Monthly Archives: December 2015

Neon Cactus

This neon cactus has been around my house for about 25 years. It serves as the night-light in the main room. Last week, it died, and that led to some investigation of the details…

The cactus is made from a “neon” sign tube, which glows brilliant green. The pale green color in the photo is an artifact of my camera. The transformer is hidden within the metal pot, and covered with a large amount of fish tank gravel, to support the transformer and the base of the neon tube. The toothiness along the neon tube is achieved by little bits of clear silicone caulk applied along the glass.

Inside the metal pot, there are two sets of connections: from the transformer to the neon tube, and from the power cord to the transformer input. The supply to the tube (output of the transformer) is rated at 3000 volts and 20 milliamps — enough to give someone a dangerous jolt, I believe. We are fortunate that we have never had a house guest who emptied his beer into the metal pot.

The cause of the failure last week was the transformer overheating. After 25 years, it deserves a rest — and has taken a permanent break. Power consumption of about 50 watts x 24 hours = 1.2 kwh/day = approx operating cost 30 cents/day at modern rates. Less than a single movie admission ticket per month. So it costs to operate a neon cactus, yes, but it is not out of line with other entertainment.

Rather than replace this neon cactus art with another neon construction, or hunt all over for a new transformer, I think it will be more in tune with the times to look for some LED or other pixel-based art. One might, for instance, mount a photo frame where the cactus was, and set up a suitable permanent or slowly cycling slide show fed from a USB stick. A bit of consideration and searching on the net and asking friends, should produce an abundance of good ideas.

Best wishes,
Ken Roberts
28-Dec-2015

Solar Cells 3

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In this post I’ll talk about the one diode model for solar cells. A solar cell produces a current because, under illumination, exciton pairs — electrons and holes — are produced, and before the electrons and holes can recombine, they flow towards and out through the contact terminals of the solar cell.

A solar cell can be described by a “lumped parameters” model. There is a current source, which represents the current which the solar cell’s material would produce, under standard illumination, in the absence of any losses due to reverse flows or contact resistance. The reverse flow, or recombination of excitons, is represented as a diode in parallel with a shunt resistance. There is also another resistance in the model, due to the contact resistance and other limitations on the flow of electrons to outside the solar cell. That latter resistance is modelled as a resistor in series with the current source / diode / shunt resistor triplet.

Here is a circuit diagram. It’s easier to see an unambiguous circuit than to follow the verbiage in the previous paragraph. The output current and voltage produced by the solar cell are I and V. These two related quantities are measurables. One actually has several values of I and V — for instance Isc the short circuit current (when V is zero), and Voc the open circuit voltage (when I is zero), both under standard illumination. As well one can obtain a value Idk, the dark current when there is no illumination and the solar cell has a bias voltage V applied. However, the other parameters in the circuit diagram are components of the model, to be determined by choosing those parameters to fit the actual I-V curve measurements (see characteristic curve in prior post) against the model. Rs is a series resistance, Rp is a shunt resistance, Iph is the photocurrent, and I0 and n are diode parameters — to be discussed in the next paragraph. Actually, given the current direction convention in the circuit, the output of the solar cell is -I not I. That is just a convention, but it accounts for the shape of the I-V curve in previous post, which looks like a letter J. Solar cells (good ones, at least) are described as having a J-shaped I-V curve, so I have adopted a current direction convention which makes the curve look like a letter J.

So, what about the diode? The standard diode model is called the Shockley model, and is described in many places on the net and in books. For an extended discussion focused on the particular context of solar cells see, for instance, “The Physics of Solar Cells”, by Jenny Nelson, chapters 1 and 6. The basic idea of a p-n junction diode is that current flows in one direction based upon the bias voltage across the diode, but it is offset by a thermal counterflow. The current through and voltage across the diode, say Idiode and V1, are related by the equation
Idiode = I0 [exp(q V1 / n k T) – 1]
where T denotes absolute temperature (degrees Kelvin), k denotes Boltzmann’s constant, and q is the magnitude of the electron charge.

There are two parameters in the Shockley diode model: I0 and n. The parameter n is called the “ideality factor”, and is 1 in an “ideal” diode, on the order of 1.05 for a real diode, and somewhere between 1 and 2 for a diode model used for a solar cell. As you can see, the assumption that a solar cell can be modelled using a diode is a bit of a stretch, but it is good enough for most practical purposes. It has the great advantage that there is lots of circuit modelling software for diode circuits. The parameter I0 is chosen to fit the data.

