Fresnel lens for photovoltaic solar cells

Solar Cells, 3 (1981) 149 - 161

149

F R E S N E L LENS F O R PHOTOVOLTAIC SOLAR CELLS

F. DEMICHELIS and E. MINETTI-MEZZETTI Istituto di Fisica Sperimentale del Politecnico di Torino, Torino (Italy) G. F E R R A R I and M. PALAZZETTI Centro Ricerche Fiat, Torino (Italy) (Received April 3, 1980; accepted May 21, 1980)

Summary An efficient Fresnel lens particularly designed for use with photovoltaic solar cells is described. The lens accomplishes different objectives: the whole energy band corresponding to the spectral response of the cell is focused on the cell; superposition of the spectra from various prisms yields a nearly uniform distribution of energy per unit area and unit time on the focal plane; the fraction of IR in the range where the cell has no response does n o t fall on the cell. The predicted performance of a methacrylate lens, assuming an air mass one spectral density, provides greater reliability.

1. Introduction Fresnel lenses offer remarkable design flexibility for optical concentrators to be used with solar cells. Fresnel lenses giving the highest transmission, the lowest optical dispersion and the smallest f number have been studied previously and have been realized experimentally [1]. The design m e t h o d takes advantage of the discrete prisms which compose the Fresnel lens and the possibility of directing a single facet anywhere on the focal plane. It is then possible n o t only to change image shapes b u t also (and it seems to be important) to compensate for chromatic aberrations and to optimize the lens performance with respect to the cell spectral response. In this paper we report the design of a concave grooves-down spherical lens for circular photovoltaic solar cells. The concave shape adds high mechanical stability and strength to the lens and its concentration increases [2]. The grooves-down feature reduces accumulation of dust and dirt on the side of the lens exposed to the sun. Taking into account the spread due to chromatic aberration and the cell spectral response (only a portion of the solar spectrum can be used by solar cells) the design is accomplished by adjusting the Fresnel prisms as follows. 0379-6787/81/0000-0000/$02.50

© Elsevier Sequoia/Printed in The Netherlands

150 (i) The whole wavelength band of response of the cell is totally utilized. (ii) IR below the band gap is not allowed to fall on the cell, thereby reducing the heating of the cell and hence maintaining a high cell conversion efficiency. (iii) Since the maximum spectral efficiency of a cell corresponds to a given peak wavelength kp, the lens is designed for that wavelength. The parameters in the calculations must be fixed according to the required concentration, allowable tracking error, type of cell etc. We applied our calculations to cells with spatial uniformity of the spectral response. The method can be extended to cells whose response is n o t uniform, e.g. the cells used by Fabre and Mautref [3], by imposing suitable conditions. The proposed lens which is specifically designed for use with photovoltaic cells is able to provide a particularly good performance. By superimposing the spreads due to the chromatic dispersion of individual prisms an almost uniform energy density distribution may be achieved on the focal plane. Furthermore, since the lens can be considered to be an inexpensive optical concentrator, it is possible to reduce the costs of concentrator-solar cell systems.

2. Design calculations Our design calculation is as follows. We compute the inclination angle e of the first prism at the edge of the lens by imposing the requirements that IR rays of wavelength 1.0 vm fall on the edge of the solar cell (for the IR rays we assume a refractive index n = n(k)~ = 1 vm corresponding to k = 1 vm) and that UV rays of wavelength 0.4 pm fall on the cell at Xuv so that 0 < Xuv ~< x m

(1)

(for the UV rays we assume a refractive index n = n ( k ) k = o . 4 p m ) . Proceeding from the edge to the centre of the lens, we give the same inclination angle e to the contiguous subsequent prisms as long as condition (1) holds true. When condition (1) is no longer satisfied, a new value of the inclination angle e is determined by imposing the above requirements and the procedure is repeated for various prisms until the centre of the lens is reached. Let us consider the first spherical refracting surface (Fig. 1). Because of spherical symmetry it is sufficient to take into consideration a plane through the optical axis. The refractive index of air is unity whereas the refractive index of the lens is n. The equation of the refracting surface is x 2 +z 2 --R 2 = 0

