**3. Theoretical design of MCA**

meet global energy needs [1–4]. However, solar energy, being the energy source with the highest energy potential [5–7], can provide a sustainable source of future primary energy supply if captured with high efficiency and simple configuration. Photovoltaic system provides the most simple configuration to convert solar energy into high-grade energy form, i.e., electricity. However, in the current photovoltaic market with 99% share of single-junction solar cells, the multi-junction solar cells (MJCs) have yet to exploit their potential of the highest solar energy conversion efficiency. This is due to the fact that the multi-junction solar cell (MJC) can respond to a full spectrum of solar radiation, with less loss and as a result higher efficiency [8–12]. On the other hand, due to their high material cost, MJCs are not available in the form of flat-plate panels like conventional single-junction solar cells. But rather, they are utilized in the form of concentrated photovoltaic (CPV) system where low-cost solar concentrators concentrate solar radiations onto the small area of solar cell material, thereby reducing the use of expensive solar cell material by 500 or 1000 times [13–17]. This is possible because MJC can withstand high concentrations. However, all of the commercial CPV modules, available hitherto, can only accommodate single solar cell per concentrator [18, 19]. Such conventional system requires increased assembly efforts. Therefore, by keeping the cell size same, this chapter discusses a novel CPV

module design which can accommodate multiple MJCs with single solar concentrator.

The simple concept of proposed multicell concentrating assembly (MCA) is shown in **Figure 1**, for novel CPV module design. The concentrating assembly is based upon a multi-leg homogenizer concept which allows the rays, collected from specially arranged pair of solar concentrators, to be uniformly distributed and transferred to four MJCs. In a simple form, received parallel solar radiations are transformed into a concentrated collimated beam, with the help of a pair of parabolic concentrators which are arranged in Cassegrain form. The concentrated collimated beam strikes at the inlet aperture of multi-leg homogenizer where it is uniformly split and distributed among MJCs, placed at the four outlet apertures of the homogenizer. In conventional CPV module design, the homogenizer is placed to accommodate the tracking errors

**2. Multicell concentrating assembly (MCA)**

112 Solar Panels and Photovoltaic Materials

**Figure 1.** Schematic of novel multicell concentrating assembly (MCA) for CPV module.

The theoretical design of proposed multicell concentrating assembly (MCA) is explained in **Figure 2**. It can be seen that the design is based upon the Cassegrain arrangement of two parabolic concentrators, with the intention of achieving the area concentration. Such concentrating assembly acts as a collimating reflector where the parallel solar radiations are converted into concentrated collimated beam. Both parabolic reflectors are arranged such that their focal points coincide with each other. In addition, one of the reflectors uses its inner surface for reflection, while the other uses its outer surface. The primary reflector converges the incoming parallel rays at its focal point. However, these converging rays interact with the outer surface of secondary parabolic reflector which is placed in their way to the focal point. The secondary reflector introduces the cancelation effect which converges rays and diverges them as parallel rays. However, due to smaller contact surface of secondary reflector, the diverged collimated rays become concentrated before they hit the inlet aperture of homogenizer. In order to design multicell concentrating assembly (MCA), the edge ray is traced such that it hits point 'b' of secondary concentrator after being reflected by point 'a' of primary concentrator. As the foci of both reflectors are coinciding, therefore, the edge ray become parallel again after being reflected by the outer surface of secondary reflector and enter the homogenizer through its inlet aperture. This edge ray, after entering into the homogenizer, hits point 'c', which is located at its lower tapered portion. This lower tapered portion of multi-leg homogenizer is designed such that the edge ray, after being reflected by total internal reflection from point 'c', hits point 'e' of outlet aperture of homogenizer and falls on the MJC, placed there. The distribution of such parallel ray distribution is easily explained in simple schematic shown in **Figure 2(b)**. The back surface of MJC is attached to the heat spreader and heat sink assembly, to effectively dissipate the heat during CPV operation. It can be seen that the rays, after being reflected by secondary reflector, become concentrated over an area and size of such area concentration depends upon the size of secondary reflector.

