Solar Panel design

SUMMARY: The angle of inclination of a solar panel array is key to its efficiency, among other factors.

Solar panels are interconnected assemblies of photovoltaic cells that collect solar energy as part of a solar power system, either on Earth or in space. Typically, several solar panels will be used together in a photovoltaic array along with an inverter and batteries to store collected energy. Photovoltaic cells convert the energy of sunlight into electricity via the photovoltaic effect (the creation of electric current in a material when it is exposed to electromagnetic radiation), which was observed by French physicist Alexandre-Edmond Becquerel in 1839. Prior to that time, many scientists and mathematicians built and researched parabolic burning mirrors, which are another way to focus solar energy. Diocles of Carystus showed that a parabola will focus the rays of the sun most efficiently. Archimedes of Syracuse may have built burning mirrors that set ships on fire. George LeClerc, Comte de Buffon, apparently tested the feasibility of such a mirror by using 168 adjustable mirrors in order to vary the focal length to ignite objects that were 150 feet away. It was also investigated experimentally in the early twenty-first century on the television program Mythbusters. Mathematics teacher Augustin Mouchot investigated solar energy in the nineteenth century and designed a steam engine that ran on sun rays. Some consider this invention to be the start of solar energy history. The first working solar cells were built by the American inventor Charles Fritts, in 1883, using selenium with a very thin layer of gold. The energy loss of Fritts’s cells was enormous—less than 1% of the energy was successfully converted to electricity—but they demonstrated the viability of light as an energy source. Engineer Russell Ohl’s semiconductor research led to a patent for what are considered the first modern solar cells, and Daryl Chapin, Calvin Fuller, and Gerald Pearson, working at Bell Labs in the 1950s, developed the silicon-based Bell solar battery. There were fewer than a single watt of solar cells worldwide capable of running electrical equipment at that time. Roughly 50 years later, solar panels generated a billion watts of electricity to power technology on Earth, satellites, and space probes headed to the far reaches of the galaxy. Scientists and mathematicians continue to collaborate to improve solar panel technology. One such focus is creating scalable systems that are increasingly efficient and economically competitive with various other energy technologies.

Physics and Mathematics of Solar Panels

In 1905, Albert Einstein published both a paper on the photoelectric effect and a paper on his theory of relativity. His mathematical description of photons (or “light quanta”) and the way in which they produce the photoelectric effect earned him the Nobel Prize in Physics in 1921. In general, the photons or light particles in sunlight that are absorbed by semiconducting materials in the solar panel transfer energy to electrons—though some is lost in other forms, such as heat. Added energy causes the electrons to break free of atoms and move through the semiconductor. Solar cells are constructed so that the electrons can move in only one direction, producing electrical flow. A solar panel or array of connected solar panels produces direct current, like chemical batteries, which can be stored. An inverter can convert the direct current to alternating current for household use.

Mathematics is involved in many aspects of solar panel design, operation, and installation. For example, the perimeter of an array of multiple solar panels may change with rearrangement of the panels, but the area stays the same. Since area is one critical variable for power collection, this suggests different optimal arrangements for surfaces where solar panels might be arranged, like walls and roofs. Satellites often use folding arrays of solar panels that deploy after launch, and folding portable solar panel arrays have been designed for applications like camping and remote or automated research and monitoring stations. Space scientist Koryo Miura developed the Muria–Ori map folding technique, which involves mathematical ideas of flexible polygonal structures and tessellations. It has been incorporated into satellite solar panels that can be unfolded into a rectangular shape by pulling on only one corner.

Arrays

A solar panel array may be fixed, adjustable, or tracking. Each method has trade-offs in installation cost versus efficiency and energy over the lifetime of the installation, which can be analyzed mathematically in order to optimize an individual setup. Fixed arrays are solar panels that stay in one position. Optimal positioning of such arrays usually involves facing the equator (true south, not magnetic south, when in the northern hemisphere), with an angle of inclination roughly equal to their latitude. Using an angle of inclination slightly higher than the latitude has been shown in some studies to improve energy collection in the winter, which can help balance shorter days or increased heating energy needs. Setting the inclination slightly less than the latitude optimizes collection for the summer. Adjustable panels can have their tilt manually adjusted throughout the year. Tracking panels follow the path of the sun during the day, on either one or two axes: a single-axis tracker tracks the sun east to west only, while a double-axis tracker also adjusts for the seasonal declination movement of the sun. Tracking panels may lead to a gain in power, but for some users, the cost trade-off might suggest adding additional fixed panels for some applications instead. Solar power companies and other entities provide maps showing the yearly average daily sunshine in kilowatt hours per square meter of solar panel. Combined with the expected energy consumption of a building, this data helps determine how many solar panels and batteries will be needed for an installation. Science and mathematics teachers often have students build solar panels and collect data to facilitate mathematical understanding and critical thinking, as well as make mathematics, science, and technology connections.

Bibliography

Anderson, E. E. Fundamentals of Solar Energy Conversion. Reading, MA: Addison Wesley Longman, 1982.

Hull, Thomas. “In Search of a Practical Map Fold.” Math Horizons 9 (February 2002).

Kryza, F. The Power of Light: The Epic Story of Man’s Quest to Harness the Sun. New York: McGraw-Hill, 2003.