Now, one can write an equation to relate I and V for the one diode model of a solar cell. The voltage drop across the entire solar cell is V = I*Rs plus V1; that is, V1 = V – I*Rs. The current through the shunt resistance is V1/Rp. The current through the diode is Idiode as per the formula above. Putting everything together, and taking account of the sign conventions, one ends up with the equation
I = I0 [exp(q (V – I Rs) / n k T) + (V – I Rs)/Rp – Iph.
This equation is exact, at least insofar as the Shockley diode model and the other components in the one diode model are adequate as a description of actual solar cells. If one chooses a value of V, one can solve for the corresponding value of I. And vice versa. So the I-V curve corresponding to this equation can be drawn, and one can adjust the five parameters I0, Rs, Rp, n and Iph to fit the solar cell model to the actual I-V curve for a solar cell. (The parameters q and k are constants, and the parameter T reflects the working context of the solar cell, so they do not have to be adjusted to fit the model to the data.)

However, choosing a value of V and solving for I means solving an implicit equation. Likewise, choosing a value of I and solving for V also means solving an implicit equation. It would be nice to have an explicit equation, I = f(V) or V = f(I), either one. The difficulty is that I and V each appears both in linear terms and in exponential terms in the model equation above.

Those sorts of equations, involving exponentials and linear terms in a variable, often can be solved using the Lambert W function, and this equation is one which can be solved. The basic solutions were found by Jain and Kapoor in 2004 (see refs in working paper for the journal reference), and I will not bother to transcribe them here. J&K give explicit equations for both alternatives, I = f(V) and V = f(I). Because it is the V = f(I) form which turns out to be best for extending to the two diode model to be considered later, I will show the V = f(I) form here:
V = f(I) = I Rs + (I + Iph + I0) Rp
– (n k T / q) LamW((q / (n k T)) I0 Rp exp[(q / n k T) Rp (I + Iph + I0) ] )
where LamW() denotes the principal branch of the Lambert W function.

That’s a mess, I know. But at least it is an explicit expression. Given a value of I, one can calculate V. That calculation process is much less time-consuming than iterative refinement to solve an implicit equation of the form function(V,I) = 0.

However, there’s a problem. That particular formula V = f(I) can experience arithmetic overflow. And that’s where our working paper comes in. I’ll discuss some computational considerations, for the one-diode model, in the next post.

By the way … Merry Christmas !

Best wishes,
Ken Roberts
25-Dec-2015

Snow Crystals 2

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Imagine my surprise to see a mainstream media article about snow crystals! Here is the link.

https://www.rt.com/news/326738-designer-snowflakes-lab-california/

It describes the snow crystals grown by Ken Libbrecht in his Caltech lab, and has some great photos.

The text of the article is also worth a read.

Best wishes,
Ken R.
22-Dec-2015

Solar Cells 2

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A solar cell produces an electric current when its material surface is illuminated. In order to standardize the methods of measuring solar cell performance, and comparing different cells, certain conventions are adopted. For instance, standard illumination, which is the amount and wavelengths of the light falling on the solar cell’s surface.

A typical solar cell might have an area of 2.2 cm^2. Sometimes the current produced by a solar cell is quoted as so-many amps (or milliamps or microamps) under standard illumination. That statistic is appropriate when considering a particular solar cell or comparing two models of solar cells. Other times the current produced by a solar cell is quoted as a “current areal density”, that is, so many amps per square centimeter of solar cell material. That statistic is appropriate when considering a proposed solar cell material, or comparing two types of material or two choices of processing options for preparing solar cells using a particular material.

Solar cells are assembled into modules, which are groups of individual solar cells connected together (eg, in series) pre-packaged for easy handling. Modules are built into arrays, which make up solar panels; the panel includes the structural framework. Solar panels are the structures one notices on roofs and in fields.

The context of our recent work has been a mathematical model which is used for describing individual solar cells. As you can see there are many other aspects of solar cells and solar panels which are worth investigation (and have been investigated). Our work is just one part of the efforts of thousands.

I’m no expert on solar power. But I want to share the bits of information (or misunderstanding!) which I’ve gathered during my work. In the rest of this post I’ll describe some general aspects of solar cell performance. The details of the math model I’ll discuss in other posts — or you can go directly to the working paper at Researchgate if you want.

There are two measurements on the performance of a solar cell which are relatively easy to make: Isc = the short circuit current, and Voc = the open circuit voltage. Isc is determined by the amount of current the solar cell will push when it is under standard illumination and its contacts are shorted (no voltage difference); that current is largely determined by the series resistance within the solar cell. Voc is determined as the voltage difference between the contacts of the solar cell, again under standard illumination, when the load is missing — ie, an open circuit (no current flowing), or the voltage into an infinite load. Under other loads, there is a current less than Isc, and a voltage difference less than Voc.