There are three phases in which the design of proposed multicell concentrating assembly (MCA) is divided. The first phase is related to the calculations needed for the sizing of primary reflector. The main factor determining the size of primary reflector is the concentration requirement, depending upon the MJC specifications. The second phase is related to the form

higher at the prototyping stage. Therefore, a reasonable concentration ratio of 165 is chosen to prove the concept. This concentration is only chosen for prototype design purpose, while the MCA can be designed for any concentration ratio of ×500 or ×1000, depending upon the specification of MJC. In order for the rays to enter homogenizer, a center hole of square size

From **Figure 3**, it can be seen that the effective area of MCA, to capture the solar radiations, is

Effective area of concentrating assembly *ACA* = *CRg* × *AM* × 4 (1)

Parameter 'd' defines the dimension of square parabolic reflector. It must be noted that the square shape of primary reflector is firstly considered because of uniform distribution of

<sup>2</sup> = *ACA* + *AH* (2)

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(50 × 50 mm) is cut in the primary parabolic reflector, as demonstrated in **Figure 3**.

the one highlighted in white, which is calculated by Eqs. (1) and (2).

Area of primary concentrator = *d*<sup>1</sup>

**Sr. no. Parameter Symbol Units Value** 1 Geometric concentration ratio CR<sup>g</sup> — 165

**Table 1.** Primary reflector design parameters.

**Figure 3.** Simple schematic for primary reflector size calculations.

2 Area of MJC AM m<sup>2</sup> 0.00003025 (5.5 × 5.5 mm) 3 Area of center square hole AH m<sup>2</sup> 0.0025 (50 × 50 mm)

**Figure 2.** (a) Design and (b) ray distribution of proposed multicell concentrating assembly (MCA).

and the design of multi-leg homogenizer, which is also depending upon the specification of MJC as the outlet apertures of homogenizer must have dimensions same as MJC size. After finalizing the design of multi-leg homogenizer, the third phase is related to the secondary reflector sizing as it depends upon the size of inlet aperture of homogenizer. The detailed procedure associated with the overall design of multicell concentrating assembly (MCA) will be explained in this section.

The design parameters associated with the sizing of primary parabolic reflector are given in **Table 1**. It can be seen that the concentration ratio selected for the targeted prototype of MCA is 165, for the MJC of size (5.5 × 5.5 mm). Concentration ratio of 165 is only chosen to limit the overall size of the primary reflector due to budget limitations as larger reflector will cost higher at the prototyping stage. Therefore, a reasonable concentration ratio of 165 is chosen to prove the concept. This concentration is only chosen for prototype design purpose, while the MCA can be designed for any concentration ratio of ×500 or ×1000, depending upon the specification of MJC. In order for the rays to enter homogenizer, a center hole of square size (50 × 50 mm) is cut in the primary parabolic reflector, as demonstrated in **Figure 3**.

From **Figure 3**, it can be seen that the effective area of MCA, to capture the solar radiations, is the one highlighted in white, which is calculated by Eqs. (1) and (2).

$$\text{Effective area of concentrated assembly assembly}\,A\_{\odot A} = \text{CR}\_{\ddagger} \times A\_{\al} \times 4 \tag{1}$$

$$\text{Area of primary concentration} = d\_1^2 = A\_{\odot A} + A\_H \tag{2}$$

Parameter 'd' defines the dimension of square parabolic reflector. It must be noted that the square shape of primary reflector is firstly considered because of uniform distribution of


**Table 1.** Primary reflector design parameters.

and the design of multi-leg homogenizer, which is also depending upon the specification of MJC as the outlet apertures of homogenizer must have dimensions same as MJC size. After finalizing the design of multi-leg homogenizer, the third phase is related to the secondary reflector sizing as it depends upon the size of inlet aperture of homogenizer. The detailed procedure associated with the overall design of multicell concentrating assembly (MCA) will

**Figure 2.** (a) Design and (b) ray distribution of proposed multicell concentrating assembly (MCA).

The design parameters associated with the sizing of primary parabolic reflector are given in **Table 1**. It can be seen that the concentration ratio selected for the targeted prototype of MCA is 165, for the MJC of size (5.5 × 5.5 mm). Concentration ratio of 165 is only chosen to limit the overall size of the primary reflector due to budget limitations as larger reflector will cost

be explained in this section.

