Narration script

Unit eight is a data lab. A spectrum is really just a table of numbers: the irradiance per nanometre at each wavelength. If you add up height times width slices, which is what integration means, that table turns into the totals engineers actually use: total irradiance, the share of power in any band, photon flux, and the maximum current any band gap can deliver. You are going to do this yourself, with the real standard table.

The goals: understand what irradiance per nanometre means, apply the trapezoid rule to unevenly spaced data, compute band fractions, photon flux and maximum current for any band gap, and check your numbers against the reference values on this page.

Questions like what fraction of sunlight can this material use, or what is the most current a cell with this band gap could ever give, are all integrals of the spectrum. The same operation, with one extra factor inside, gives spectral mismatch corrections, quantum-efficiency-to-current calculations, and tandem current matching.

First, the units. The table gives irradiance per nanometre of bandwidth. At five hundred nanometres, the global spectrum is about one point five five watts per square metre per nanometre. That means a one-nanometre-wide slice around five hundred carries about one point five five watts per square metre. To get the power in a band, you add up slices. Power is the area under the curve.

The standard table has two thousand and two rows from two hundred and eighty to four thousand nanometres, with columns for space, global and direct sunlight. Here is the trap. The spacing is not uniform. It is half a nanometre steps in the ultraviolet, one nanometre through the visible and near infrared, and five nanometre steps beyond about seventeen hundred. Any method that assumes equal spacing will give the wrong answer. You can get the table from the national lab's reference spectra page, or as a C S V file inside the open-source pvlib Python library, which is the copy this site used.

The trapezoid rule handles uneven spacing automatically. Between two neighbouring rows, draw a straight line. The slice is a trapezoid, and its area is the average of the two heights times the width. Then add up all the slices.

Here is that rule applied to real rows between five hundred and five hundred and twenty nanometres. Each blue trapezoid is one slice. Together they total about thirty watts per square metre for that twenty nanometre band.

On the page you will find two recipes. In a spreadsheet, one formula in a helper column computes each slice, and a sum gives the total. In Python, numpy's trapezoid function does it in one line, and a boolean mask selects any band you like.

Here are reference results, computed on this site, for you to check yourself against. Total irradiance: thirteen forty-seven point nine for space, one thousand point four for global, and nine hundred point one for direct. Ultraviolet share of the global spectrum, four point six percent. Visible share, about fifty percent. Beyond silicon's cutoff, nineteen point two percent. And the global photon flux, four point three one times ten to the twenty-one per second per square metre. If you can reproduce this table, you have mastered the method.

This running total shows how the power builds up with wavelength. About five percent of the power lies below four hundred nanometres. About forty-eight percent below seven hundred. Eighty-one percent below silicon's eleven hundred and seven nanometre cutoff. And ninety percent below fifteen hundred.

Now the most useful calculation in the unit. Convert every row from watts to photons, add up all photons shorter than the band gap's cutoff wavelength, and multiply by the charge of an electron. In words: count every photon energetic enough to be absorbed, and give each one an electron.

The table shows the results. Germanium, with a small gap, could deliver about sixty-one milliamps per square centimetre. Silicon, forty-three point eight. Gallium arsenide, thirty-two. A one point seven five electron-volt top cell for a tandem, only twenty-one. Smaller gaps catch more photons and give more current. The last column, the ultimate efficiency, assumes every absorbed photon delivers exactly the band gap energy. For silicon, that is forty-nine percent. It counts only the two unavoidable spectral losses, which we will meet again in unit eleven.

A quick sanity check. Forty-three point eight milliamps per square centimetre corresponds to about two point seven three times ten to the twenty-one photons per second per square metre. If each delivered one point one two electron-volts, that is about four hundred and ninety watts per square metre, which is forty-nine percent of one thousand. It matches.

Now it is your turn. Download the table and plot all three columns. Reproduce the reference results to within half a percent. Compute the maximum current for band gaps from zero point six to two point four electron-volts and plot it. Work out the share of power between three hundred and twelve hundred nanometres. You should get about eighty-four percent. And for the extension, multiply by a simple made-up quantum efficiency curve and integrate. You should get about forty-two point four milliamps per square centimetre. If your totals are off by ten to thirty percent, you almost certainly assumed uniform spacing.

Misconceptions. You cannot just add up the column unless every step is exactly one nanometre. Power fraction is not photon fraction: beyond silicon's cutoff lies nineteen percent of the power, but about thirty-six percent of the photons. And a higher maximum current does not mean a better band gap, because voltage falls as the gap shrinks.

In industry, integrating a quantum efficiency curve against this spectrum is how labs cross-check flash tester currents. The spectral mismatch factor used to calibrate production testers is a ratio of four integrals just like these. And this is a gentle first taste of the Python data work that factory engineering roles ask for.

Do the review questions, then actually run the lab. Next unit, we look at angles: how the Sun's position and reflection at surfaces decide how much light gets into a panel.