(2)

where R is the radius of curvature. A monochromatic ray p given by

p =p,,i + p z k

p = (Px 2 + p z 2 ) 1/2

(3)

151

Z V

it

I

iI

I

II il II II

/,

/

/

/

/

, /

×

×T

Fig. 1. Cross section of the Fresnel lens and a ray path through the lens.

incident at P(xp, Zp ----(R 2 - - X p 2 ) 1/2) on the surface gives rise to an emergent ray r given by

r = rxi + rzk - I l-4zP(xppz-zPpx)+2xP~li+ ~2 np 1 }4xp(~p= + ~¢np

-z~x)_+ 2 ~ f k

(4)

where the normal N to the surface is given by N = 2Xpi + 2zpk

N 2 - 4R 2

(5)

and

+=21R2

1

n2p2 (Xppz --zppx) 2

1/2

}

(6)

The emergent ray is then given by Z--Zp

rz

= --(X--Xp)

rx

(7)

152

N e x t we consider r e f r a c t i o n at the s e c o n d surface. If s is the thickness of the lens, a second spherical surface should have the e q u a t i o n x 2 +z 2-(R-s)

2 = 0

(8)

Let us consider a p o i n t M at this s e c o n d spherical surface with XM = Xp. T h e c o o r d i n a t e s o f the p o i n t M are t h e n x M - Xp

zM

= ((R - - s ) 2 - - x M 2 } 1/2

(9)

L e t M be a p o i n t on a prism whose inclination with respect to the z axis is e. The cross section of the plane surface o f the prism t h r o u g h the x z plane is given by the straight line t h r o u g h the p o i n t M with the e q u a t i o n Z--ZM - - ( t a n e ) ( X - - X M ) = 0

(10)

T h e ray e m e r g e n t f r o m the first spherical surface is n o w the ray incident at p o i n t Q o f the prism. By substituting eqn. (7) into eqn. (10) and taking into a c c o u n t t h a t XM -- Xp we have the following c o o r d i n a t e s for Q: Z M --Zp X Q : X M d- r x

rz - - rx tan e

(11)

Z M - - Zp ZQ = Zp +

rz

rz - - rx tan e

A t Q the ray suffers a s e c o n d r e f r a c t i o n and b e c o m e s the ray Q~2. Taking into a c c o u n t t h a t the o r d e r o f the refractive indices is n o w reversed because the e m e r g e n t ray r* goes f r o m the lens into the air we o b t a i n r* = r x * i + r~*k

-

1 N 2 (n(r~ + r z tan e ) - - (tan e ) ~ } i + 1 + N----~ (n(r~ + rz tan e)tan e + ~ } k

(12)

where N = - - ( t a n e)i + k

N 2 = 1 + tan 2 e

(13)

and = ((1 - - n 2 r x a) - - 2 n 2 r ~ r z tan e + (1 --n2r~ 2) tan 2 e} 1/~

(14)

When e is k n o w n , eqn. (12) gives the d i r e c t i o n of the ray e m e r g e n t f r o m the lens. T h e e q u a t i o n o f the e m e r g e n t ray is t h e n Z--ZQ----

rz * . (X--XQ)

r~

(15)

Setting z = z a , the c o o r d i n a t e s o f the p o i n t T on the solar cell are given by

153 r~ XT = XQ + - - ~ ( Z n - - Z Q )

(16)

rz

Z T = Z~

Moreover, f r o m eqn. (12) e can be d e t e r m i n e d . L e t us i m p o s e the c o n d i t i o n s t h a t IR rays (k = 1.0 p m ) emerging f r o m p o i n t Q reach the p o i n t ~2 on the cell. T h e n , f o r z = z n a n d x = x n , eqn. (15) becomes * rz

zn --ZQ = ~

rx

( x n --XQ)