114 Solar Panels and Photovoltaic Materials

**Figure 3.** Simple schematic for primary reflector size calculations.

rays over cell area as MJC is of square shape. Secondly, the square shape primary reflector provides maximum packing density and perfect arrangement within CPV module, without leaving any empty pocket.

In Eq. (1), factor '4' is used because of four outlet apertures of homogenizer, with the size same as MJC. By using the concentration ratio of 164, the size of square primary parabolic reflector is calculated as 15 × 15 cm. Before proceeding to the design of multi-leg homogenizer, it is important to trace the position of edge ray. It must be noted that, as per **Figure 3**, the primary parabolic reflector is actually a square-cut piece of circular shape, and the diagonal of square reflector gives the diameter of the circular shape, as given by Eq. (3):

$$D\_1 = \sqrt{d\_1^2 + d\_1^2} = \sqrt{2d\_1^2} \tag{3}$$

Besides the size of primary reflector, it is also very important to keep the overall height of CPV modules minimum as it can make shipping and handling easy and cheap. It can also lessen the mechanical problems faced during CPV operation. Therefore, in order to keep the overall height of MCA minimum, the focal point of primary parabolic reflector is kept same as its depth 't'. By using the parabolic curve equation, the focal point 'f<sup>1</sup> ' of primary parabolic concentrator can be calculated by Eq. (4):

$$f\_1 = \frac{\{D\_i/2\}^2}{4\ t\_1} = \frac{D\_i^{\cdot 2}}{16f\_1} \quad \{f\_1 = t\_1\}\\ f\_1^2 = \frac{D\_i^{\cdot 2}}{16} \tag{4}$$

Diameter 'D<sup>1</sup> ' of corresponding circular shape of primary reflector can be calculated as 21.2132 cm, by using the 15 cm size in Eq. (3). As a result, Eq. (4) gives the focal point value of primary parabolic concentrator as 5.3033 cm. The simple parabolic Eq. (5) can now be used to calculate the coordinates of the parabolic surface of primary reflector:

$$\mathbf{x}^2 = \mathbf{4}\mathbf{\!}\mathbf{\!}\mathbf{y} \tag{5}$$

of homogenizer. Such a ray must exit through edge point 'c' of the outlet aperture of the homogenizer, to ensure uniform distribution of solar radiations over the cell area. Therefore, the dimensional parameters of lower tapered portion of homogenizer, i.e., height 'h' and slope angle 'θ' of side line 'bl', are of prime importance here. As explained, the value of these two parameters must be tuned such that the parallel edge rays, after passing through the homogenizer, must exit through end point 'c' of outlet aperture. **Figure 4** shows the angular position of such tapered portions as defined by the trigonometric laws. The angle between line 'bl' and the hypothetical line 'bg' is taken as 'θ'. As these lines are perpendicular to each other, therefore, the edge ray also makes same angle 'θ' with the hypothetical line 'bg'. By considering the triangular section ∆ckb, the height 'h' of lower tapered portion of homog-

\_\_\_\_\_\_ *ID* − *OD*

It can be seen that there are two unknowns in Eq. (6), i.e., 'θ' and 'h'. Therefore, such equation cannot be solved for the mentioned unknowns. For the solution of such equation, the triangle

<sup>2</sup> )].tan(2*<sup>θ</sup>* <sup>−</sup> 90) (6)

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enizer is given by Eq. (6):

