Narration script

Unit nine, angles. A panel only collects the part of the sunbeam that meets it face-on; that is the cosine law. At steep angles, more light bounces off the glass; that is Fresnel reflection. And bifacial modules also collect light reflected from the ground through their back side. We will cover just enough geometry to understand how light gets into a module, without going into system design.

By the end, you can describe the Sun's position, estimate the noon Sun angle from latitude and season, apply the cosine law, calculate reflection at an interface, and explain bifacial modules and albedo.

Front glass coatings, cell texture, the incidence angle modifier on datasheets, and the boom in bifacial glass-glass modules are all angle-and-reflection engineering. In Lab B, you will measure the cosine law with a real cell.

The Sun's position is given by its zenith angle, measured from straight up, and its azimuth, the compass direction. Because Earth's axis is tilted by about twenty-three and a half degrees, the latitude where the Sun is overhead at noon swings between twenty-three and a half north in June and twenty-three and a half south in December. At solar noon, the Sun's angle from overhead is simply your latitude minus that declination.

In Chicago, at about forty-two degrees north, the June noon Sun is about eighteen and a half degrees from overhead. At the equinox, about forty-two. And in December, about sixty-five degrees, low in the southern sky.

Now the cosine law. A beam striking a surface at an angle spreads its power over a larger area. If the angle from the surface's normal, the line perpendicular to it, is theta, the irradiance on the surface is the beam irradiance times cosine theta.

In this diagram, the angle of incidence is measured from the normal. The same beam covers more panel area when it comes in at a slant, so each square metre gets less. Part of the beam also reflects off the surface.

Example. A south-facing panel in Chicago, tilted thirty degrees. At the equinox noon, the beam is about twelve degrees off the panel's normal, so a direct beam of nine hundred watts per square metre delivers about eight hundred and eighty-one. In December the angle grows to about thirty-five degrees, and the panel would get about seven hundred and thirty-four, even before accounting for the weaker winter beam.

Whenever light crosses from one refractive index to another, part reflects. At normal incidence, the reflectance is the difference of the two indices divided by their sum, squared. From air into glass, that is four percent. From air into bare silicon, about thirty-five percent. At slanted angles, reflectance stays near its straight-on value up to about forty or fifty degrees, then climbs steeply toward one hundred percent at grazing angles.

This pair of plots shows both effects. On the left, the relative light collected versus angle: the pure cosine law as a solid line, and cosine times glass transmission, dashed. They barely differ until large angles. On the right, the reflectance of glass and of bare silicon versus angle. Glass starts at four percent and stays low until about sixty degrees, then shoots up. Bare silicon starts at a ruinous thirty-five percent.

The table puts numbers on it. At sixty degrees, glass reflects about nine percent. At eighty degrees, nearly thirty-nine percent. Datasheets capture this with an incidence angle modifier: the extra loss at an angle, beyond the cosine factor.

A thirty-five percent reflection loss from bare silicon would be unacceptable. So cells use pyramid texture, so that reflected light hits a neighbouring pyramid and gets a second chance, plus a silicon nitride anti-reflection coating. Together with encapsulation, reflection drops to a few percent. Module glass often adds its own porous anti-reflection coating as well.

Now bifacial modules. These have a transparent back, either glass or a clear backsheet, and cells that collect light on both faces. The back side sees light reflected from the ground. Albedo is the fraction the ground reflects: roughly zero point two for grass, zero point three for concrete, and up to about zero point eight or more for fresh snow, all approximate. Bifaciality is the ratio of rear efficiency to front efficiency, roughly seventy percent for PERC, eighty for TOPCon, and eighty-five to ninety for heterojunction, again estimates.

A rough example. Front irradiance nine hundred watts per square metre. Suppose the rear gets ten percent of that over grass, ninety watts. With a bifaciality of zero point eight, that adds seventy-two watts, an eight percent boost. Over snow, the boost might reach about twenty percent. These are assumed inputs, not measurements.

Misconceptions. Tilting a panel ten degrees off the Sun does not lose ten percent; it loses one and a half. Glass reflecting four percent does not mean the module loses exactly four percent. And bifacial modules do not double output; typical gains are single digit to low double digit percent.

In production, glass coating, cell texture and nitride thickness are monitored daily with reflectance measurements. Bifacial cell designs are now mainstream for TOPCon and heterojunction. And incidence angle and bifaciality are measured during module qualification.

Try the review questions, and then do Lab B to see the cosine law with your own cell. Next unit: standard test conditions, and why a real panel almost never produces its rated power.