Solar Cells, 3 (1981) 355 - 368 355

THEORETICAL BASES OF PHOTOVOLTAIC CONCENTRATORS FOR EXTENDED LIGHT SOURCES

ANTONIO LUQUE

Instituto de Energia Solar, Escuela Tdcnica Superior de Ingenieros de Telecomunica- ci6n, Universidad Politdcnica de Madrid, Ciudad Universitaria, Madrid 3 (Spain)

(Received November 25, 1980; accepted January 5, 1981)

Summary

The theoretical bases for casting maximum energy on a cell placed in a static concentrator of minimum entry aperture are derived. Several concen- trators are analysed according to this theory and the gain achievable with each is obtained. When practical materials are used the maximum gain is about 9 for the direct beam source and about 4.5 for diffuse radiation. One of the concentrators analysed allows for reaching these figures.

I. Introduction

Conventional concentrating photovoltaic devices require some kind of tracking to keep the cells illuminated while the sun position varies. Further- more, they do not use diffuse radiation which is important even in clear climates. The purpose of this paper is to develop the theoretical bases which show the feasibility of static concentrators as well as their operating limit; the extent to which diffuse radiation can be concentrated by these devices is also considered.

For this, the sun's direct beam is regarded as an extended source occupying the region of the sky in which it can be found at some moment throughout the year. Diffuse radiation is considered to be hemispherical and isotropic and the principles of non-imaging optics [1, 2] are used to analyse the conditions leading to maximum concentrat ion from the extended source. Four concentrators, which are straightforward adaptations of well-known concentrators to the theory presented here, are analysed with respect to both radiation sources.

One of the techniques for maximum concentrat ion is to use bifacial cells, i.e. cells which are active when illuminated on either face. The avail- ability of bifacial cells is a key point of optimal concentrators. Earlier bifacial cells were n* -p -n ÷ made by planar technology [3] . A low cost procedure for manufacturing bifacial p÷-n-n ÷ solar cells has been developed

0379-6787/81/0000-0000/$02.50 © Elsevier Sequoia/Printed in The Netherlands

356

[4] recently with an average bifacial efficiency (the mean of the front and back efficiencies) of 12% for a pilot series of 200 cells.

An important feature of these cells is that they can be manufactured like conventional back surface field cells. The only different step is the delineation of a metal grid on the back face. The technology for this step is not critical and can be the same as that used for the front grid. No mask alignment step is required.

2. Fundamentals of photovoltaic concentration

2.1. Theoretical background Let S be a lambertian source placed at infinity with a constant angular

energy flux density Ps in the direction normal to the source. The power collected by a concentrator is [5]

f W Ps~ J cos 0 sin 0 d~ d0 dx dy (1)

De

where De is the domain of points (x,y) and existing directions (¢, 0 ) at the entry aperture surface Ze of the concentrator. The factor cos 0 takes into account the projection of the surface element on the plane normal to the ray. The element of solid angle is sin 0 de dO.

The direction cosines of each ray with respect to the x and y axes multiplied by the refractive index n at (x, y) are now taken as direction coordinates. We can write these new variables as

p = n sin ¢ cos 0

q = n cos ¢ cos 0

and

dp dq = n 2 sin 0 cos 0 de d0

so that for air (n = 1) eqn. (1) is

(2)

(3)

Ps f dx dy dp dq = PsEe W De

(4)

where the integral is called dtendue by Welford and Winston [2] according to an old Poincar~ nomenclature.

Not all the rays entering the entry aperture reach the collector. A function t(x, y,p,q) can be defined as uni ty for rays reaching the collector and as zero for the remaining rays. The energy reaching the collector for a lossless concentrator can be formally written as

357

We =Ps f t(x,y,p,q) dx dy dp dq (5) D e

When a ray proceeds through the concentrator suffering reflections and refractions and reaches the cell, a new set of coordinates (x',y',p',q') can represent the ray there. It can be demonstrated [2] that

dx dy dp dq = dx' dy' dp' dq' (6)

If variables x,y,p, q are changed into x',y',p', q', eqn. (3) can be written a s

Wc = P s / - 1 , , , , ( i x ' t s ( x , y , p , q ) dy' dp' dq' = VsEc (7) ~ c ~g) l c

o - - 1 ~ t P t t x where ~s (x ,y ,p ,q ) is defined as uni ty for every ray issuing from the cell and reaching the source and as zero for the remaining rays. The entry aper- ture must be chosen so that every ray leaving the cell and reaching the source passes through it. It is clear that every time that t(x,y,p,q) is unity,

- -1 ' t t t ts ( x , y , p , q ) is also unity. Z c ® I c is now the domain of possible rays issuing from the cell surface £c in all possible directions Ic.

