Optimum Wiring Pattern for Solar Panel =========Warren D Smith===July 2015========== (Also a highly related problem, is optimum pattern of drainage ditches for a farm field.) Consider a plane region (interior of Jordan curve, usually rectangle or circle or semicircle). It is made of silicon. On top of the silicon layer we place a wiring pattern made of sheet metal, forming a tree. The root of the tree must lie on the boundary of the region. Silicon has some constant resistivity, and metal has a much smaller constant, say K=10^4 times smaller. Metal is opaque. Imagine that the silicon is a current source (generates some constant current per exposed square cm) at constant voltage. The goal is to design the metal pattern, to get the most power out of the solar cell. The two loss mechanisms are (1) joule loss (proportional to R*I^2 where R=resistance and I=current) (2) loss due to metal covering up the silicon, reducing the exposed area. BROWNIAN MOTION REFORMULATION: It also is possible to reformulate this problem in the language of brownian motion (continuum limit of random walks). Imagine that each square-cm of exposed silicon is equally likely to spontaneously generate a randomly-walking drunk. Once generated the drunk brownian-motions until hitting metal. He also brownian motions inside metal, but with a much greater diffusion constant. The goal is to minimize the expected time before the drunk hits the tree root. ASYMPTOPIA: To make the problem easier, consider the limit of large size-scaling of the solar panel, and ignore subleading terms when considering optimality. TWO PROBLEM VERSIONS: Also, we could consider allowing the metal to have variable sheet thickness, or demand it has constant thickness. With variable thickness assume the metal casts a shadow proportional to its height. EXAMPLE: Let us first consider a fairly standard and boring kind of metal tree pattern -------*------- | -------*------- | -------*------- | -------*------- | -------*------- | -------*------- | root fitting inside a 2N wide by N high rectangle, with vertical periodicity length P. Any electron can reach the tree root by traveling at most distance P/2 in silicon then at most distance 2N in metal. With constant metal thickness and constant wire widths: it pays to make the metal wires thinner and P smaller in proportion until either (a) the wire widths become comparable to the metal's vertical (sheet) thickness, or (b) until the resistance within the metal becomes comparable to the resistance within the silicon (both as experienced by typical electron). If case b, that would suggest it is optimal to have P*K of the same order as 2*N*W where W is the wire width (assuming K is the ratio of sheet resistivities for Si and metal). You then realize that it is optimal for the left-right wires to have varying widths linearly increasing toward the center, and for the up-down wire to have width linearly increasing toward the root. More precisely, the width of each wire optimally should be proportional to the current it carries. With variable metal thickness: It is optimal for the metal thickness to be comparable to the width of a wire, i.e. all wires have fixed cross-sectional shape. In that case the wire radius should vary proportional to the 2/3 power of the current it carries (wires thicker near tree root than near leaves, as usual). BETTER PATTERN: I believe the following kind of pattern will enjoy improved efficiency compared to the pattern above. In fact I think the losses should be at least 20% smaller. Imagine rays of metal all emerging radially from center at the root, like a child's drawing of sun with rays. Equally space them angularly. So far, what we have described would involve distance between adjacent rays, that grows proportionally to distance-to-root. That in general is undesirable. We instead want the inter-ray spacing to grow proportionally to the resistance (at that distance-to-root) of the metal trace leading to the root. But actually, what we have described IS desirable, if we insist on using constant-width and constant-thickness metal wires. If we allow variable width wires (and/or variable sheet-thickness), then modify pattern as follows. The wire widths W will vary as a function of distance-to-root, so that W is proprotional to current-carried for the consant-thickness case, or to current^(2/3) for the variable thickness case (constant cross sectional shape for wires) We can vary the inter-ray spacing as a function of distance-to-root by making the rays "split" in the manner of a binary tree. E.g if we wanted constant inter-ray spacing, then every time we go out to 1+epsilon times the old distance to the root, split a fraction epsilon of the rays into two. There are some local readjustments of the pattern near each split point. Also we could terminate some rays instead of splitting them. These two maneuvers allow us get essentially any inter-ray spacing law we want, as a function of distance-to-root, with losses that will be negligible (subleading terms) versus simpler-to-think-about scenarios where all rays go exactly straight. The optimal spacing law will involve inter-ray spacing distances proportional to the metal ray's electrical resistance from there to the root. I have never seen a solar cell involving this kind of "sunburst" wiring pattern. I conclude that this is an easy way to improve solar cell efficiency. This is most natural for semicircle-shaped solar cells. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)