Analysis of Fundamental Light Receiving Characteristics of Spherical Solar Cells

HIROKI ISHIKAWA, TAKAYUKI FURUNO, SATORU KAMBE, and HARUO NAITOHGifu University, Japan

SUMMARY

Spherical solar cells are claimed to have some advan-tageous characteristics superior to those of ordinary planarsolar cells. The most significant one is that the sphericalsolar cells have no directivity to light. This paper examinesthe characteristics based on geometrical considerations.The authors prove that a single spherical cell has no direc-tivity as a whole. In practical use, many cells are used in anarray configuration, where an individual cell receives theshadows cast by other cells around it. The adjacent shad-ows, so named in this paper, cause directivity. Their effectsare evaluated geometrically and the theoretical considera-tions are verified through experiments. © 2008 Wiley Peri-odicals, Inc. Electron Comm Jpn, 91(6): 34–46, 2008;Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecj.10119

Key words: spherical silicon micro solar cell; ad-jacent shadow.

1. Introduction

Given global warming and other environmental prob-lems, and the depletion of fossil fuels, the development ofenergy sources to replace fossil fuels is important. As such,expectations for wind power and solar light as renewableenergy are high.

The majority of solar cells in use at present are flat.These cells will be referred to as planar solar cells. A planarsolar cell is manufactured from silicon crystals with a highpurity, and as a result, the process of removing impuritiesis complex and costly. The same holds for the processes ofmaking them thin and flat, and of making them into mod-ules. Because of these factors, manufacturing costs arehigh. This is one major reason impeding the spread of solarcells.

In addition to planar solar cells, there are also spheri-cal solar cells [1–3]. These cells will be referred to asspherical silicon micro solar cells. A spherical silicon micro

solar cell is purified to a high degree by removing impuritiesby repeatedly melting and fusing silicon with inexpensivelow-purity metals. In recent years, methods to make spheresby taking advantage of the surface tension in a zero-gravityenvironment and at the same time increasing purity havebeen proposed, and spherical silicon micro solar cells cannow be produced inexpensively.

A spherical silicon micro solar cell receives light onits entire surface as a single unit (referred to as a unit cell).As a result, it has no directivity with respect to light [1, 2].Consequently, light from all directions, including reflectedlight and scattered light, and not just direct light, can beused. No quantitative verification of this nondirectivity hasbeen performed. Multiple spherical silicon micro solar cellsare connected in parallel and used in practice [4–6]. Thissetup is referred to as a module. There have been fewanalyses of the characteristics of such modules.

In this paper the authors analyze the above effects andcharacteristics through a geometrical discussion of thelight-receiving characteristics of a spherical silicon microsolar cell. First, they clarify the characteristics of the incom-ing light on a unit cell. Next, they discuss modules. Forconvenience, they compare this discussion with the planarsolar cells currently in use and analyze the strengths andweaknesses of each. They confirm the validity of theseissues through experiments. The conditions shared in theanalyses above include the following assumptions.

• Incident light consists only of parallel lines oflight; scattered light is not considered.

• The reflected light on the surface of a solar cell isnot considered.

2. Effective Area for a Planar Solar Cell and a UnitCell

2.1 Effective area of a planar solar cell

In order to compare the light-receiving charac-teristics for both a unit cell and a planar solar cell, theauthors analyzed the effective area as defined below.

© 2008 Wiley Periodicals, Inc.

Electronics and Communications in Japan, Vol. 91, No. 6, 2008Translated from Denki Gakkai Ronbunshi, Vol. 127-D, No. 4, April 2007, pp. 441–450

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Figure 1 shows an overview of the incident light ona planar solar cell. The angle of incidence θ is the angle ofthe incident light described from the cell surface, as shownin the figure. The intensity of the incident light per unit areais λ, and the surface area of the incident surface is S. If theamount of light reaching the surface is Λ, then

Equation (1) shows that the surface area is equivalent towhen the light of intensity λ is incident perpendicularly tothe planar solar cell of S sin θ. This equivalent area isdefined as the effective area for the area receiving light. Fora planar cell, this is written as SPEFF :

Equations (1) and (2) show that SPEFF is dependent on theangle of incidence, that is, it has directivity with respect tothe angle of incidence.