The “I-V characteristic” for a solar cell is the curve which relates the current and the voltage. The graph shows a typical example, for a silicon solar cell manufactured about 3 decades ago. The current is shown as a negative number because of the convention used in the test circuit. What you can see in this curve is that this solar cell has Isc about 102 milliamps and Voc about 520 millivolts. The solar cell puts out a fairly constant current, between 102 milliamps and 95 milliamps, regardless of the load resistance, as long as the voltage drop does not exceed say 420 millivolts. That is, the load resistance can be up to say 420/95 = 4.3 ohms. As the load resistance goes higher, the solar cell’s current output drops rapidly towards zero. A load resistance of about 4.3 ohms, for this solar cell, is the “sweet spot” when the I*V product is a maximum. Power equals I*V (for direct current circuits like a solar cell), and hence that sweet spot is known as the “maximum power point” or MPP of the solar cell. The “fill factor” or FF of the solar cell is the ratio between the I*V product at the MPP, and the Isc*Voc product.

The maximum power point can shift if the illumination on the solar cell changes. For example, if the solar cell falls into shadow. The calculations to determine the MPP for an array of solar cells, for load balancing, can represent a lot of work. The MPP can be estimated by searching, varying the load slightly, but sometimes the power curve will have multiple peaks, and automated searching may not find the optimal operating conditions. It is important to have a good mathematical model for the I-V characteristic, in order to rapidly find the MPP under changed illumination. Further, that model should be suitable for implementation on very inexpensive microprocessors, such as may be used in a field installation. That’s where the method which S. R. Valluri and I recently described in our working paper is likely to be most useful.

That’s it for this post. More later!

Best wishes,
Ken R.
20-Dec-2015

Solar Cells 1

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This is the first of a series of posts I wish to make about solar cells. I am particularly interested in the diode models which are used to calculate the I-V characteristic curve of a solar cell. These models have an implicit equation relating current I and voltage V, and that implicit equation can be solved to give voltage as an explicit function of current, for instance. The solution uses the Lambert W function, which is what first led me to the topic of solar cell models.

The calculation of V = f(I) for a solar cell, using the exact formula, can be difficult using computer hardware arithmetic, such as in Fortran or C. Overflow of the arithmetic hardware may occur, because some intermediate numbers in the calculation are extraordinarily large. That poses a problem, since for many applications, such as load balancing of solar cell array panels, it is desirable to be able to perform solar cell calculations on inexpensive computer hardware, such as micro-controllers.

That situation led me in Spring 2015 to write a brief note (Arxiv 1504.01964) suggesting that a variant function y = g(x) = log(W(exp(x))), where W() is the principal branch of the Lambert W function, might be a better way to perform solar cell calculations. That article, though correct, was perhaps a bit terse. Recently S. R. Valluri and I have prepared a working paper which sets out the solar cell application of the y = g(x) function in detail, with example calculations for two actual silicon solar cells (one-diode model) and one actual organic solar cell (two-diode model).

Our working paper is now available on Researchgate at
https://www.researchgate.net/publication/287195509

My intent is, via a sequence of posts in this blog, to walk through that working paper. It may be enjoyable (for me, at least) to go through the material gradually, perhaps explaining some of the details which had to be condensed for the working paper itself.

Best wishes,
Ken Roberts
16-Dec-2015

Climate Change Agreement Text

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The purpose of this post is to provide a link to the text of the recent Paris agreement regarding climate change. I’m not sure how this text and its followups are going to be referred to in web links in future, but it appears that search terms (Paris Agreement) and (Framework Convention on Climate Change) are relevant. Perhaps (Paris Agreement on Climate Change). The website http://unfccc.int/ appears to be the online presence of the Framework Convention on Climate Change.

The URL for the text of the agreement itself is at…

Click to access l09.pdf

I have not yet read the text. I post the link at this early date (the text was just adopted yesterday) because the great interest in this topic may lead to an excess of interpretation and commentary, some of which is likely to be grounded in emotion rather than consider the substance. I will read the text.

My own interest in Climate Change is more towards the physical data. To that end, I am enrolled in an intro geography department course (2nd year undergrad level) on climate change, as an auditor. That course is oriented towards a study of the physical data. There are many significant social and political aspects, and I do not mean to trivialize them. In my personal life, I have found political work is much like washing dishes — necessary, and needs to be re-done every day after the dishes have been used as intended. Washing dishes can be calming and contributes to good health of oneself and others. It is an appropriate use of a part of one’s time, even if not one’s core role in society.

