## Abstract

The limit of concentration for cylindrical concentrators whose collector rejects rays arriving at grazing angles is calculated. Cylindrical concentrators having as cross section a 2-D optimal and ideal concentrator (CPC type) with restricted exit angle are analyzed. These concentrators do not achieve, in general, the upper limit but are very close to it. It is also shown that the upper limit is achievable. Finally, two 2-D concentrators with restricted exit angle are shown for the case when the collector is bifacial and specularly reflects the rejected rays.

© 1984 Optical Society of America

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### Equations (15)

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(1)
$${C}_{g}\le \frac{\pi {n}^{2}}{\widehat{a}}={C}_{gm}(\widehat{a}).$$
(2)
$$g\le \frac{\pi {A}_{c}{n}^{2}\hspace{0.17em}{\text{sin}}^{2}{\varphi}_{m}}{\pi {A}_{c}{n}^{2}}={\text{sin}}^{2}{\varphi}_{m},$$
(3)
$${C}_{g}\le {\text{sin}}^{2}{\varphi}_{m}{C}_{gm}(\widehat{a})={\text{sin}}^{2}{\varphi}_{m}\frac{{n}^{2}\pi}{\widehat{a}}.$$
(4)
$${g}_{m}({C}_{g},n)=\frac{2}{\pi {n}^{2}}\left\{{C}_{g}\hspace{0.17em}{\text{sin}}^{-1}{\left(\frac{{n}^{2}-1}{{C}_{g}^{2}-1}\right)}^{1/2}+{n}^{2}\times {\text{sin}}^{-1}\left[\frac{1}{n}{\left(\frac{{C}_{g}^{2}-{n}^{2}}{{C}_{g}^{2}-1}\right)}^{1/2}\right]\right\}.$$
(5)
$$g=\frac{1}{{A}_{c}{{\pi}_{n}}^{2}}{\int}_{x=0}^{x=1}dx\hspace{0.17em}{\int}_{p=-1}^{p=1}{E}_{2-D}(p)dp,$$
(6)
$${E}_{2-\text{D}}(p)={\int}_{B(p)}dydq\le 2{A}_{e}{(1-{p}^{2})}^{1/2},$$
(7)
$${E}_{2-\text{D}}(p)={\int}_{{B}^{\prime}(p)}d{y}^{\prime}d{q}^{\prime}\le 2{A}_{c}{({n}^{2}\hspace{0.17em}{\text{sin}}^{2}{\varphi}_{m}-{p}^{2})}^{1/2},$$
(8)
$${g}_{mr}({C}_{g},n,{\varphi}_{m})={\text{sin}}^{2}{\varphi}_{m}{g}_{m}({C}_{g},n\hspace{0.17em}\text{sin}{\varphi}_{m}),$$
(9)
$$\frac{\partial {n}^{2}{g}_{m}({C}_{g},n)}{\partial n}>0$$
(10)
$$\frac{\partial {n}^{2}\hspace{0.17em}{\text{sin}}^{2}{\varphi}_{m}{g}_{m}({C}_{g},n\hspace{0.17em}\text{sin}{\varphi}_{m})}{\partial (n\hspace{0.17em}\text{sin}{\varphi}_{m})}>0,$$
(11)
$$\frac{\partial {g}_{mr}({C}_{g},n,{\varphi}_{m})}{\partial {\varphi}_{m}}>0.$$
(12)
$${C}_{g}\le {g}_{mr}({C}_{g},n,{\varphi}_{m}){C}_{gm}(\widehat{a}).$$
(13)
$${C}_{o}\le {g}_{mr}\left(\frac{{C}_{o}}{I},\hspace{0.17em}n,{\varphi}_{m}\right)\hspace{0.17em}{C}_{OM}(I).$$
(14)
$${C}_{g}=\frac{{A}_{e}}{2\overline{AC}}=\frac{n}{\text{sin}{\varphi}_{a}}.$$
(15)
$$\frac{{A}_{e}}{\overline{OC}}=\frac{2n\hspace{0.17em}\text{sin}{\varphi}_{m}}{\text{sin}{\varphi}_{a}},$$