2.2 Effective area of a unit cell

Figure 2 shows a schematic [1] of the basic structureof a unit cell. A part of the light-receiving surface is theelectrode, but in this paper the authors assume that the entiresphere is a light-receiving surface and do not take theelectrode into consideration. Because a lead wire runs tothe electrode, its shadow may strike the unit cell. The leadwire is extremely thin compared to the radius of the cell,and so its shadow is ignored.

Figure 3 shows the incidence of light on a unit cell.For the light L1 incident at the vertex of the sphere, the light

is incident perpendicularly to the surface (referred to as thecontact surface) in contact with the sphere at the point ofillumination (the vertex). The light L2 incident at pointsother than the vertex does not strike the contact surfaceperpendicularly. If a unit cell is considered to have a verysmall effective area, then anywhere but the vertex is thesame as when light is incident on a planar solar cell at adiagonal, and so has directivity with respect to the incidentlight. Given this, the effective area of the entire unit cell canbe calculated.

Taking into consideration the parallel lines of lightthat illuminate a hemisphere of the unit cell at all times, theauthors first calculate the effective area of the region shapedlike a ball and shown in Fig. 3. The effective area of theentire unit cell is then found by integrating this over theupper half of the entire sphere. Consider the infinitesimalarea dS in the region in the circle, as shown in the figure.The angle formed by the light and the contact surface is theangle of incidence θ.

Here, R is the radius of the unit cell. The infinitesimaleffective area dSEFF is given by

If φ in Eq. (4) is integrated from 0 to 2π and θ is integratedfrom 0 to π/2, then the effective area SUEFF of the unit cellcan be found:

Based on Eq. (5), SUEFF is πR2, in other words thearea of the great circle of the sphere. The fact that θ is notincluded in Eq. (5) proves that there is no directivity to theincident light for the unit cell. Given the analysis above andbased on the perspective of the effective area, the unit cellcan be considered as equivalent to a planar solar cell of anarea equal to the great circle when light is incident perpen-dicularly at all times.

Fig. 1. Effective area of planar solar battery.

Fig. 2. Cross-section view of spherical solar cell.

Fig. 3. Incidence of light to a unit cell and infinitesimalarea of the cell.

(1)

(2)

(3)

(4)

(5)

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2.3 Comparison of the effective area of aplanar cell and a unit cell

Let us consider a rectangular planar solar cell tangen-tial to the circle with radius R shown in Fig. 4. Based on Eq.(2), its effective area SPEFF can be written as

Based on Eqs. (5) and (6), if the angle of incidence atwhich SUEFF and SPEFF are equal is θEQ1, then the followingrelationship holds:

Based on this result, when θ is small, the morning andevening are better for the unit cell for solar power genera-tion, and when θ is large, in other words during midday, theplanar solar cell is better.

In the unit cell, the light incident on the region shownby the diagonal lines in Fig. 4 does not contribute as muchto power generation as in the planar solar cell. When thereflected light considered in the next section is not used, theunit cell is at a disadvantage for that amount of light.

2.4 Effect of reflection on the unit cell

In order to make use of the effect of light passingthrough the region shown by the diagonal lines in Fig. 4, areflective plate may be considered when putting the spheri-cal silicon micro solar cells into practical use. In this sectionthe authors evaluate the effect of reflected light using areflective plate. In order to compare the unit cell with aplanar cell, the reflective plate is a rectangle with one sidemeasuring 2R and inscribed in the unit cell. The coefficientof reflectivity for the reflective surface is assumed to be 1.

As can be seen in Fig. 5(a), the light is reflected bythe reflective plate and strikes the lower surface of the unitcell.

The light should be taken as directly striking themirror image of the sphere marking the boundary of thereflective surface, as shown in Fig. 5(b), when calculating

the effective area due to the reflective surface. The actualbody of the unit cell is SP1, and the mirror image is SP2.The light illuminating SP1 is blocked by SP1, and creates ashadow on SP2. In this paper this shadow is called anadjacent shadow. Let us consider the circle (small circle) atthe cutoff sliced for both spheres using the flat surface PΨincluding any incident light for the parallel lines of light fora given incident angle θ. When the small circle SC1 for SP1

is larger than the small circle SC2 for SP2, the small circle2 is completely in the adjacent shadow of the small circle1, and no light hits it. When SC2 is larger, the portionblocked by SC1 is the adjacent shadow, as can be seen inFig. 5(c).