Likely my subsequent posts on the topic of climate change will be regarding physical data, as interesting tidbits of information come by.

Best wishes,
Ken Roberts
13-Dec-2015

Soil Loss – Grantham

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Nearly one-third of the world’s arable land has been lost to erosion or pollution in the last four decades. That comes from a briefing note prepared by the Grantham Centre, entitled — A sustainable model for intensive agriculture. It’s a quick read. The pdf file is only four pages; just two pages of content, plus cover sheet and backgrounder on the Grantham Centre and the paper’s authors. I recommend it.

A couple of observations, off the top of my head:

— The Haber-Bosch process used to produce inorganic nitrogen, consumes 3-5 percent of the world’s natural gas production (as hydrogen feedstock to make NH3), equivalent to 1-2 percent of the world’s annual energy supply. I remember thinking about the Haber-Bosch process while studying thermodynamics, and set the topic aside, but never got back to it. There is clearly an opportunity to develop replacement processes, feasible on an industrial scale, taking advantage of more recent discoveries.

— I once killed the soil around my house, which has exterior log walls, by treating the walls with a fungicide: disodium octaborate tetrahydrate (DOT). Rain caused DOT washoff into the soil, and all the mycorrhizal fungi in the soil died. I ended up with dust, not capable of sustaining life, about 1 meter around the house to a depth of half a meter. That was a lot of wheelbarrows of moving earth! My point is that such an extreme of what-not-to-do, may give some guidance in identifying what-better-to-do.

These are just initial observations. You will probably see more opportunities. The briefing note is full of ideas.

Some links:

http://grantham.sheffield.ac.uk/ — Grantham Centre for Sustainable Futures

http://grantham.sheffield.ac.uk/soil-loss-an-unfolding-global-disaster/ — The page for the briefing note, with link to PDF of the note itself.

https://en.wikipedia.org/wiki/Disodium_octaborate_tetrahydrate — Wikipedia re disodium octaborate tetrahydrate.

https://en.wikipedia.org/wiki/Haber_process — Wikipedia re Haber-Bosch process.

Best wishes,
Ken Roberts
03-Dec-2015

Snow Crystals

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A snow crystal is a single crystal of snow, whereas a snow flake is a clump of snow crystals. Snow crystals show a six-sided symmetry, and often lie in a single plane. There is a book by W. A. Bentley, who photographed snow crystals for about five decades; his 2,400-some photos are in a 1931 book which has been republished by Dover. See the figure for an example, and see the end of this post for pointers to the book and some online resources.

Why are snow crystals symmetrical? There are a couple of hypotheses. (A) One, from Ken Libbrecht at Caltech Physics, is this: “Branches begin to sprout from the six corners of the hexagon… Since the atmospheric conditions (eg, temperature and humidity) are nearly constant across the small crystal, the six budding arms all grow out at roughly the same rate.” Secondly, Libbrecht notes, symmetrical crystals are rare — irregular crystals are much more common.

Another hypothesis (B) is that snow crystals grow upon a charged core, and the charge promotes growth which fills in gaps in the structure. That is, symmetrical growth is a lower energy state, hence encouraged.

These two alternative hypotheses might be tested, by a statistical examination of snow crystal photographs. Consider two arms separated by 60 degrees; call that type 60 symmetry. Or consider two arms separated by 120 degrees; that is type 120 symmetry. Or, opposite arms, type 180 symmetry. If hypothesis A (external conditions) holds, we would expect the context to be roughly constant across the crystal; hence type 60, vs 120, vs 180 symmetry should be about the same across a population of crystals. If hypothesis B holds (charge migration, energy minimum) we would expect type 60 symmetry to be stronger than type 120 symmetry, and type 120 to be stronger than type 180.

So there is interesting work to be done. Opportunity beckons.

Here are some pointers to get you started…

Book: Snow Crystals, by W. A. Bentley and W. J. Humphreys, 1931, Dover reprint 1962. A beautiful book to browse.

Website: http://snowcrystals.com — this website was written by Ken Libbrecht of Caltech Physics, and links to his academic pages at Caltech.

Website: http://www.its.caltech.edu/~atomic/snowcrystals/faqs/faqs.htm — this web page has the hypothesis A which was quoted above, as well as other interesting info, and links to other comparable pages.

Best wishes,
Ken Roberts
01-Dec-2015