(17)

Setting z n - - Z q = a a n d x n - - X Q = b, eqn. (17) r e d u c e s to (18)

arx* = brz*

S u b s t i t u t i n g r~* and rz* f r o m eqn. (12) in this e q u a t i o n , squaring and using eqn. (14) f o r 4 , we o b t a i n {nUb2rz 2 - - a 2 ( 1 --n2rz2)} tan 4 e + ( 2 n 2 b 2 r x r z - - 2 a b

+ 2 n 2 a 2 r x r z ) tan 3 e +

+ {n2bUr~ 2 + n2a2rz 2 - - b2(1 - - n2rz 2) -- a2(1 -- n2rx2)} tan 2 e + + ( 2 n 2 a 2 r x r z + 2 n 2 b 2 r ~ r z - - 2 a b ) t a n e + {n2a2r~ 2 - - b2(1 -- n2rx2)} = 0

(19) Setting n 2 r z 2 ( a 2 + b 2) - - a 2 = A o 2 { n 2 r x r z ( a 2 + b 2) - - a b } = A 1

(n 2

A2

-- 1)(a 2 + b 2) =

(20)

2 { n 2 r ~ r z ( a 2 + b 2) - - a b } = A 3 - A 1 n 2 r x 2 ( a 2 + b 2) -- b 2 = A4

eqn. (19) b e c o m e s Ao tan 4 e + A1 tan 3 e + A2 tan 2 e + A1 tan e + A4 = 0

(21)

F u r t h e r m o r e , if we set A1 C1

-

A2 C2

Ao

-

C3

Ao

=

C1

C4 -

A4

Ao

(22)

we m a y finally write eqn. (21) in t h e f o r m tan 4 e + C 1 tan a e + C2 tan 2 e + C 1 tan e + C 4 = 0

(23)

E q u a t i o n (23) gives the r e q u i r e d angle e. N o w we d e t e r m i n e the e n d p o i n t S o f a prism which is the start o f the n e x t prism. L e t us assume t h a t l (the w i d t h o f the prism) is the c h o r d subt e n d e d to the arc MS. T h e p o i n t S can be o b t a i n e d f r o m the e q u a t i o n s

154

Xs 2 + Zs 2 = (R -- s) 2

(XM --Xs) 2 + (ZM --Zs) 2 = 12

(24)

Solving for Xs and Zs we obtain {2(R - - s ) 2 - - 12}XM + Z M l { 4 ( R - - s ) 2 - X S =

2(R

--

12}1/2

s) 2

(2(R -- s) 2 -- 12} _ 2XMXs ZS =

(25)

2ZM

In order to c o m p u t e the losses at the non-active face of the prism, it is convenient to introduce the ratio Pl

-

XH 2 --

XS 2

XM2 _

Xs 2

(26)

which gives the fraction of incident light lost for total reflection at the HS face of the prism. For the poi nt H we have x 2 + z 2 = (R - - s ) 2

Z - - (Z M + (tan e)(x s - x M )

(27) rz ) = -- (x --XN)

rx

Thus we obtain the expression XH

=

X s ( G / r x ) -- (ZM + (tan e)(x s --XM)} 1 + r,2/rx 2

+

((1 + rz2/r~2)(R - - s ) 2 -- [XN(G/rx) -- (ZM + (tan e ) ( X s - XM)}] )1/2 1 + rz2/rx 2

(28)

The radiation incident on the system will not, in general, be in the meridian plane and therefore off-meridian incidence must be considered in the calculation of the acceptance angle. A knowledge of the acceptance angle is indeed necessary to determine the performance of the tracking system. The calculation in two dimensions to fix the inclination of the prisms does n o t solve the problem of determining the acceptance angle. We now e x t e n d our c o m p u t a t i o n s to three dimensions. The first refracting surface is x 2 +y2 +z 2 _R 2 = 0

(29)