*h* = [*OD* + (

**Figure 4.** Multi-leg homogenizer design and schematic.

∆blk can be considered to get Eq. (7):

The multi-leg homogenizer is consisting of four similar sections, joined together as a single unit. Therefore, for easy understanding, the design of single section will be explained in detail first, which will be combined together to form the proposed multi-leg homogenizer. The simple 2D schematic of multi-leg homogenizer is shown in **Figure 4** where only its two sections are shown. Each single section of multi-leg homogenizer has two tapered section, i.e., above and below line 'eb'. The tapered section below line 'eb' is designed such that it uniformly distributes solar radiations over the cell area in case of parallel rays. However, the tapered section above line 'eb' is designed to accommodate the tracking error by guiding the non-parallel incoming solar radiations towards the outlet aperture of homogenizer. In order to discuss the homogenizer design, it is better to discuss the case for parallel rays first, for easy understanding.

Edge ray 'j' which is parallel to the primary reflector axis, as shown in **Figure 4**, is the same edge ray explained in **Figure 2(a)** that is coming from secondary reflector. However, after entering into the homogenizer, it hits point 'b' located at the start of lowered tapered portion

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**Figure 4.** Multi-leg homogenizer design and schematic.

rays over cell area as MJC is of square shape. Secondly, the square shape primary reflector provides maximum packing density and perfect arrangement within CPV module, without

In Eq. (1), factor '4' is used because of four outlet apertures of homogenizer, with the size same as MJC. By using the concentration ratio of 164, the size of square primary parabolic reflector is calculated as 15 × 15 cm. Before proceeding to the design of multi-leg homogenizer, it is important to trace the position of edge ray. It must be noted that, as per **Figure 3**, the primary parabolic reflector is actually a square-cut piece of circular shape, and the diagonal

> \_\_\_\_\_\_ *d*1 <sup>2</sup> + *d*<sup>1</sup> <sup>2</sup> = √

Besides the size of primary reflector, it is also very important to keep the overall height of CPV modules minimum as it can make shipping and handling easy and cheap. It can also lessen the mechanical problems faced during CPV operation. Therefore, in order to keep the overall height of MCA minimum, the focal point of primary parabolic reflector is kept same

\_\_\_\_ 2*d*1

' of corresponding circular shape of primary reflector can be calculated as

<sup>2</sup> (3)

' of primary parabolic

<sup>16</sup> (4)

of square reflector gives the diameter of the circular shape, as given by Eq. (3):

as its depth 't'. By using the parabolic curve equation, the focal point 'f<sup>1</sup>

<sup>1</sup> <sup>=</sup> (*D*<sup>1</sup> /2)<sup>2</sup> \_\_\_\_\_\_ 4 *t* 1

calculate the coordinates of the parabolic surface of primary reflector:

<sup>=</sup> *<sup>D</sup>*<sup>1</sup> 2 \_\_\_\_ 16 *f* 1 (*f* <sup>1</sup> = *t* 1) *f* 1 <sup>2</sup> <sup>=</sup> *<sup>D</sup>*<sup>1</sup> 2 \_\_\_

21.2132 cm, by using the 15 cm size in Eq. (3). As a result, Eq. (4) gives the focal point value of primary parabolic concentrator as 5.3033 cm. The simple parabolic Eq. (5) can now be used to

*x*<sup>2</sup> = 4*fy* (5)

The multi-leg homogenizer is consisting of four similar sections, joined together as a single unit. Therefore, for easy understanding, the design of single section will be explained in detail first, which will be combined together to form the proposed multi-leg homogenizer. The simple 2D schematic of multi-leg homogenizer is shown in **Figure 4** where only its two sections are shown. Each single section of multi-leg homogenizer has two tapered section, i.e., above and below line 'eb'. The tapered section below line 'eb' is designed such that it uniformly distributes solar radiations over the cell area in case of parallel rays. However, the tapered section above line 'eb' is designed to accommodate the tracking error by guiding the non-parallel incoming solar radiations towards the outlet aperture of homogenizer. In order to discuss the homogenizer design, it is better to discuss the case for parallel rays first, for easy

Edge ray 'j' which is parallel to the primary reflector axis, as shown in **Figure 4**, is the same edge ray explained in **Figure 2(a)** that is coming from secondary reflector. However, after entering into the homogenizer, it hits point 'b' located at the start of lowered tapered portion

leaving any empty pocket.