Let the radii of SC1 and SC2 be r1 and r2. Thediscussion above can be summarized by dividing the light-receiving region for the reflected light for SP2 is divided, asseen in Fig. 6(a). For SP2:

• Region 1: r2 < r1. Entirely in the adjacent shadow,and no light reaches it.

(6)

Fig. 4. Circumscriptive square of unit cell.

Fig. 5. Overview of and treatment for reflection.(7)

Fig. 6. Effective area of reflected light for a unit cell.

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• Region 2: 0 < r1 ≤ r2. Parts other than the adjacentshadow are struck by light.

• Region 3: r1 = 0. No adjacent shadow is created,and light strikes everywhere directly.

• Regions 4, 5, 6: Light does not illuminate SP2.

Given the above, regions 1, 4, 5, and 6 are not relevant,and regions 2 and 3 should be considered.

A way to find the effective area is given below. Aswas pointed out in Section 2.2, the illumination by light onthe great circle GC2 for SP2 and the great circle GC1 for SP1

orthogonal to the effective area should be used to calculatethe effective area resulting from parallel lines of light.

If these two circles are seen from the direction of theincident light in Fig. 6(a), they would appear to overlap, asshown in Fig. 6(b). The overlapping region (shown by agrayed out area in the figure) is the region that light doesnot reach SP2 due to the adjacent shadow of SP1. As for thearea marked with vertical lines, because there is no reflec-tive plate in region 4, light does not reach SP2 directly.

For a given incident angle θ, the effective area SP2 isequal to the gray region and the area with vertical linessubtracted from the area of GC2. As can be seen in Fig. 6(c),the gray area should be double the area S1arc for the arc(diagonal) in which the triangular area O1AB is subtractedfrom the fan-shaped area O1AB given its symmetry. S1arc

can be found using

The area S2arc for the arc for the area with vertical lines canbe calculated using this idea, specifically

However, α = arc cos(sin θ – cos θ).Based on Eqs. (8) and (9), the effective area SREFF

obtained due to reflection can be represented as

Figure 7 shows the relationship between SREFF and θ.R is set to 0.925 mm, used in the experiments in the latersections of this paper. Based on Fig. 7, the following is clear.

1 θ = 0°: light does not reach the reflective plate ∴SREFF = 0

2 0° < θ < 90°: • Reflected light reaches SP1, and an adjacent

shadow occurs on the mirror image.

• When the angle of incidence is small, the adjacentshadow is also small. As the angle of incidenceslowly rises, the adjacent shadow also becomeslarger.

• The effective area increases for an angle of inci-dence from 0° to 55.2°, and then decreases.

3 θ = 90°: The entire mirror image is in the adjacentshadow ∴ SREFF = 0

Figure 8 shows a comparison of the effective areaSPEFF for a planar solar cell and the effective area SAEFF (=SUEFF + SREFF) for a unit cell combining direct light andreflected light. The angle of incidence θEQ2 at which thesetwo areas are equal is

In other words, if reflection is taken into consideration,when θ < 75.3°, the unit cell is more advantageous than theplanar solar cell.

(8)

(9)

Fig. 7. Effective area of reflected light alone for a unitcell.

(11)

Fig. 8. Effective area of whole irradiation light for aunit cell.

(10)

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3. Analysis of the Spherical Silicon Micro Solar CellModule

Figure 9 shows an external view of the module. Thisis the module also used in the experiments in the nextsection. Light is assumed to come from the left and right inthe figure. Because no adjacent shadow occurs between thecells above or below, only cells arranged horizontally areconsidered for the module. One cell among them (called thesphere of interest) is focused on, and the shadow of the othercells (adjacent shadow) is evaluated. The amount of lightreceived on the module overall can be found by multiplyingthe number of cells in a module by the amount of lightreceived on the sphere of interest.

3.1 Evaluation of direct light

If several unit cells are arranged adjacent to eachother, an adjacent shadow due to adjacent spheres withrespect to the direct light appears on the sphere of interest.This may be considered similar to the case of the real bodyand mirror image of the cell described in Section 2.4. Thedifference between this and Section 2.4 is that cases inwhich the distance (d) between cells is greater than zeroshould also be considered, and that the adjacent shadow fortwo or more adjacent spheres must be taken into considera-tion.

