the list). In practice the high-concentration systems work at about 300 to 700 suns, which is the more likely working range, and the 1000-sun figure is a nominal class boundary (my reading). The gain from 500 to 1000 suns is only 1.00 point in my model (computed, six junctions: 63.38 to 64.38 %). Reference: Wikipedia, ‘Concentrator photovoltaics’; Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Fraunhofer ISE press release ‘World record: 46 percent’ (1 December 2014; Soitec, CEA-Leti, Fraunhofer ISE).

Ques: Is the efficiency gain at 1000 suns 20 to 30 % or 21.0 to 32.5 %?

Wikipedia says that concentration gives 20 to 30 % more efficiency at 1000 suns, while its table gives 32.5 % for a 550 mV multicrystalline silicon cell, 25.5 % for 700 mV and 21.0 % for 850 mV. With $k_{B} T / q$ = 25.85 mV the relative gain is 25.85 mV × ln(1000)/V $_{oc}$ , which reproduces all nine entries (computed). The text is a rounded range and the table the more exact figure. The 850 mV of the triple-junction row is the voltage per junction, since a stack of three has about 3 V (my reading). For the six-junction cell the formula predicts 13.8 % at 143 suns, while the records give 1.202 (direct-beam against global one-sun) or 1.051 (with the factor 0.8746), so the realised gain is smaller than the voltage gain (computed; different devices). Reference: Wikipedia, ‘Concentrator photovoltaics’; Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024).

Ques: Is the limit for five junctions 65 %, 66 % or 57 %?

Wikipedia gives a theoretical limit approaching 65 % for five junctions in its CPV article, and in its multi-junction article 86.8 % for an infinite number. My model gives 56.78 % for five junctions at one sun (global), 66.01 % at 500 suns (direct) and 69.90 % for six junctions with fixed gaps at 46 200 suns (computed). The 65 % is therefore close to a concentrated value of 500 suns and not to a one-sun limit, and the extract does not state the concentration (my reading). My values assume emission into $2 π$ , which lowers the voltage at high concentration, so a limit with restricted emission would be higher (background knowledge). I take 65 % as the more likely value for several hundred suns and 57 % for one sun. Reference: Wikipedia, ‘Concentrator photovoltaics’; Wikipedia, ‘Multi-junction solar cell’; III-V Multi-Junction Solar Cell page of this vault.

Ques: What is the CPV module efficiency: 41.4, 38.9, 36 or 33 %?

Fraunhofer ISE reported 41.4 % for a 10-cell minimodule in November 2018 (121.8 $c m^{2}$ , 230 suns) and Green et al. list 43.4 % for a lens-cell minimodule (18.2 $c m^{2}$ ), 38.9 % for the Soitec module (812.3 $c m^{2}$ ) and 35.9 % for the Amonix module. Wikipedia gives about 40 % for production modules, 36 % from a 44 % cell with 75 to 90 % optics, an outdoor value above 33 %, and a system efficiency of 30 % (25 to 28 % AC). The values refer to different sizes and conditions: minimodules are small, the 36 % assumes 850 $W/ m^{2}$ , and system values include inverters and losses. For a full-size module 36 to 39 % is the more likely level, and 43.4 % is a minimodule (my reading). The module is 0.84 to 0.94 of the best cell efficiency (computed). Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Fraunhofer ISE press release #25, ‘Concentrator photovoltaics shows peak performance: module efficiency reaches 41.4 percent’ (22 November 2018); Wikipedia, ‘Concentrator photovoltaics’.

Ques: How close are the concentrator cells to their limits?

With the printed gaps my model gives 35.23 % for GaAs at 61 suns and 61.58 % for the six-junction cell at 143 suns (computed), so the cells reach 0.874 and 0.765. For the cells without gap sets the best N-junction values give 0.744 for four junctions at 665 suns, 0.734 at 234 suns, 0.755 for three junctions at 302 suns and 0.720 for two junctions at 38 suns (computed). For the equal split of the photons from 300 to 1750 nm that Fraunhofer describes, the model gives 63.01 % at 508 suns, so the 46.0 % cell reaches 0.730 (computed). Multi-junction cells therefore lose about a quarter of the ideal value, while the GaAs cell loses 0.126 (1 - 0.874). The model has no reflection or resistance, so a ratio of 1 cannot be reached. Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Fraunhofer ISE press release ‘World record: 46 percent’ (1 December 2014; Soitec, CEA-Leti, Fraunhofer ISE); Gallium Arsenide (GaAs) Single-Junction Solar Cell page of this vault.

Ques: Is a tracking accuracy of 0.1 to 0.3 degrees compatible with the acceptance angle?

Wikipedia requires trackers and module supports to stay within 0.1 to 0.3 degrees for modern high-concentration CPV and gives no acceptance angle. An ideal concentrator in air for X suns accepts a half-angle of $arcsin (1/ X)$ , 2.56 degrees at 500 suns and 2.22 degrees at 665 suns (computed), but the sun itself fills 0.27 degrees (background knowledge) and real optics lose light well inside the ideal angle. The tolerance of 0.3 degrees is only 0.135 of the 2.22 degrees (computed), so it is a design tolerance on the sum of tracking, mounting and optical errors, not the acceptance angle (my reading). The ideal angle shrinks to 1.81 degrees at 1000 suns, so the same fraction would be about 0.24 degrees (computed, assumed proportional), and to 0.27 degrees at 46 200 suns (computed). Reference: Wikipedia, ‘Concentrator photovoltaics’.

Ques: How much CPV is installed: 350 MW at the end of 2015 or 2016, and is it still active?

Wikipedia states the cumulative CPV capacity as 350 MW in 2016 and also as 350 MW by the end of 2015, which cannot both describe a growing market. It adds that this is under 0.2 % of 230 000 MW, that installations peaked in 2012 and have been near zero since 2018, and that Suncore, Soitec, Amonix and SolFocus stopped manufacturing, although Soitec and Suncore appear as vendors of later plants (Fort Irwin 2015, Desert Green 2014) and a 4 MW system with storage dates from 2023. The closure dates are not given. The five main plants add up to about 237 MWp (computed), so a total of about 350 MW is plausible; the year 2016 is the more likely one because most plants were built by 2016 (my reading). The market is small, but research continues. Reference: Wikipedia, ‘Concentrator photovoltaics’.

Ques: Is 47.6 % still the concentrator record?

Green et al. (version 64, 2024) list 47.6 ± 2.6 % at 665 suns as the highest concentrator cell, and Wikipedia states that 47.6 % for four or more junctions is the laboratory record as of April 2026. Versions 65 to 68 of the efficiency tables could not be read here, so I cannot check them directly. The Fraunhofer release of 2022 calls 50 % a goal of the project, and Semiconductor Today says that more than 50 % is achievable for the six-junction approach, with series resistance as the main hurdle. Statements of aims are not results. The 47.6 % is the more likely record value according to the sources read, with an uncertainty of 2.6 points, and 47.1 % is within it. A newer table may contain a better value, which the extracts do not show. Reference: Green, Dunlop, Yoshita, Kopidakis, Bothe, Siefer, Hinken, Rauer, Hohl-Ebinger and Hao, ‘Solar cell efficiency tables (Version 64)’, Prog. Photovolt. 32(7), 425-441 (2024); Wikipedia, ‘Concentrator photovoltaics’; Fraunhofer ISE press release on the 47.6 % concentrator cell (30 May 2022); Semiconductor Today, report on the six-junction inverted metamorphic cell (14 April 2020; Geisz et al., Nature Energy 2020).

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Concentrator Photovoltaic (CPV) Solar Cell