116 Solar Panels and Photovoltaic Materials

*D*<sup>1</sup> = √

concentrator can be calculated by Eq. (4):

*f*

Diameter 'D<sup>1</sup>

understanding.

of homogenizer. Such a ray must exit through edge point 'c' of the outlet aperture of the homogenizer, to ensure uniform distribution of solar radiations over the cell area. Therefore, the dimensional parameters of lower tapered portion of homogenizer, i.e., height 'h' and slope angle 'θ' of side line 'bl', are of prime importance here. As explained, the value of these two parameters must be tuned such that the parallel edge rays, after passing through the homogenizer, must exit through end point 'c' of outlet aperture. **Figure 4** shows the angular position of such tapered portions as defined by the trigonometric laws. The angle between line 'bl' and the hypothetical line 'bg' is taken as 'θ'. As these lines are perpendicular to each other, therefore, the edge ray also makes same angle 'θ' with the hypothetical line 'bg'. By considering the triangular section ∆ckb, the height 'h' of lower tapered portion of homogenizer is given by Eq. (6):

$$h = \left[ \text{OD} + \left( \frac{\text{ID} - \text{OD}}{2} \right) \right] \tan(2\theta - 90) \tag{6}$$

It can be seen that there are two unknowns in Eq. (6), i.e., 'θ' and 'h'. Therefore, such equation cannot be solved for the mentioned unknowns. For the solution of such equation, the triangle ∆blk can be considered to get Eq. (7):

$$h = \left(\frac{\text{ID} - \text{OD}}{2}\right) \tan \theta \tag{7}$$

is 1° deviation in the incident ray from the primary reflector axis, such deviation increases to 2° after being reflected by the primary reflector. Similarly, after being reflected by secondary reflector, this deviation increases to 4°. Therefore, to handle such 1° deviation in the incident

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In the third phase of MCA design, the size of secondary reflector is calculated. As shown in **Figure 2(A)**, the edge ray, after being reflected by point 'b' of secondary reflector, hits point 'c' of the homogenizer which is the edge point of lower tapered portion of 'ID'. Therefore, for the calculation of the focal point of secondary reflector, the value of D<sup>1</sup> = ID should be used in Eq. (4). The main reason for this consideration is that the focal point of both reflectors is coinciding as the parallel ray pattern is needed to be achieved after secondary reflection, which must hit the lower tapered portion of the homogenizer with size 'ID = 16 mm'. However, the radius of secondary reflector will be according to 'ED' which is found to be '19 mm'. The coordinates for parabolic surface of secondary reflector can be found using Eq. (5), for the value of 'x' being varied from 0 to 19. The size of secondary reflector is same as that of the inlet aperture

To verify and analyze the field performance of proposed and designed MCA, a prototype of CPV is fabricated as per designed concentration ratio of ×165. The prototype of fabricated MCA is shown in **Figure 6** and the MCA-based novel CPV unit is shown in **Figure 7**. The primary and secondary reflectors are machined from aluminum blocks. However, to improve the surface quality and reflection characteristics, the reflecting surface of both reflectors was coated with thin optical graded reflecting aluminum layer, using sputter-coating method. The multi-leg homogenizer was fabricated in the form of four symmetrical pieces which were joined together to form a single unit. Each homogenizer piece is similar to the half of schematic shown in **Figure 4**. The individual pieces of homogenizer were machined using N-BK7 glass material. To form a single homogenizer unit, all four pieces were joined together using optical graded UV glue. It must be noted that the machining method of fabrication is an expensive fabrication technique, and that is why smaller concentration ratio was chosen at the start to keep the overall cost minimum. However, for mass production of reflectors and homogeniz-