The ratio of power reaching the cell to power entering a lossless concen- trator is called the optical intercept factor I, i.e.

I = Ec/E e (8)

where I ~< 1.

2.2. Conditions at the solar cell; optimal concentrators Coordinates p and q are restricted to vary inside a circle of radius n as

can easily be seen from eqns. (2) and (3). Each point of the p-q plane corre- sponds to two rays symmetrical with respect to the 0 = 0 plane. Formally Ic can be considered as the union of two circles in the p-q space corresponding to the rays in both hemispheres. The maximum value of E c (for

- 1 X ' i , p ts ( , Y ,P ,q ) = 1 in eqn. (7)) is Ec = 2~rn2Ac, where Ac is the bifaeial cell area and where the factor 2 accounts for the rays coming from both hemi- spheres. To achieve this ~tendue Ec isotropic illumination of the cell is necessary. A degree of isotropy g is now defined as

Ec - - < 1 ( 9 )

g - 2~n2Ac representing the fraction of energy received at the cell with respect to the maximum energy possible. Concentrators with g = 1 are called optimal. With

1 conventional monofacial cells only g = -~ can be obtained.

Three rules can be derived to achieve maximum energy on the solar cell: (a) use bifacial cells; (b) submerge them in a transparent medium of highest n; (c) achieve the highest degree of isotropy for light incident on the

358

cell. For this, any ray issuing from any point of the cell should reach the source.

To analyse specific concentrators it can be useful to define a local degree of isotropy as

1 [ " - z , , , , gL(X',Y')- 2n2~ -- ts ( x , y , p , q ) dp' dq' (10)

Xc

It is easy to show that g is the average value of the local degree of isotropy:

1 f gL(x' ,y ') dx' dy' (11)

g= A-~ ~:c

Optical concentration can be defined as the ratio between the power in a cell placed in a lossless concentrator and the maximum power of the cell outside it. The maximum power of a cell outside a lossless concentrator is

w, -- P E, -- e, : 3 dy f dq -- P, AoA (12) 2~ c 2; s

where Z¢ and A¢ are the cell surface and its area; Z a is the region of existing rays in the p-q plane, i.e. the source region, and As is its area. The optical concentration is

E¢ 21rn2g Co - - (13)

AcAs As This value is independent of the entry aperture whose definition in a given concentrator is sometimes ambiguous. The maximum value of Co is obtained f o r g = 1.

For a concentrator that is oriented towards the intersection of the local meridian and the celestial equator, the region of the p-q plane where the sun can be found throughout the year is represented in Fig. 1. This region com- prises the points in the circle of radius uni ty fulfilling the condition --sin ~ n~ < q < sin 5 m where 6m = 23.45 ° is the maximum sun declination. It constitutes the direct beam solar source if some shadowing at two of the corners is neglected. This shadow is cast by the plane of the horizon for locations outside the equator.

The direct beam solar source area Ash is 1.549 (see Appendix A). Using this value in eqn. (13) for optimal concentrators (g = 1) we obtain an upper bound for the direct beam optical gain. This value is 13.14 for n = 1.8 and 9.13 for n = 1.5.

The hemispherical diffuse radiation fills the full circle of radius uni ty in the p-q space, and its area Asd is It. The upper bound for the diffuse radia- tion optical gain is 2n 2. For n = 1.8 this value is 6.48 and for n = 1.5 it is 4.5.

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================================================== •

Fig. I. Direct beam solar source (cross-hatched) and diffuse solar source (dotted) in the p-q plane. The acceptance regions of the concentrators in Figs. 2(a), 2(b) and 5(c) are also represented: they are the ellipses MNPQ, M'NP°Q and the circle M"NP"Q respectively.

2.3. Conditions for the entry aperture; ideal concentrators An increase in the energy on the cell is not the only factor in reducing

the cost of the concentrating system: the cost of the optical parts must also be reduced. For this it can be assumed that the concentrator entry aperture is flat and that it must have a minimum area for a given energy reaching the cell. The ~tendue at the entry aperture which is proport ional to the power W is

Ee---- f d dy f dpdq=AeA, (14) Ze ~s

where Z, is the entry aperture and Ae is its area. We can write

Ec A, - (15)

IAs

Since E¢ and As are operating data, the minimum value of Ae is obtained for I = 1. According to Welford and Winston [2] these concen- trators are called ideal. A fourth rule to decrease the cost of the concentrat- ing system is as follows: (d) all rays entering the entry aperture must be cast on the cell so that I = 1.