First, the authors analyzed the case of one adjacentsphere. Let us consider a case in which the angle of inci-dence is slowly decreased from 90°. An adjacent shadowcannot be formed on the sphere of interest SP1 up to theangle of incidence θTL parallel to the line segment in contactwith both SP1 and SP2, as shown in Fig. 10. In other words,the effective area SEFF for θTL ≤ θ ≤ 90° is constant for thearea of the great circle for SP1. θTL can be found as

An adjacent shadow can be formed on SP1 for θ < θTL. Theeffective area SEFF can be calculated using the same think-ing found in Section 2.4:

However,

β = arccos

sin θsin θTL

The space normalized by dividing d by the diameter2R of a cell is defined as D, and is called the normal space.Figure 11 shows the effective area SEFF for incident lightfor SP1 when performing the calculation using Eq. (13) forwhen θ is varied from 0° to 90°, and D from 0.0 to 2.0 inincrements of 0.5. As the angle of incidence θ increases, theadjacent shadow becomes smaller, and as a result, SEFF

becomes larger. When θ reaches θTL, the adjacent shadowis not generated, and SEFF is constant at the area of the greatcircle for the sphere. The larger the cell space d becomes,the small θTL is, and the effective area becomes larger.

Next, the authors analyze whether or not the adjacentshadow for more than two spheres must be taken intoconsideration.

Let us consider a sphere of interest SP1 and twoadjacent spheres, SP2 and SP3, as shown in Fig. 12(a). Whenthe three great circles GC1, GC2, and GC3 perpendicular tothe direction of the incident light are viewed, they appearas shown in Fig. 12(b). As the figure makes clear, theoverlapping part (part with vertical lines) for GC3 and GC1

is enclosed within the overlapping part (gray part) for GC2

and GC1. In other words, the light blocked by SP3 is blocked(12)

Fig. 9. Tested spherical cell module.

Fig. 10. Evaluation of direct light with an adjacent cell.

Fig. 11. Effective area of direct light for cell module.

(13)

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by SP2. As a result, SP3 is not of interest for calculating theeffective area. The same holds for when there are more thanthree adjacent spheres: only one of the adjacent spheresassociated with the direct light needs to be taken intoconsideration.

3.2 Evaluation of reflected light

As was the case in Section 2.4, the effect of reflectedlight should be thought of as direct light on the mirror imageof the sphere forming a boundary with the reflective sur-face. The difference here is that for a module, as shown inFig. 13, the shadow of the other real spheres SP21 to SPn1

on the mirror image sphere SP12 for the sphere of interestSP11 must be taken into consideration.

3.2.1 Method to calculate the effective area

The effective area for SP12 is found using the re-flected light with the cell space d taken into considerationfor n (n ≥ 2) unit cells. As was the case in Section 2.4, thegreat circle of spheres perpendicular to the incident light isused. For the sake of comparison with planar solar cells, thedimensions of the reflective plate represent a rectangle thathas one side at 2n(R + d/2) = 2nx and the other side at 2(R+ d/2) = 2x, corresponding to the n cells included in d. Theangle of incidence θ is defined by dividing it intoθTL1, θCC2, . . . , θCC(m−1), θCCm, . . . , θCCn, θTL2, θCCR, asshown in Fig. 13. θTL1, θCCm, θTL2, and θCCR can be repre-sented as

Based on the angles defined above, the effective area for thesphere of interest SP12 with respect to the real sphere SP11

by dividing them into the six regions is explained in theprocedure below.

(1) Region 1: 90° ≥ θ ≥ θTL1

An adjacent shadow is formed on SP12 only for thesphere of interest SP11. The same holds for the unit cell inSection 2.4. In Section 2.4, the amount of decrease [verticallines in Fig. 6(b)] for the effective area in SP12 due to theabsence of a reflective plate does not have to be taken intoconsideration. Based on Eq. (10), the equation below with-out this portion subtracted represents the effective area S12

for 1:

(2) Region 2: θTL1 > θ ≥ θCC2

An adjacent shadow is formed on SP12 by SP11 andSP21. If the great circles for SP11, SP12, and SP21 which areorthogonal to the incident light are designated GC11, GC12,and GC21, then based on the changes to the angle of inci-dence θ for the light, these three great circles overlap, thatis, the formation of adjacent shadows changes. To evaluatethis, GC11, GC12, and GC21 are observed using the lines ofsight parallel from the light source to the incident light. Atthis point, if the great circles with good circumstances forthe regions are fixed in place, another two circles appear tomove along with changes to θ. The effective area is calcu-lated based on the way the three circles overlap. The aboveidea also holds for 3 through 5.

