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Model check
The science
What you see. A 156 mm silicon solar cell lies on a test chuck under a lamp. The silver fingers and busbars on top collect the current that the cell generates; the blue dots are electrons travelling to the nearest finger and then along the busbar to the ribbon and probe bar. You choose the grid, the light and the temperature; the page computes the full current-voltage curve.
Diode model. The cell is a current source (the photocurrent Iph, proportional to the light that gets through the grid) in parallel with a diode and a leakage (shunt) resistance Rsh, all behind a series resistance Rs: I = Iph − I0(e(V+IRs)/nVt − 1) − (V+IRs)/Rsh. The implicit equation is solved exactly with the Lambert W function. The maximum power point is found by golden-section search; fill factor FF = Pmp/(IscVoc) and efficiency η = Pmp/(G A).
The metal grid. Fingers shade the cell (shaded fraction = N w / L plus the busbars) but they also shorten the path of the current. Between two fingers a distance S apart the emitter carries the current sideways and wastes Rsheet S²/12 (Ω cm², integrating (Jx)²Rsheet from the finger to the mid-point). Along a finger of length ℓ the current grows linearly toward the busbar, which wastes ρ S ℓ²/(3 w h). More fingers: less emitter loss, more shading, so the efficiency has a maximum, which the page finds numerically.
Temperature. The cell temperature follows the NOCT rule Tc = Tair + (NOCT − 20) G / 800. The saturation current grows as T³exp(−Eg/nkT), so the open-circuit voltage falls by about 2 mV per kelvin while the photocurrent rises by 0.05 % per kelvin: a hot cell makes less power.
Honest simplifications. One diode with ideality factor 1 and a constant series resistance (real cells have a second diode, voltage-dependent recombination and lumped contact effects); the emitter and finger losses use the uniform-current-density approximation at the maximum power point, and the busbar and the ribbon resistance are left out; the photocurrent density at one sun (41 mA/cm² before shading) and the other material numbers are typical textbook values for an aluminium back-surface-field silicon cell, not a particular product; the fingers are drawn wider than their true 45 µm so they can be seen, and the electron paths are a sketch, not a drift-diffusion calculation. Educational physics for trends, not a cell design tool.
Checks. (1) The Lambert-W current equals a bracketed Newton solution of the diode equation and the open-circuit voltage equals nVt ln(Iph/I0+1); (2) the ideal fill factor equals the closed-form FF0 expression; (3) the numerical temperature coefficient of Voc equals the analytic one; (4) the finger and emitter resistances equal direct numerical integrations of the dissipated power; (5) the optimum number of fingers equals a brute-force sweep, and the areas of the wafer, busbar and finger integrated from the Blender GLB triangles equal the nominal dimensions.