The concentrator 's geometrical gain Cg is AJAc. This value can be related to the optical gain by using eqns. (13) and (15):

Co = ICg (16)

Equation (16) requires that the concentrator entry aperture and the cell outside it have the same orientation.

We should ment ion that the definition of Z e is ambiguous in a given concentrator. The only condit ion to be fulfilled is that the rays leaving the cell and reaching the source must pass through the entry aperture. Several entry apertures can be defined for a given concentrator leading to different

3 6 0

values of Cg and I. However, the optical gain Co is a clearly defined magni- tude independent of the entry aperture.

The optical intercept factor is closely related to the directional inter- cept factor a (p,q) used by Jones [5] . In fact

1 a(p,q) = ~ee f t(x,y,p,q) dx dy

Ze

and it can easily be shown that I is the average value of a(p,q):

(17)

1 f a(p,q) dp dq (18)

I= A--~ ~s

The inequality in Jones ' work linking a(p,q) and the geometrical gain can be written, looking at eqns. (13), (16) and (18), as

2~n2g Cg f ~(p,q) dp dq (19)

2~ s

The concentrat ion function defined by Guay [6] is in our nomenclature

Co(P,q) = Cga(p,q) (20)

whose average value is Co. The function Co(p,q) does not depend on the entry aperture definition.

In concentrators with losses the function t(x,y,p,q) can take values below unity. The function Co(p, q) can now be called the directional effective gain Ce~f(p,q). Its average is the effective gain Ce~,. The Ce~(P,q) function is of an easily measurable magnitude for analysing a practical concentrator.

3. Analysis of some specific concentrators

3.1. Two-dimensional concentrators The two-dimensional concentrator whose outline is represented in Fig.

2(a) is a trough with two stages [7] . The first one is an angle transformer [8] and the second one is a biracial compound parabolic concentrator (CPC) [9] filled with a transparent medium of high refractive index. Its geometrical gain is

2n Cg - (21)

sin Cm

where ~m is the acceptance angle for meridian rays. The condition for accep- tance of off-meridian rays (for an infinite length concentrator) is that their projections on the meridian plane are accepted rays. In the p-q plane the region of accepted rays is an ellipse whose semi-axes are unity and sin q~m. This is the MNPQ ellipse represented in Fig. 1; a(p,q) is unity for all points inside this region and zero for points outside it. According to eqn. (18) the

(a)

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(b)

Fig. 2. Two-dimensional optimal concentrators.

optical intercept factor is the area of the source region inside this ellipse divided by the source area (see Appendix A). The intercept factor and optical gain are represented as functions of Cm in Fig. 3 for the solar beam source. With Co and using eqn. (13) it is possible to calculate g for n = 1.5 (in this concentrator ng is constant). The same calculations are made for the hemispherical diffuse radiation source in Fig. 4.

In Fig. 2(b) we represent the outline of a bifacial two-dimensional CPC [9] completely filled with a dielectric. The geometrical gain of this concen-

1 - 1 0 . . . . ' r ~ 1.0 -

8 ~ g d l , _ _ .8 E

i 'I ~ 1 1 1 -Id,d ~ i

o

5 l C d 3 d

,. o , i . ~ o I C d l l

.2 ~

' i 1 1o ~ o , i: ==

lo 30 so 70 90 lo 30 so 70 Qo (~)m {degrees) 4~ m Idegreesl

Fig. 3. Optical gain C, degree of isotropy g and intercept factor I of the two-dimensional one-stage (subscript le) and two-stage (subscript 2e) concentrators of Fig. 2 and of the three-dimensional concentrator (subscript 3d) of Fig. 5(b) for the direct beam source (subscript b) as functions of the meridian acceptance angle ~n.