If CG11 is fixed in place and observed, then GC12 andGC21 appear to move toward the bottom and the top as theangle of incidence θ decreases slowly. This positional rela-tionship is shown in Fig. 14. The arrows in the figurerepresent the direction of this movement. The same holdsfor Fig. 16, shown later.

The area of the portion enclosed in the thick line inFig. 14, where GC12 does not overlap with GC11 or GC21,is the effective area S12 of SP12. This method is differentfrom where GC12 and CG21 are in contact as shown in (a),that is, the range θTL1 > θ ≥ θTL for the angle of incidencein which GC12 and GC21 do not overlap, and the range

Fig. 12. Irradiation with three cells.

(14)

(15)

(16)

(17)

Fig. 13. Evaluation of reflected light for a cell module.

(18)

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θTL > θ ≥ θCC2 for the angle of incidence from (b) to (d)where the two great circles overlap completely. As a result,the two can be separated and calculated as shown below.

First, let us consider θTL1 > θ ≥ θTL. The effective areaS12 for SP12 is represented as follows, with the areas S11 andS12 which are adjacent shadows due to SP11 and SP21 beingsubtracted from the area SGC12 for GC12:

However,

β1 = arccos

sinθtanθCC2

− cos θ

Next, let us consider the angle of incidenceθTL > θ ≥ θCC2 [Figs. 14(c) and (d)] where the two greatcircles GC12 and GC21 overlap. In this range if Eq. (19) isused as is, when the area of the adjacent shadow decreaseswith respect to GC12, caution is required because the partSoverlap2 duplicating GC11 and GC21 [horizontally linedregion in Fig. 14(c)] may be subtracted twice. Thus, theeffective area S12 for θTL > θ ≥ θCC2 is given by

However,

β2 = arccos

sin θtan θCC2

(3) Range 3: θCC(m−1) ≥ θ ≥ θCCm

m: a natural number for which 3 ≤ m ≤ n

This range is the range for the angle of incidence forthe lined space that connects the center of SP12 and thecenter of SP(m-1) as shown in Fig. 13. It can be subdividedin n – 2 ways based on n. However, for two spheres (n = 2),this range does not exist.

Let us take m = 3 as an example. In range 3, the lightirradiates as shown in Fig. 15(a). If the direction of theincident light is seen from GC11, GC12, and GC21, then theoverlaps shown in (b) occur. The area S11 (in the figure,vertically shaded) where GC11 and GC12 overlap includesthe area S21 where GC21 and GC12 overlap. As a result, GC11

is not a target for consideration. On the other hand, the realsphere on the left of SP31 is excluded from the calculationof the effective area because the overlapping portion ofthese great circles and GC12 are all within GC31, as shownin Fig. 12 and was the case when evaluating direct light inSection 3.1. Extending this results in SPm1 and SP(m-1)1

forming an adjacent shadow with respect to SP12. In otherwords, GC(m-1)1 and GCm1 are outside consideration. Be-cause there is always overlap in GC(m-1)1 and GCm1, theoverlap can be found using the same thinking for thearrangement (b) through (d) in Fig. 14. If the position ofGC(m-1)1 is fixed and the angle of incidence θ is slowlydecreased, then GC12 and GCm1 shift as shown in Fig. 16.

Thus, in range 3, S12 has the same form as Eq. (20)for range 2, and can be represented using the followingequation where the angle correspondence goes θ → γ1, β1

→ γ2, β2 → γ3:

However,

(19)

(20)

Fig. 14. Calculation of effective area for range.

(21)

Fig. 15. Irradiation for region 3.

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This range can be divided n – 2 ways, as was previouslyshown. If 3 to n is substituted in sequence for m in Eq. (21)for each subdivided range, then the effective area for thetotal range for 3 can be found.