We take as reference the incident ray at the vertex V(0, 0, R) of the lens (Fig. 2); the direction of this ray is given by P = pxi + pzk

This ray represented by the straight line x

z--R -

Px

(30)

Pz

155 \

\

\

it

\ \ ^ \

// //

\\

i//

\

I \

/

y/7

/

I

I \ I \// /\ / \ // \ / ~\

//

--x

\

I

\1 .~_~_

~-

T

Fig. 2. View of the lens showing the behaviour of off-meridian ray refraction at the lens and intersection with the focal plane.

falls o n the cell placed in the focal plane z = z a at T0{XT0, z~2) where Px

XT0---- - -

(31)

{Z~--R)

Pz Let us n o w consider a ray parallel to the direction p o f the reference ray and incident at PK. We assume that the PK p o i n t s lie o n concentric circles o f radius p and satisfy the c o n d i t i o n s x p K = p cos 6 ypK = p Zp K =

sin 6

(32)

{R 2 _ _ p 2 ) I / 2

where 6 = K ~ / 8 with K = 0, 1, 2, . . . , ray has the e q u a t i o n X--Xp K _ y--ypK_ rx

ry

Z--ZPK

15 and p2 = XPK2 + ypK2. The refracted (33)

rz

Let us consider refraction at the s e c o n d surface o f the lens which consists o f the prism distribution calculated above. For the three-dimensional case eqn. ( 1 0 ) has the form

156 (Z + X M

tan

e --ZM) 2 --

(tan 2 e)(x 2 + y2) = 0

(34)

The ray e m e r g e n t f r o m the first refracting surface falls on the prism at Q. Combining eqns. (33) and (34) we obtain the c o o r d i n a t e s o f the p o i n t Q:

--t3 + (~2 _ ~7)1/2 ZQ =

rx XQ = XPK

(35)

(XQ - - Z p K )

+--

rz ry

YQ = YPK + ~ (XQ - - ZpK) where Ot = (rx 2 + ry 2) t a n 2 e - - r z

2

= (rx(XPKrZ --ZPKrx ) + ry(YPKrZ --ZPKry )} tan 2 e --rz2(XM tan e --ZM) 7 = ((XPKrz - - Z P K r x ) 2 + (YPKr~ --ZPKr~)2} tan 2 e --(XM tan e --ZM) 2 rz 2

(36) Finally for the ray emerging f r o m the lens we o b t a i n 1 N e 2 [ n ( N e 2 r x + 2xQ(tan 2 e ) ( N e ' r ) } - - 2Xq(tan 2 e ) ¢ ]i +

r* +

1

2 [n(N~2ry + 2YQ( tan2 e ) ( N e ' r ) } - - 2YQ( tan2 e)~ ]] +

Ne +

1

2 [ n ( N e 2rz -- 2ZQ(Ne'r)} + 2Zq~ ] k

(37)

Ne

where N~ = - - 2 ( t a n 2 ¢)xQi -- 2(tan 2 e)yQ] + 2(ZQ + Ne2 = 4((XQ2

+

yO2) tan 4 e

+

XM

tan e - - Z M ) k

(ZQ - - Z M + XM tan e) 2 }

(38)

and N e . r = 2{(ZQ - - z M + x M tan e)r z -- (XQrx + yQry) tan 2 e}

(39)

¢ = ((1 -- n2)Ne 2 + n 2 ( N e ' r ) } 1/2

(40)

T h e ray emerging f r o m the lens is r e p r e s e n t e d b y the e q u a t i o n X --XQ

y --yQ

Z --ZQ

(41) rx *

ry *

rz*

T h e c o o r d i n a t e s o f the p o i n t T(XT, YT, Z T )

on

the cell are given by

157

rx X T=xQ

(Za --ZQ)

+ - rz *

ry

,

YT = YQ + - -

(Za --ZQ)