Fig. 4. Optical gain C, degree of isotropy g and intercept factor I of the two-dimensional one-stage (subscript le) and two-stage (subscript 2e) concentrators of Fig. 2 and of the three-dimensional concentrator (subscript 3d) of Fig. 5(b) for the diffuse source (sub- script d) as functions of the meridian acceptance angle Cm-

362

trator is the same as that of the previous one and is also given by eqn. (21), but in this case the acceptance' condition indicated before is only valid for the rays once they have entered the dielectric and have suffered refraction. The acceptance region (rays inside this region are accepted, rays outside it are rejected) is now, as demonstrated elsewhere [ 2], an ellipse whose semi- axes are n and sin Cm. This is the M'NP'Q ellipse represented in Fig. 1. In this region a(p,q) = 1 and outside it a(p,q) = 0. Calculation o f / , Co and g can be performed as before for n = 1.5 (again ng is constant). Their values are represented as functions of Cm in Fig. 3 for the solar beam source and in Fig. 4 for the hemispherical diffuse source.

We notice from these curves that the isotropy factor is not unity, i.e. the concentrators are not optimal, and the reason is that the rays emitted by the cell with q ~ 0 are turned back by total internal reflection at the inter- face of the dielectric and do not reach the source.

For a small acceptance angle the geometrical gain becomes very high but the optical gain remains moderate owing to the low value of the intercept factor. If a concentrator is made with this small value of Cm the energy on the cells depends very little on the value of Cm but the entry aperture increases and so does the cost of the concentrator. Moreover, since the direct beam solar source is an average of the direct beam orientation, low values of Cm lead to a very uneven time distribution of the energy collected, which is usually a drawback. It is interesting to note that for angles Cm ~> 31.19° (for n = 1.5) the one-stage concentrator in Fig. 2(b) has an intercept factor I = 1, meaning that the direct beam is collected throughout the year. For this condition the optical gain is 5.79 for the direct beam and 3.43 for diffuse radiation. The optical intercept factor for the diffuse radiation source is 59.28% which is the fraction of diffuse radiation collected.

A final comment is now needed: two-dimensional concentrators with one stage show a better performance than ones with two stages; however, two-stage concentrators use a smaller amount of dielectric, thus reducing their cost.

3.2. Three-dimensional concentrators Let us consider a concentrator, represented in Fig. 5(a), formed by the

intersection of two two-dimensional CPCs whose collector is rectangle ABCD. A second stage is formed by half a cylinder of circular section which presents both faces of the bifacial cell to the first stage. Both stages are filled with a medium of high refractive index.

The geometrical gain of this concentrator is

2n 2 Cg - (22)

sin Cp sin Cq

where Cp and Cq are the acceptance angles of the vertical and horizontal CPCs respectively.

Since this concentrator is formed by two CPCs filled with a dielectric, apparently all the rays in the intersection region VRWNXSYM (Fig. 6) of

(a) (b)

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Fig. 5. Three-dimensional concentrators for bifacial cells.

T

O IP

U

Fig. 6. The region MVRWNXSY is the assumed acceptance region of the concentrator of Fig. 5(a). The ellipse MRNS is the acceptance region of an optimal concentrator with an optical gain equal to the geometrical gain of that concentrator.

the equatorial (PNQM) and meridian (RUST) CPCs acceptance ellipses are accepted rays. However, some of these rays are rejected, as shown by ray- tracing procedures [2] . To explain this let us assume a source filling the region VRWNXSYM of area As. The maximum achievable optical gain (see eqn. (13)) is

2~n 2 Co ~< (23)

As

The maximum value of the optical intercept factor is then (see eqn. (16))

sin Cp sin Cq I~< (24)

As

where the numerator coincides with the area of the criss-crossed ellipse RNSM which is lower than the area As of the intersection region, showing that the concentrator is no t ideal.

364

Nevertheless it seems interesting to use concentrators of this kind to reduce the useless acceptance region which corresponds to the non-existing rays outside the circle of radius unity. For this Op = 90 °.

We shall consider another type of bifacial three-dimensional concen- trator represented in Fig. 5(b). The first stage consists of a three-dimensional CPC, and the second stage is hemispherical with a cell shaped as a half circle placed in its meridian plane. The whole concentrator is filled with a medium of high refractive index. The objective of the second stage is to present both faces of the bifacial cell to the first stage as if it were a fully circular mono- facial wafer.

Three-dimensional CPCs were studied long ago. In fact the first publica- tions on CPCs were devoted to them [1, 10]. The geometrical gain of this concentrator is

2n 2

Cg - (sin era) 2 (25)

where Cm is the semi-acceptance angle. According to these studies a(p,q) is unity for p - q points well inside the circle of radius sin Cm and zero for points outside it. There is a boundary region near the circle where a(p,q) varies from unity down to zero. The width of this region is less than 3 ° . The integral of ~(p,q) inside the circle of radius ~b m is more than 96% of the circle's area. If we integrate over the whole p - q space (circle of radius unity) we obtain a value very close to the area of the circle of radius era.