(4) Range 4: θCCn ≥ θ ≥ θTL2

The range 4 is related to the sphere on the left edgeof the row to be considered. θTL2 is the angle of incidenceparallel with the tangent to SP12 subtracted from the edgeLE on the reflective surface. Figure 17(a) shows the irradia-tion of light in range 4. In ranges 2 and 3, both the cellsgenerating adjacent shadows in the upper part of the greatcircle as in GCm1 in Fig. 16 and the cells generating theadjacent shadows in the lower part of the great circle as inGC(m-1)1 are real spheres. In contrast, in range 4 the cellsgenerating adjacent shadows in the lower part of the greatcircle is the real sphere SPn1, as can be seen in Fig. 17(b).The sphere generating adjacent shadows in the upper partof the great circle is the mirror image sphere SP22. There-fore, calculating the effective area S12 requires consideringGC22 and GCn1. At this point, the overlapping of the threegreat circles is the same as in range 2 and range 3. In otherwords, S12 in range 4 can be found using the equationbelow where the corresponding angle in Eq. (20) for range2 goes θ → δ1, β1 → δ2, β2 → δ3. Note that for n = 2, θCC(n-1)

= θCC1 = 90°.

However,

(5) Range 5: θTL2 > θ ≥ θCCR and range 6: θCCR

> θ > 0°

The ranges 5 and 6 represent the ranges for theangles of incidence where the reduction in the effective areafor SP12 due to there being no reflective plate must be takeninto consideration. This reduction is shown by vertical linesin Fig. 18. In range 5, as can be seen in (a), the reductionis included in the adjacent shadow resulting from SP22.Consequently, the effective area S12 shown in gray in (a) canbe found using Eq. (22). In contrast, in range 6, the regionin gray and the vertical lines overlap, as can be seen in (b).As a result, the effective area can be calculated by subtract-ing this overlapping area from S12 in Eq. (22).

(6) Calculation example and evaluation of thenumber of actual calculations

Figure 19 shows the results of calculating the effec-tive area for a sphere with respect to the reflected light whenthere are 2, 4, 7, and 10 unit cells and the standard gap D is0.0.

The number of unit cells arranged in a row in a realmodule can reach dozens or even hundreds. With respect tothe direct light, only one adjacent sphere needs to be con-sidered. For the reflected light, calculations in the priorsections must be performed for all spheres, to be precise.As can be seen in Fig. 19, the effect as the number of cellsrises is greatly reduced, and limited to where the angle ofincidence is small. Thus, the calculations can be truncatedat the number which provides the precision required for thepurpose of the analysis. In this paper the authors use n = 10for the sake of convenience and proceed with the analysisin the next section.

Fig. 16. Calculation of effective area for region 3.

Fig. 17. Irradiation for region 4.

(22)

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3.3 Optimum cell gap

The previous sections provided a method for calcu-lating the direct light and the reflected light. In this sectionthe authors find the optimum cell gap based on the totaleffective area for a unit cell by totaling both kinds of light.

Figure 20 shows the total effective area combiningthe effective area for the direct light and reflected light. Inthe figure, the cell gap is taken to be the normal gap D. Asthe cell gap is increased, the adjacent shadow decreases, theeffective area rises, and the amount of light received by theunit cell increases. However, the number of cells that canbe placed in a single region decreases. The reverse happenswhen the cell gap is decreased. There is an inverse relation-ship between the amount of light received by a single celland the number of cells in the same region. For a module,the total effective area for one region, in other words thecell gap at which the total amount of light received is at amaximum, is the optimum gap. If the cell gap is designated

d, then the unit cell has the area S0 given by the followingequation in the module:

This is called the dominant area. If the amount of lightreceived in the unit cell is divided by S0, then the densityργ of the amount of light received in the unit cell results. Ifthe area of the entire module is multiplied by ργ, then thetotal amount of light received by the module results, andbased on this d for which ργ is at a maximum is the optimumgap. The relationship between d and ργ is found by dividingthe effective area in Fig. 20 by S0. Figure 21 shows theresults. Specifically, we can conclude that arranging thecells tightly together without any gap is optimal for themodule.

3.4 Comparison and evaluation of sphericalmodules and planar modules

In Fig. 21, the light-receiving density characteristicsfor a planar solar cell are also shown. Based on a compari-son of this with the characteristics for the optimum gap D

Fig. 19. Characteristics of effective area for reflectedlight.

