## Abstract

Static and quasi-static concentrators present interesting characteristics for obtaining photovoltaic solar energy. In this work we study the characteristics of the crossed compound parabolic concentrator, formed by the intersection of two cylindrical compound parabolic concentrators (CPC). Bifacial cells are used in this concentrator as a requirement for obtaining higher concentrations. Static and quasi-static concentrators see the sun as an extended source, so a simplified source model of radiance for the sky of Madrid is used. The figures of merit of a lossless concentrator are studied and the most important parameters influencing its optical behavior are discussed. We conclude that these concentrators obtain results that lead to a decrease in the cost of photovoltaic energy.

© 1984 Optical Society of America

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### Equations (21)

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(1)
$${h}_{q}/W=\frac{n+\text{sin}{\varphi}_{q}}{{\text{sin}}^{2}{\varphi}_{q}}{({n}^{2}-{\text{sin}}^{2}{\varphi}_{q})}^{1/2},$$
(2)
$${h}_{p}/W=\frac{n+\text{sin}{\varphi}_{p}}{2\hspace{0.17em}{\text{sin}}^{2}{\varphi}_{p}}{({n}^{2}-{\text{sin}}^{2}{\varphi}_{p})}^{1/2}*(L/W),$$
(3)
$${h}_{q}/W={h}_{p}/W.$$
(4)
$${C}_{g}=\frac{2{n}^{2}}{\text{sin}{\varphi}_{p}\hspace{0.17em}\text{sin}{\varphi}_{q}}.$$
(5)
$$\frac{L}{W}=2\frac{1-{f}_{q}}{1-{f}_{p}}\frac{{\text{sin}}^{2}{\varphi}_{p}}{{\text{sin}}^{2}{\varphi}_{q}}\frac{n+sin{\varphi}_{q}}{n+\text{sin}{\varphi}_{p}}\frac{{({n}^{2}-{\text{sin}}^{2}{\varphi}_{q})}^{1/2}}{{({n}^{2}-{\text{sin}}^{2}{\varphi}_{p})}^{1/2}}.$$
(6)
$${C}_{g}=2\hspace{0.17em}\left(\frac{h}{W}\xb7\text{tan}{\varphi}_{\text{aq}}^{\prime}-1\right)\left(2\xb7\frac{h}{L}\xb7\text{tan}{\varphi}_{ap}^{\prime}-1\right),$$
(7)
$$I={A}_{e}\hspace{0.17em}\int {\sum}_{s}\alpha (p,q)dpdq,$$
(8)
$$I={I}_{b}{f}_{b}+{I}_{d}{f}_{d},$$
(9)
$${C}_{o}=I{C}_{g}.$$
(10)
$${C}_{o}=g\gamma {C}_{om}(I),$$
(11)
$${C}_{om}(I)=\frac{2\pi {n}^{2}I}{{\psi}_{c}^{-1}[I{\psi}_{e}(\pi )]},$$
(12)
$$r=f/[{\text{sin}}^{2}(\psi /2)],$$
(13)
$$f=W(1+\text{sin}{\varphi}_{q}^{\prime}).$$
(14)
$$h=r\hspace{0.17em}\text{cos}{\varphi}_{aq}^{\prime}=W\frac{1+\text{sin}{\varphi}_{q}^{\prime}}{{\text{sin}}^{2}\hspace{0.17em}\left(\frac{{\varphi}_{q}^{\prime}+{\varphi}_{aq}^{\prime}}{2}\right)}\text{cos}{\varphi}_{aq}^{\prime}.$$
(15)
$$hq=W\frac{(1+\text{sin}{\varphi}_{q}^{\prime})}{{\text{sin}}^{2}{\varphi}_{q}^{\prime}}\text{cos}{\varphi}_{q}^{\prime}.$$
(16)
$$h=\frac{L}{2}\frac{1+\text{sin}{\varphi}_{p}^{\prime}}{\text{sin}\hspace{0.17em}\left(\frac{{\varphi}_{p}^{\prime}+{\varphi}_{ap}^{\prime}}{2}\right)}\text{cos}{\varphi}_{ap}^{\prime}.$$
(17)
$${\varphi}_{aq}^{\prime}=2\hspace{0.17em}{\text{tan}}^{-1}\hspace{0.17em}\left(\frac{-\text{sin}{\varphi}_{q}+{\left\{\frac{4W}{h}(n+\text{sin}{\varphi}_{q})\hspace{0.17em}\left[\frac{2W}{h}(n+\text{sin}{\varphi}_{q})+2{({n}^{2}-{\text{sin}}^{2}{\varphi}_{q})}^{1/2}\right]\right\}}^{1/2}}{4\hspace{0.17em}\left[\frac{2W}{n}(n+\text{sin}{\varphi}_{q})+n+{({n}^{2}-{\text{sin}}^{2}{\varphi}_{q})}^{1/2}\right]}\right).$$
(18)
$${\varphi}_{ap}^{\prime}=2\hspace{0.17em}{\text{tan}}^{-1}\hspace{0.17em}\left(\frac{-\text{sin}{\varphi}_{p}+{\left\{\frac{2L}{h}(n+\text{sin}{\varphi}_{p})\hspace{0.17em}\left[\frac{W}{h}(n+\text{sin}{\varphi}_{p})+2{({n}^{2}-{\text{sin}}^{2}{\varphi}_{p})}^{1/2}\right]\right\}}^{1/2}}{4\hspace{0.17em}\left[\frac{W}{h}(n+\text{sin}{\varphi}_{p})+n+{({n}^{2}-{\text{sin}}^{2}{\varphi}_{p})}^{1/2}\right]}\right)$$
(19)
$${W}^{\u2033}=2(h\hspace{0.17em}\text{tan}{\mathrm{\Phi}}_{aq}^{\prime}-W),$$
(20)
$${L}^{\u2033}=2(h\hspace{0.17em}\text{tan}{\mathrm{\Phi}}_{ap}^{\prime}-L/2),$$
(21)
$${C}_{g}=4\hspace{0.17em}\left(\frac{h}{W}\text{tan}{\mathrm{\Phi}}_{aq}^{\prime}-1\right)\left(\frac{h}{L}\text{tan}{\mathrm{\Phi}}_{ap}^{\prime}-1/2\right).$$