To calculate I we can use eqn. (18) and it can be done accurately if we consider a(p,q) to be uni ty for points inside the circle of radius sin ~b m and zero for points outside it. This means that the circle NP"QM" in Fig. 1 can be considered as the acceptance region. In this way I is the area of the inter- section of the source and this circle divided by the source area (see Figs. 3 and 4 for direct and diffuse radiation respectively). C O and g have been cal- culated from these values and can be seen in Figs. 3 and 4.

It can be observed from the above-mentioned figures that this concen- trator is optimal for the diffuse source and only for angles below 23.45 ° is it optimal for the direct source. In this case the optical gain reaches the highest possible value, 9.13, for the direct beam and 4.5 for diffuse radiation if n = 1.5. However, the optical intercept factor is very small, so that a large entry aperture is required to achieve these values of optical concentration. An acceptance angle of 37.39 ° ensures 8 h of sun collection at the solstices and 7 h at the equinoxes. An optical gain of 7.02 can then be obtained for the direct beam and 4.5 for the diffuse source, with a geometrical gain of 12.20.

4. Conclusions

The concept of geometrical gain often used when defining concentrat- ing systems is relatively meaningless since it is not directly related to the

365

power cast on the cell. The concept of optical gain describes better the merits of a given concentrator. An upper bound exists for the concentrat ion if the source is directionally extended. To achieve this upper bound bifacial cells submerged in a transparent dielectric of very high n must be used and light must be cast isotropically by the concentrator on these cells.

These concepts can be applied to the design of static concentrators for the yearly averaged solar beam or for diffuse radiation. The analysis of two- stage and one-stage two-dimensional concentrators and of a three-dimensional concentrator shows that the three-dimensional concentrator allows for the highest optical gain, attaining the above-mentioned upper bound, but also requires a more expensive concentrator. The lowest optical gain is obtained with the cheaper two-stage two-dimensional concentrator. The information in this paper should help to achieve a cost-optimal design of a static con- centrator for direct and diffuse solar radiation.

Acknowledgment

The author would like to thank Mr. M. G0mez-Agost and Mr. J. Eguren for their helpful comments in the preparation of this paper.

Nomenclature

A c

As Asb Asd Ceff Cg Co

De

Ec Ee Ef g gL(X', y ' ) I lc n

P,q p ' ,q '

Ps

t ts-1 W wc

cell area source region area in the p - q space area of the solar beam source area of the diffuse radiation source optical gain in a concentrator with losses geometrical concentration: ratio of the entry aperture area to the cell area optical concentration: ratio of the power at the cell in a lossless concentrator to the power at the cell in a flat panel domain of points (x,y) at the entry aperture Y~e of the concentrator and existing directions of rays in the x, y, ~b,e space or in the x, y ,p ,q space ~tendue at the cell (eqn. (7)) ~tendue at the entry aperture (eqn. (4)) ~tendue at a flat panel degree of isotropy at the cell (eqn. (9)) local degree of isotropy (eqn. (10)) optical intercept factor (eqn. (8)) domain of directions, which includes all of them index of refraction of the medium surrounding the cell angular coordinates of a ray at the entry aperture angular coordinates of a ray at the cell radiance, i.e. angular density of the energy flux, in the direction normal to the source transmission function inverse transmission function power collected by the concentrator power reaching the cell

366

Wf x,y x ' ,y ' o~ 6m Zc 2~e Zs Cq

¢; ~,0

power reaching a fiat panel position coordinates of a ray at the entry aperture position coordinates of a ray at the cell directional intercept factor maximum sun declination, 23.45 ° cell surface entry aperture surface (usually a plane region) source region in the p-q plane meridian acceptance angle of a three-dimensional concentrator meridian acceptance angle equatorial acceptance angle of a three-dimensional concentrator angles of polar coordinates

References

1 R. Winston, J. Opt. Soc. Am., 60 (1970) 245. 2 W. T. Welford and R. Winston, The Optics of Non-imaging Concentrators, Academic

Press, New York, 1978. 3 A. Luque, A. Cuevas and J. M. Ruiz, Double-sided n÷-p-n ÷ solar cell for bifacial con-

centration, Sol. Cells, 2 (2) (1980) 151. 4 A. Luque, J. Eguren, J. M. Ruiz, A. Cuevas, J. del Alamo, J. M. Gomez, M. Acufia and

G. Sala, New concepts for static concentration of direct and diffuse radiation, Proc. 3rd Eur. Communities Conf. on Photovoltaic Solar Energy, Cannes, October 27- 31, 1980, Reidel, Dordrecht, 1981, p. 396.