Four MJCs were attached at the four outlet apertures of the homogenizer. However, the back side of MJCs was attached to the heat spreader and heat sink to dissipate the heat during CPV

In order to test the performance of developed novel MCA-based CPV module, a twoaxis solar tracking unit was used with tracking accuracy of 0.1° [20, 21]. The developed CPV module was mounted onto the top frame of solar tracker. The tracking system is based upon the hybrid tracking algorithm which defines the solar position through both active and passive techniques. After calculating the position of sun, based upon the solar geometry model, the actual position of the sun is verified by taking the real-time feedback

ray, by the homogenizer, its half width is found to be 19 mm, by the graphical method.

of the homogenizer, to account for the tracking error.

**4. Development of MCA-based CPV prototype**

ers, injection molding techniques for plastic and glass materials are used.

operation and to keep the cell temperature within optimum limit.

If Eqs. (6) and (7) are compared, then we can find the expression for the lower slope angle 'θ' as given by Eq. (8):

$$
\left[OD + \left(\frac{ID - OD}{2}\right)\right] \tan(2\theta - 90) = \left(\frac{ID - OD}{2}\right) \tan\theta\tag{8}
$$

Parameter 'OD' represents the dimension for the outlet aperture of homogenizer, which is same as the size of MJC, i.e., 5.5 mm. On the other hand, a suitable value of '16' is also considered for parameter 'ID' so that ray can easily be propagated to the outlet aperture. If the value of 'ID' is small, then the height 'h' will also be small, same for 'OD'. Otherwise, the ray will have multiple total internal reflections (TIR) inside the homogenizer if the value of 'h' is higher for smaller 'ID'. Or, there may be a chance that the ray can propagate back if the value of 'h' is very big. However, the large value of 'ID' also requires a larger value of 'h', same for 'OD'. For the given value of 'ID' and 'OD', the value of 'θ' and 'h' can be calculated as 81.33o and 34.43 mm, respectively.

After finalizing the design of lower tapered portion of the homogenizer, now the upper tapered portion of homogenizer is considered, which is needed to be designed to accommodate the rays which are not parallel to the axis of concentrating assembly. From the design point of view, a tracking error of 1o is selected which can be handled by the homogenizer. This means that the homogenizer design will be able to handle the deviated ray for maximum angle of 1°. By using the graphical method and trigonometric laws, it has been shown in **Figure 5** that if there

**Figure 5.** Acceptance angle calculation for multicell concentrating assembly.

is 1° deviation in the incident ray from the primary reflector axis, such deviation increases to 2° after being reflected by the primary reflector. Similarly, after being reflected by secondary reflector, this deviation increases to 4°. Therefore, to handle such 1° deviation in the incident ray, by the homogenizer, its half width is found to be 19 mm, by the graphical method.

In the third phase of MCA design, the size of secondary reflector is calculated. As shown in **Figure 2(A)**, the edge ray, after being reflected by point 'b' of secondary reflector, hits point 'c' of the homogenizer which is the edge point of lower tapered portion of 'ID'. Therefore, for the calculation of the focal point of secondary reflector, the value of D<sup>1</sup> = ID should be used in Eq. (4). The main reason for this consideration is that the focal point of both reflectors is coinciding as the parallel ray pattern is needed to be achieved after secondary reflection, which must hit the lower tapered portion of the homogenizer with size 'ID = 16 mm'. However, the radius of secondary reflector will be according to 'ED' which is found to be '19 mm'. The coordinates for parabolic surface of secondary reflector can be found using Eq. (5), for the value of 'x' being varied from 0 to 19. The size of secondary reflector is same as that of the inlet aperture of the homogenizer, to account for the tracking error.