Fig. 18. Irradiation for region 5 and 6.

Fig. 20. Total effective area of a module.

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Fig. 21. Light-receiving density characteristics.

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= 0.0, the module with an optimum gap and the planar solarcell clearly have very similar light-receiving characteristics.This means that for a planar solar cell, there is no directivityin the unit cell. But for a module, almost the same directivityis seen as found in a planar solar cell. This is caused by theadjacent shadows. Within the total range of the angle ofincidence, the light-receiving density for the planar solarcell is clearly greater.

4. Experimental Evaluation

Because of the strong correlation between the amountof light received and the amount of electricity generated ina solar cell, a proportional relationship between the two isassumed, and in the experiments the amount of light re-ceived is evaluated using the amount of electricity gener-ated. The irradiation is assumed to be constant at 1.3kW/m2.

4.1 Evaluation of no directivity

Figure 22 shows the cells used in the experiment, andan expanded view of one cell. A halogen lamp was used asthe light source. The light source was placed at the focalpoint of a convex lens to create parallel light rays. Theparallel light rays struck the spherical silicon micro solarcells, which were titled at angles from 0° to 90° in incre-ments of 10°, and the amount of electricity generated wasmeasured using the maximum power at each angle ofincidence.

Evaluation of no directivity should be performedusing one cell, but because the amount of electricity gener-ated is minimal, several cells connected in series were used,as shown in the figure. The light was shown from a directionorthogonal to the cell string so that adjacent shadows wouldnot be created.

Figure 23 shows the experimental results. As theangle of incidence grew smaller, a slight decrease in theamount of electricity generated was seen, but the amountwas almost constant, and the lack of directivity was con-firmed. This small decrease is thought to be a result of theeffective area decreasing as the angle of incidence grew

smaller, which affected the contact area of the electrodes,as can be seen from the magnified view in Fig. 21.

4.2 Comparison of a planar solar cell and aspherical module

The spherical module used in the experiment isshown in Fig. 9, and its parameters as well as the parametersof the test planar solar cell are given in Table 1.

Figure 24 shows the angle of incidence θ and theoutput electrical characteristics obtained in the experiment.The vertical axis for both the planar solar cell and thespherical module represents the power per unit surface arearesulting from dividing the electricity generated by the area.In contrast, the mathematical calculation of the effectivearea derived in the previous sections is compared here. Theconditions for the calculations are the same as they were forthe spherical module used in the experiment. Figure 25shows the results of the calculations for the effective areaper unit area. The characteristics curves in the figure areclearly similar.

The measured results in Fig. 24 represent the electri-cal power per unit area, and the results of numerical calcu-lations in Fig. 25 represent the effective area. These

Fig. 23. Experimental results for no directivity of asingle cell.

Table 1. Specifications of tested module

Fig. 22. A cell string tested.

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physical quantities are different, and as a result cannot becompared directly. However, as was described at the begin-ning of this paper, there is a strong correlation between theamount of light received and the amount of electricitygenerated in a solar cell. Consequently, by normalizingthese quantities, a direct comparison can be performed.Figure 26 shows the results of comparing the experimentalresults and the numerical calculations for the effective areaderived in the previous section. The experimental resultsrepresent the amount of electricity generated in a planarsolar cell with an angle of incidence θ of 90° in Fig. 24, andthe results of numerical calculations represent the effectivearea for a planar solar cell with the same angle of incidenceθ of 90° in Fig. 25. Both these quantities are normalized.Based on the comparison, the authors were able to confirmthat the method derived for numerical calculations is validbased on the very close match between the experimentalresults and the results of the numerical calculations.

5. Conclusion

In this paper the authors derived a geometrical calcu-lation method for the basic characteristics related to thelight-receiving characteristics of a spherical silicon microsolar cell.

First, for a unit cell, the authors clarified the follow-ing:

• The effective area is the size of the great circle.• The effect of the reflective plate can be evaluated

using the adjacent shadow created by the directlight on the cell mirror image and the real sphere.

Next, the authors extended their discussion of the unitcell to a module of spheres connected and arranged in a row,and then derived a method for calculating the effective areafor the module. The results showed that:

• For the arrangement of the cells in a module,packing them together without gaps is optimal notonly for maximizing the adjacent shadow but alsofor increasing the effective area to the utmostpossible.

