Solar Energy Vol. 69, No. 5, pp. 357–362, 2000 2000 Elsevier Science Ltd  Pergamon PII: S0038–092X(00)00109–2 All rights reserved. Printed in Great Britain 0038-092X / 00 / $ - see front matter www.elsevier.com / locate / solener EXACT SOLUTIONS FOR MULTILAYER OPTICAL STRUCTURES. APPLICATION TO PV MODULES ² N. FRAIDENRAICH and O. C. VILELA Research Group on Alternative Sources of Energy (FAE)–Nuclear Energy Department (DEN), Federal University of Pernambuco–UFPE, Av. Prof. Luiz Freire, 1000 50740-540 Recife (PE), Brazil Received 2 November 1999; revised version accepted 13 July 2000 Communicated by JEFFREY GORDON Abstract—The analysis of multilayer optical devices is important in solar technology as they can be found in a large variety of equipment, particularly in photovoltaic modules. Absorbed and transmitted light by a set of thick layers are usually obtained by ray tracing procedures. When the optical structure is formed by more than one layer, the calculation of both the absorbed and transmitted light is rather complex due to multiple interactions between the several interfaces of the optical device. This paper describes a general analytic procedure, valid for any number of layers, to calculate the absorbed and transmitted light, e.g. solar radiation, through a set of thick optical layers. Consideration of incoming and outgoing light fluxes at each interface, and the assumption that the last interface acts as a light sink, leads to a closed system of equations that can be solved sequentially. Results are applied to analyze the optical behavior of an encapsulated solar cell and a photovoltaic module.  2000 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION This paper describes a new procedure to obtain analytic solutions for the transmittance and the The analysis of multilayer optical devices is reflectance of light, e.g. solar radiation, incident important in solar technology. This type of optical on a set of thick optical layers. The method is structure can be found in a large variety of based on: equipment, particularly in photovoltaic modules. (a) the energy balance of light flux components Absorbed and transmitted light by a thick at each layer interface; multilayer optical structure is usually treated by (b) a set of necessary conditions to be satisfied ray tracing procedures. Transmittance and reflect- by the light flux components; ance of any number of transparent glass layers (c) the assumption that the last interface acts as (glass layer, air, glass layer, air, . . . ) was first a light sink; and found by Stokes (1862) (cited by Hottel and (d) a flux diagram, essential to identify the Sarofim, 1967). Using a multiple reflection pro- relevant light flux components. cedure, Stokes derived equations for a single The energy balance of light flux components at layer. Immediately, exact equations for two each interface together with the set of rules blocks of layers, of m and n plates, respectively, (statement (b) above) yield a system of equations. were obtained. More recently, Krauter et al. Using the condition that the last layer (bottom (1994), using a similar procedure, obtained ex- layer) is a light sink, a closed system of equations pressions for total transmissivity and reflectivity is obtained. Once all flux components are de- of a set of n adjacent layers. These results were termined, the light fraction that is absorbed by later improved by Sjerps-Koomen et al. (1996) each layer and converted into heat can be calcu- and Krauter and Hanitsch (1996). Multiple reflec- lated. tions procedures are used in all those papers, starting with one layer and progressively increas- ing its number. Due to multiple interactions 2. METHODOLOGY between layers, ray tracing becomes rather com- We start analyzing results obtained with a ray plex as their number increases. tracing procedure in a single layer optical device. Light bundles are classified as light flux com- ² ponents and calculated with the multiple reflec- Author to whom correspondence should be addressed. Tel.: tions procedure. 155-81-271-8252; fax: 155-81-271-8250; e-mail: nf@npd.ufpe.br Immediately, and as an alternative method, we 357