(42)

rz *

Z T ---- Z$2

3. N u m e r i c a l s o l u t i o n s We a p p l i e d o u r c a l c u l a t i o n s to a Fresnel lens o f focal length f = 60 c m , o f a p e r t u r e h = 50 c m , o f radii R = 50 c m a n d R - - s = 49 c m a n d o f p r i s m w i d t h 1 = 1 c m which is to be u s e d in a c o n c e n t r a t o r - s o l a r cell s y s t e m ; t h e lens c o n v e r g e s t h e r a y s o n t o a circular silicon cell 4.4 c m in d i a m e t e r . This c o r r e s p o n d s to a g e o m e t r i c a l c o n c e n t r a t i o n r a t i o (the r a t i o o f t h e lens area to the cell area) o f 130. T h e a d v a n t a g e o f designing a lens a c c o r d i n g to the c o n d i t i o n s i m p o s e d in S e c t i o n 1 can be p r o v e d b y c o n s i d e r i n g t h e c h r o m a t i c a b e r r a t i o n . I f we a s s u m e t h a t t h e r e f r a c t i v e i n d e x n ( o f m e t h a c r y l a t e ) is given b y n = 1.47901 +

0 . 0 0 3 80 0 . 0 0 0 18 ~2 + ha

(43)

we o b t a i n a value f o r the l o n g i t u d i n a l c h r o m a t i c a b e r r a t i o n o f fi R - - f v v = 3.7 c m ( w h e r e f m a n d f u v c o r r e s p o n d t o t h e w a v e l e n g t h s 1 p m a n d 0.4 u m r e s p e c t i v e l y ) w h e n the focal length f = 50 c m ; f o r transverse c h r o m a t i c a b e r r a t i o n aT given b y fm - fuv aT ~

2fuv

(44)

we o b t a i n a value o f aT = 1.5 c m . Since t h e Fresnel lenses to be used as solar c o n c e n t r a t o r s m u s t have wide a p e r t u r e s , a T is r a t h e r high c o m p a r e d with t h e size o f the cells. I f we c o n s i d e r t h e c o n t r i b u t i o n t o t h e transverse c h r o m a t i c a b e r r a t i o n o f the I R n a r r o w b a n d 0.7 - 1 p m we o b t a i n a value o f a b o u t 0.5 c m which is a l m o s t negligible with r e s p e c t to the size o f the cell. Since, d u r i n g focusing, such a n a r r o w b a n d o f I R ( c o r r e s p o n d i n g t o t h e m a x i m u m spectral r e s p o n s e o f t h e cell) can d i s a p p e a r o u t o f t h e edge o f t h e cell, a p o r t i o n o f t h e i n c i d e n t r a d i a t i o n can be lost. Figure 3 s h o w s t h e c a l c u l a t e d p r i s m d i s t r i b u t i o n . O n the cell we have a s u p e r p o s i t i o n o f t h e s p e c t r a f r o m t h e various p r i s m s w h o s e w a v e l e n g t h s are i n c l u d e d in t h e spectral r e s p o n s e b a n d o f the cell. M o r e o v e r , t h e I R r a y s a b o v e 1 p m w h i c h c o m e f r o m t h e marginal p r i s m s o f each set do n o t fall on t h e cell. F r o m o u r c o m p u t a t i o n s it a p p e a r s t h a t 25% o f these I R r a y s are s e n t o u t o f t h e cell. Figure 4 s h o w s t h e d i s t r i b u t i o n o f t h e e n e r g y d e n s i t y E ( e n e r g y p e r u n i t t i m e a n d p e r u n i t area) at t h e cell w h e n it is p l a c e d in t h e focal plane. This d i s t r i b u t i o n is o b t a i n e d b y a d d i n g t h e e n e r g y densities sent t o t h e cell f r o m

158

,7 ,~

~:

;iv

:~l¸¸

I

i

:

F ]

,

/

::j/]

!,i.:/

×

i Fig. 3. Paths of IR rays (k = 1 pm) (right-hand side) and UV rays (k = 0.4 pm) (left-hand side).