• When the gap is optimized, the light-receivingcharacteristics of the module of spheres are almostthe same but slightly lower than those of a planarsolar cell.

Further, based on the geometrical discussion pro-posed in this paper, the authors clarified the following:

• Unit cell (no reflective plate): no directivity• Unit cell (reflective plate): some directivity, but

not as strong as in a planar solar cell• Module: directivity

Based on the experiments, the above discussion wasverified. First, the authors confirmed that there was nodirectivity for the unit cell. They compared the module of

Fig. 24. Characteristics of generated power measured.

Fig. 25. Characteristics of effective areas of both cells.

Fig. 26. Normalized characteristics of experimentalresults and effective areas of both cells.

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spheres with a planar solar cell, and based on the similarityin the effective area characteristics resulting from calcula-tions and the power generation characteristics based on theexperiments, they confirmed the validity of the calculationmethod in this paper.

The irradiation conditions for light in this paper applyto when sunrise is due east, sunset is due west, and the angleof incidence θ is 90° at the southing time from where solarpower generation is under way. Changes in the height of thesun and cases in which Japan is not immediately below theequator were not taken into consideration. An analysis ofthe light-receiving characteristics using these issues and thecontributions of scattered light represents a topic for thefuture. The possibility that the light-receiving charac-teristics of a module of spheres exceeds those of a planarsolar cell cannot be denied. The quest for practical use of amodule of spheres, making the most of the superior manu-facturing costs for sphere cells, the structure of the module,and other advantages will be valuable for future researchand development.

REFERENCES

1. Nakata J. Spherical silicon micro solar cells. EnergyMonthly 2001;34:74–77. (in Japanese)

2. Nakata J, Kogo N, Niki R, Yamada K, Fujita D, YasutaN. Spherical Si micro solar cell. 2002 National Con-vention Record, IEE Japan, No. 7-075. (in Japanese)

3. Naitoh H, Wang D, Ishikawa H, Nakata J. Fundamen-tal properties of spherical solar cells. 2002 NationalConvention Record, IEE Japan, No. 7-074. (in Japa-nese)

4. Kikuchi H, Kogo N, Inagawa I, Nakata J. Incidentlight angle dependence of the characteristics of spe-cial micro solar cell module. 2003 National Conven-tion Record, IEE Japan, No. 4-029. (in Japanese)

5. Naitoh H, Wang D, Ishikawa H, Fujii A, Nakata J.Experimental verification of non-directivity of microsolar cells. 2003 National Convention Record, IEEJapan, No. 7-144. (in Japanese)

6. Wang D, Ishikawa H, Naitoh H, Nakata J. Optimumseries-parallel connection of spherical solar cells.2003 National Convention Record, IEE Japan, No.7-123. (in Japanese)

AUTHORS (from left to right)

Hiroki Ishikawa (member) completed the first part of his doctoral studies in electronic engineering and informatics atGifu University in 1995 and became a lecturer there. He completed his doctorate in engineering at the University of Tokyo in2006. He is primarily pursuing research related to solar light power generation systems, switched reluctance motors,soft-switching electric power conversion circuits, and circuit simulators. He is a member of IEEE and the Japan Institute ofPower Electronics.

Takayuki Furuno (student member) is currently studying to complete the first part of his doctoral work in electrical andelectronics engineering at Gifu University. He is primarily pursuing research related to solar light power generation systems.

Satoru Kambe (nonmember) completed the first part of his doctoral studies in electronic engineering at Gifu Universityin 2006. While in school he was primarily pursuing research related to spherical micro solar cells and solar light power generationsystems.

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AUTHORS (continued)

Haruo Naitoh (member) completed a doctorate in electrical engineering at the University of Tokyo in 1980 and joinedToshiba Corporation. He was dispatched by Toshiba to be a researcher at California Institute of Technology in 1984–1985. Hewas a visiting assistant professor at Virginia Institute of Technology in 1985–1986. After retiring from Toshiba in 2000, hebecame a professor at Gifu University. He is pursuing research related to power electronics. He received the Institute of ElectricalEngineers of Japan Prize for Outstanding Paper in 1983. He holds a D.Eng. degree, and is a member of IEEE.

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