5 R. E. Jones, J. Opt. Soc. Am., 67(1977) 1594. 6 E. J. Guay, Sol. Energy, 24 (1980) 265. 7 A. Luque, J. M. Rufz, A. Cuevas, J. Eguren and J. M. Gomez, Proc. International

Solar Energy Society Congr., New Delhi, January 1978, Pergamon, New York, 1978, p. 729.

8 A. Rabl and R. Winston, Appl. Opt., 15 (1976) 2880. 9 R. Winston and H. Hinterberger, Sol. Energy, 1 7 (1975) 255.

10 H. Ploke, Optik, 25 (1) (1967) 31.

Append ix A

Calculation o f the opt ical in tercep t fac tor An ellipse is represented by the parametr ic equa t ions

x = a cos a (A1)

y = b sin a (A2)

The area of an elliptic zone, i.e. of the po r t i on o f an ellipse be tween t wo hor izon ta l lines symmetr ica l with respect to its centre , is given by

Az(a, b ,~o) = ab(2~0 + sin 2a0) (A3)

The angle s0 defines the po in t (Xo, Yo) of intersect ion, in the first quadrant , o f the ellipse and the hor izon ta l lines def ining the zone. I t is calculated as

367

C~o=tan-l(x/a~ (A4) ~y/b] where a and b are the ellipse semi-axes.

For a circumference the direct beam solar source area Ads is then given by

Ads = A(1,1,6m) = 1.549 (A5)

Let us consider the direct beam source intercept factors. For the con- centrator in Fig. 2(a) the intercept factor is given by

n sin Cm I - (A6)

Ads

for Cm < 5m and by

Az(1,sin ~m,al) I = (A7)

Ads

for 5m < era. In this case a l is the angle corresponding to point X l ( x l , y l ) considered as a point of the ellipse MNPQ (see Fig. 1). It is calculated as

a, = sin_l/sin Cm 1 \ sin 8m ]

(AS)

For the concentrator of Fig. 2(b) the angle a2' corresponding to X , ~ x , ,. 2 ( 2 ,Y2 ) as a point of the ellipse M'N'P'Q' is

I n 2 - - I 11/2 Ol 2' = s i n - 1 (A9)

n 2 -- (sin era) 2

while the angle ~2' corresponding to the same point as a point of the circle of radius uni ty is

~2 ' = t a n - l ( b tan ~2' ) (A10)

If Cm < 5m, then the optical intercept factor is

~n sin Cm -- Az(n, sin era,a2') + Az(1,1,/~z') I = (All)

Ads

If Cm is increased the ellipse M'NP'Q intersects the horizontal boundaries of the direct beam solar source, This will happen if 6m < Cm < ~bc where ¢c is the meridian acceptance angle of the ellipse M'NP'Q when it is circumscribed to the solar source. Angle be is

¢c = sin-1 [ (Sin ~m) 2 ] 1 _ {~s ~m)/n}2 ] (A12)

The angle a l ' corresponds to the point Xl ' (x l ' ,Yl ' ) as a point of the ellipse (see Fig. 1). It is calculated by using eqn. (A8). For 6m < ~m < ¢c

368

Az(n, sin ~m,al ' ) -- Az(n, sin era,a2') + A~(1,1,f12') I = (A13)

Ads

Finally, for ¢c < era, I = 1. For the three-dimensional concentrator of Fig. 5(b) the intercept factor

for ~m < ~m is

n sin 2 ~m I - (A14)

Ads

For 5m < Om the angle a l " corresponding to Xl"(Xl",Yl") as a point of circle M"NP"Q is

" = s in - l ( s in 6m / al \ ~ / (AI5)

and

A~(sin era,sin Cm,~l") I = (A16)

Ads Finally, we consider the diffuse radiation source. For the concentrator

in Fig. 2(a) we have

I = sin Cm (A17)

For the concentrator in Fig. 2(b) we have

1 I = -- {~n sin @m --Az(n, sin (~m,{22') 4-Az(1,1,/~2')} (A18)

For the concentrator of Fig. 5(b) we have

I = sin 2 Cm (A19)