all the prisms of the various sets, assuming an air mass one spectral density. We can see that away from the edge the energy density is nearly uniform except for slight ripples near the centre. We can also observe that at the edge of the cell (x > 22 mm) the energy density distribution associated with IR rays (), > 1 pro) can be sent, as required, out of the cell. Regarding the lens transmittivity we can assume that our lens has the usual losses encountered in Fresnel lenses. These losses consisting of reflection from the spherical surface, reflection from the prism surface and absorption are of the order of 10%. However, in the designed lens, losses due to the

159 E (Watt/m 2 ) xlO4 20 15 10 5

&

lb

oE

1~

2~o 2~ ~ x l E

-3

x

Fig. 4. Energy density distribution on the cell.

non-active prism faces must be taken into account. By using eqn. (26) we calculated Pl as a function of the position of the prisms for the distribution shown in Fig. 3. The results are shown in Fig. 5. We can see that, as expected, the losses are greater at the edge of the lens than near the centre. In fact the contribution of the marginal prisms is predominant because the area of the marginal rings is larger. From our computations we obtain, to a good approximation, a value of the order of 6% for these losses. A reduction in the losses is possible if the prism face LS (see Fig. 1) is inclined so as to be parallel to the direction of the ray PQ [2]. If the technical realization of this inclination does n o t present excessive difficulties, the losses can be considerably reduced. Finally, using the relations (42) we c o m p u t e d the acceptance angle 4. Since we have a high concentration, a small acceptance angle is expected. Figure 6 shows the results of the calculations for four values of the accep-

.15

/ /

.10-

J .05"

J J I

0

.05

.10

.1

.20

.25

X(r ) Fig. 5. Losses due to the non-active prism face for the different prisms.

160

Y

~

=1:) c

Fig. 6. Image o f the solar disc o b t a i n e d for a c c e p t a n c e angles ~ equal t o 0 ° , 3 ° , 7 ° and 15 ° .

tance angle. The image of the solar disc in the focal plane was obtained by considering 16 parallel rays falling on the ring consisting of the first marginal prisms. We can see that for ~ = 3 ° the image is almost out of the cell already. This imposes rather stringent angular tracking requirements. We consider that the results that we obtained are a good test of our computations. 4. Conclusions From the results of our computations we can deduce that the Fresnel lens that we designed is able to provide a good performance. This lens has several attractive features which make it particularly suitable for use with solar cells. The prism distribution is such that (1) the wavelength band corresponding to the spectral response of the cell is completely utilized, (2) about 25% of the IR above 1 ~m is sent out of the cell, thus reducing the heating, and (3) the distribution of the energy per unit area on the cell is fairly uniform. Thus we have improved the efficiency of the optics of a concentratorcell system in the sense that the improvements obtainable with this lens do not require an increase in system cost. Nomenclature aT f fir fuv h l n Pl R s e

transverse c h r o m a t i c a b e r r a t i o n focal length focal length for IR radiation f o c a l l e n g t h for U V radiation a p e r t u r e o f the lens w i d t h o f the prism refractive i n d e x f r a c t i o n o f i n c i d e n t light lost for total r e f l e c t i o n radius o f curvature o f the first refracting surface t h i c k n e s s o f t h e lens inclination angle o f the prism with r e s p e c t to the optical axis wavelength a c c e p t a n c e angle

161

References 1 0 . E. Miller, J. H. McLeand and W. R. Sherwood, J. Opt. Soc. Am., 41 (1951) 11. M. Collares-Pereira, A. Rabl and R. Winston, Appl. Opt., 6 (1977) 2677. E. M. Kritchman, A. A. Friesem and G. Yekutieli, Solar En., 22 (1979) 119. M. Collares-Pereira, Solar En., 23 (1979) 409. 2 J. Bellugue, Acta Electron., 20 (2) (1977) 153. 3 E. Fabre and M. Mautref, Acta Electron., 18 (4) (1975) 332.